{ "name": "The Choice Strength Table", "description": "The 1,528 theorems in Metamath's set.mm whose proofs reach a choice principle, each labelled with the strongest of the three the database declares separately — full choice, countable choice, dependent choice — together with the raw membership in all three.", "version": "2026-08-12", "sha256": "b4fa7bbd8b69998631a472149df11f926139e8845e3405d241642ebe6408110e", "url": "https://f-keys.com/gonzalgo/choice-strength/", "seriesDoi": "https://doi.org/10.5281/zenodo.21900625", "license": "https://creativecommons.org/licenses/by/4.0/", "producedBy": { "software": "gonzalgo", "url": "https://pypi.org/project/gonzalgo/", "paper": "https://doi.org/10.5281/zenodo.21769846" }, "measured": { "Metamath": "set.mm", "theorems": 47621, "principles": "ax-ac, ax-ac2, ax-cc, ax-dc" }, "fields": [ "theorem", "strongest", "reaches_full_choice", "reaches_countable_choice", "reaches_dependent_choice", "axioms_reached" ], "displayed": [ { "key": "theorem", "label": "theorem" }, { "key": "strongest", "label": "strongest principle needed" }, { "key": "axioms_reached", "label": "axioms reached" } ], "rows": [ { "theorem": "1arithufd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "1arithufdlem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "1arithufdlem4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "2sqr3nconstr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "abrexctf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "abrexdom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "abrexdom2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "abrexdom2jm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "abrexdomjm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "ac3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "ac4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac4c", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac5", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac5b", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6c4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6c5", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6gf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6mapd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6n", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6s", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6s2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6s3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6s3f", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6s4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6s5", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6s6", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6s6f", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6sf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6sf2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac6sg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac7", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac7g", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac8", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac8prim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac9", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ac9s", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "aciunf1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "aciunf1lem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ackm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acsdomd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acsexdimd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acsinfd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acsinfdimd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acsmap2d", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acsmapd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acunirnmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acunirnmpt2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "acunirnmpt2f", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "aean", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "aleph1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "aleph1irr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "aleph1re", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "alephexp2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "alephom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "alephreg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "alephsucpw", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "alephval2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "alexsubALT", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "alexsubALTlem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "alexsubALTlem4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "algextdeg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "algextdeglem4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "algextdeglem6", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "algextdeglem8", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "assafld", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "assalactf1o", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "assarrginv", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "axac", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "axac10", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "axac2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "axac3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "axaci", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "axacnd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "axacndlem4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "axacndlem5", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "axacprim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "axdc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "axdclem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "bayesth", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "bdayfin", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-ac2 ax-dc" }, { "theorem": "bdayfinbnd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-ac2 ax-dc" }, { "theorem": "bdayfinbndlem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-ac2 ax-dc" }, { "theorem": "bdayfinbndlem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-ac2 ax-dc" }, { "theorem": "bdayfinlem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-ac2 ax-dc" }, { "theorem": "bj-grur1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "bj-pwcfsdom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "boolesineq", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "borelmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "bormflebmf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "brdom3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "brdom4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "brdom5", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "brdom6disj", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "brdom7disj", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "canth3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "carageniuncl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "carageniuncllem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "caragensal", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "caragenunicl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "caratheodory", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "caratheodorylem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "carddom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "carden", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardeq0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardeqv", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardid", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardidd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardidg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardmin", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardsdom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cardval", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "carsgclctun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "carsgclctunlem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "carsgclctunlem3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "carsgsiga", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ccfldextdgrr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cfpwsdom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cldssbrsiga", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cmpcref", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cmpfiref", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cmppcmp", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cndprob01", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cndprobin", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cndprobnul", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cndprobprob", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cndprobtot", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cnmbfm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "coinflippvt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "constrcon", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "constrelextdg2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "constrext2chn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "constrext2chnlem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "constrextdg2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "constrextdg2lem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cos9thpinconstr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cos9thpinconstrlem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ctbssinf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "ctvonmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "cyc3genpm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "cyc3genpmlem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "decsmf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "dflring3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dflring4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dflringlem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dfufd2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dfuniv2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dfz12s2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-ac2 ax-dc" }, { "theorem": "difelsiga", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dimcl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dimkerim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dimlssid", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dimpropd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dimval", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "disjinfi", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dmct", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dmovnsal", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "dmvon", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "dominfac", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "domtri", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "drngdimgt0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "drngmxidlr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dstfrvclim1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dstfrvinc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dstfrvunirn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dstrvprob", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dya2icobrsiga", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dya2iocbrsiga", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dya2iocct", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "dynkin", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "eln0s2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "elnanelprv", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "elrgspnsubrun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "elrgspnsubrunlem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "elrspunidl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "elwspths2on", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "elwspths2spth", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "elwwlks2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "elwwlks2on", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "entri2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "entri3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "entric", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "esumiun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "eulerpart", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "eulerpartgbij", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "eulerpartlem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "eulerpartlemgf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "eulerpartlemgs2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "eulerpartlemgvv", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "eulerpartlemn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ex-sategoel", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ex-sategoelel", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ex-sategoelel12", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ex-sategoelelomsuc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "extdg1b", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "extdg1id", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "extdgcl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "extdgfialg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "extdgfialglem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "extdggt0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "extdgid", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "extdgmul", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fedgmul", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ffsrn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ficard", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "filssufil", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "filufint", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fimact", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fin71ac", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "finextalg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "finexttrb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldext2chn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldext2rspun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspundgdvds", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspundgdvdslem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspundgle", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspundglemul", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspunfld", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspunlem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspunlem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspunlsp", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldextrspunlsplem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fldlring", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fnct", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fnpreimac", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fnrndomg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fodom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fodomb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fodomg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "foresf1o", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fpwrelmapffs", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fpwrelmapffslem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "frgr2wwlkeqm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "frlmdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fsumiunle", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "fsumiunss", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "gch-kn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grothprim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "grucollcld", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "gruex", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "gruina", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grumnud", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grumnudlem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grumnueq", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grur1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grur1a", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grurankcld", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "gruscottcld", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grutsk", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "grutsk1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "hausmapdom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "hauspwdom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "hgt750lema", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "hgt750leme", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "hoimbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "hoimbl2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "hoimbllem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "hspmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "iccvonmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "iccvonmbllem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "imadomg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "imambfm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "inaex", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "inagrud", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "inaprc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "inar1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "inatsk", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "indexdom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "inelsiga", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "infinf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "infinfg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "infmap", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "infxpidm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ioovonmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "islinds4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "issmfge", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "issmfgelem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "issmfgt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "issmfgtd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "issmfgtlem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "issmfle", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "issmfle2d", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "issmfled", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "issmflelem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "isvonmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "iundom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "kardcard", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "konigth", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "konigthlem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "krull", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "krullndrng", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lbsex", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lbsext", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lbslelsp", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ldgenpisys", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ldsysgenld", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lmicdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lmimdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "locfinref", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "locfinreflem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lsatdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lssdimle", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lvecdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lvecdim0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lvecdim0i", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lvecendof1f1o", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "lvecisfrlm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "madefi", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "matdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "mbfmco2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "mbfpsssmf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "mblvon", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "meascnbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measinb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measinb2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measinblem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measiun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measiuns", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measssd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measunl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measvuni", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measvunilem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "measxun2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "minregex", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "minregex2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "mpocti", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "mptct", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "mptctf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "mxidlnzrb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "n0cutlt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "nlpfvineqsn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "nmobndseqiALT", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "nmounbseqiALT", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "nsssmfmbf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "nsssmfmbflem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "numth", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "numth2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "numth3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "numthcor", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "oldfi", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "oldfib", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "omeiunle", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "omeiunlempt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "omeiunltfirp", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "omsmeas", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "omssubadd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "ondomon", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "onltn0s", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "onsfi", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "opnvonmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "opnvonmbllem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "orrvccel", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "orvccel", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "orvcgteel", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "orvclteel", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ovnome", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "ovnsplit", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "ovnsubadd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "ovnsubadd2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "ovnsubadd2lem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "ovnsubaddlem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ovnsubaddlem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "pcmplfinf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "permac8prim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "pibt2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "pmeasadd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "prob01", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "probdif", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "probdsb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "probinc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "probmeasb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "probtotrnd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "probun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "probvalrnd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "prv0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "prv1n", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "psmeasure", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "psmeasurelem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ptcls", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ptcmp", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ptpconn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "pwcfsdom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "pwinfi3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "qusdimsum", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "r1omALT", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "r1omtsk", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "r1tskina", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rabfodom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "reff", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rhmpreimacn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rlmdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rnct", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rr-groth", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rr-grothprim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rr-grothprimbi", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rr-grothshort", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rr-grothshortbi", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rrnmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "rrvadd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rrvmulc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rrvsum", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rrxdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rspectps", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rtelextdg2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "rtelextdg2lem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "s3wwlks2on", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sate0fv0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "satef", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "satefvfmla0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "satefvfmla1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sategoelfv", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sategoelfvb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "satfun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "satfvel", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sdomsdomcard", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sge0iun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sge0iunmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sge0iunmptlemre", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sge0xp", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sibfinima", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sibfof", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sigaclci", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sigaclcuni", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sigainb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sigaldsys", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sigapildsys", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sigapisys", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sitgaddlemb", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sitmcl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sitmf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "smf2id", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfadd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfaddlem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfco", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfconst", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfdiv", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfdivdmmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfdivdmmbl2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfdmmblpimne", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfinf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfinfdmmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfinfdmmbllem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfinflem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfinfmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smflim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smflim2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfliminf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfliminflem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfliminfmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smflimlem6", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smflimmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smflimsup", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smflimsuplem3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smflimsuplem8", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smflimsupmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfmbfcex", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfmul", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfmulc1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfmullem4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfneg", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimbor1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimbor1lem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimbor1lem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimcc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimgtmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimgtxr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimgtxrmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimgtxrmptf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimioo", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimioompt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimltxr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimltxrmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimltxrmptf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimne", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpimne2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpreimage", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpreimagt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpreimagtf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfpreimale", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfrec", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfres", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfresal", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfsup", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfsupdmmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfsupdmmbllem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfsuplem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfsuplem3", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfsupmpt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "smfsupxr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "snvonmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "ssdifidl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ssdifidlprm", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ssmxidl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "subsaliuncl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "subsalsal", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "subsaluni", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "sxbrsiga", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sxbrsigalem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sxbrsigalem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sxbrsigalem4", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sxbrsigalem5", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "sxbrsigalem6", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tgoldbachgt", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "tgoldbachgtd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "tgoldbachgtda", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "tgoldbachgtde", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "tngdim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "totprob", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "totprobd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "trisecnconstr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tskcard", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tskint", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tskmap", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tskun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tskuni", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tskurn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tskwun", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "tskxp", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ttukey", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ufilen", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ufileu", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "ufinffr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "umgrwwlks2on", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "unidifsnel", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "unidifsnne", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "unidmvon", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "unidom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "uniimadom", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "uniimadomf", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "unirnfdomd", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "unsnen", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "vitali2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "von0val", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "voncl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "voncmpl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonct", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonhoi", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonhoire", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonicc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonicclem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonicclem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonioo", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonioolem1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonioolem2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonmblss", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonmblss2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonmea", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonn0hoi", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonn0icc", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonn0icc2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonn0ioo", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonn0ioo2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonsn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonvol", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonvol2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonvolmbl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonvolmbl2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "vonxrcl", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2 ax-cc" }, { "theorem": "wdomac", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "weth", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "wfac8prim", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "wlkiswwlksupgr2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "wlkiswwlkupgr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "wlklnwwlklnupgr2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "wlklnwwlknupgr", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "wwlks2onsym", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zarcls1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zarclssn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zarcmp", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zarcmplem", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zarmxt1", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zart0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zartop", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zartopn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zartopon", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zfac", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "zfcndac", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac" }, { "theorem": "zorn", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zorn2", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "zornn0", "strongest": "full choice", "reaches_full_choice": true, "reaches_countable_choice": false, "reaches_dependent_choice": false, "axioms_reached": "ax-ac2" }, { "theorem": "0bdop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "0cnop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "0hmop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "0leop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "0leopj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "0lnop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "1stccn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "1stccnp", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "1stcelcls", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "1stckgen", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "2ndcctbss", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "2ndcsep", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "3factsumint", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "3factsumint1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "3factsumint3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "3oai", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "3oalem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "5oai", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "acncc", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "adj0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "adjbd1o", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "adjbdln", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "adjbdlnb", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "adjcoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "adjeq0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p7", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p7d1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p8", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "aks4d1p9", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "areaquad", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "arearect", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atabs2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atabsi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atcv0eq", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atcv1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atcvat2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atcvat2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atcvat3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atcvat4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atcvati", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atcvatlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atdmd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atdmd2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atexch", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atmd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atmd2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atnemeq0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atnssm0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atom1d", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atoml2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atomli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atord", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atordi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "atssma", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axcc2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axcc2lem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axcc3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axcc4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axcc4dom", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axccd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axccd2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axccdom", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axgroth3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axgroth4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axpjcl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "axpjpj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bddibl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bddmulibl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bdopadj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bdopcoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bdopssadj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bra11", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bracnln", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bracnlnval", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "bracnvbra", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ch1dle", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chabs1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chabs1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chcon1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chcon2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chcon3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chcv1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chcv2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmj1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmj1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmj2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmj2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmj3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmj3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmj4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmj4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmm1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmm1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmm2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmm2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmm3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmm3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmm4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chdmm4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chirred", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chirredi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chirredlem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chirredlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chirredlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chirredlem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chj0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chj00i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chj0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chj12", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chj12i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chj4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chj4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chjass", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chjassi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chjatom", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chjidm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chjidmi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chjjdiri", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chlejb1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chlejb1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chlejb2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chlub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chlubi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chlubii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chnle", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chnlei", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chpssati", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chpsscon1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chpsscon2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chpsscon3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chrelat2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chrelat2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chrelat3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chrelat3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chrelat4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chrelati", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chsscon1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chsscon1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chsscon3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chsscon3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chsup0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chsupid", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chsupsn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "chsupval2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "circlemeth", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "circlemethhgt", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "circlemethnat", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "circlevma", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cm0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cm2j", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cm2ji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cm2mi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmbr2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmbr3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmbr3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmbr4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcm2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcm2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcm2ii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcm3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcm3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcm3ii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcm4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcmi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcmii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmcmlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmdmdi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmidi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmj1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmj2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmm1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmm2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cmmdi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnbdibl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cncombf", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cniccibl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnioobibld", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjeu", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjeui", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjlem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjlem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjlem6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjlem7", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjlem8", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnadjlem9", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnlnssadj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnvbrabra", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnvbracl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnvbramul", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cnvbraval", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "csmdsymi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cssbn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "csschl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvati", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvbr4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvcon3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvdmd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvexch", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvexchi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvexchlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvmd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvmdi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "cvp", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "df0op2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dfch2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dfiop2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dfpjop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "dirkeritg", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdbr2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdbr3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdbr4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdbr4ati", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdbr5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdbr5ati", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdbr6ati", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdbr7ati", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdcompli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdi4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdmd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdoc1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdoc2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmdsym", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmvlsiga", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dmvolsal", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "dominf", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "domtriom", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "domtriomlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "eighmorth", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "eighmre", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "eigvalcl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "eigvec1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "eleigvec2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "eleigveccl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "elmbfmvol2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "elpjch", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "elpjhmop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "elpjidm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "elpjrn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "elspansn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "elspansn2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "elspansn3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "elspansn4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "elspansn5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "elspansni", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "etransc", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "etransclem18", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "etransclem23", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "etransclem46", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "etransclem47", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "etransclem48", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ex-ovoliunnfl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fdvnegge", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fdvneggt", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fdvposle", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fdvposlt", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fh1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fh1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fh2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fh2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fh3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fh4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fin41", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourier", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourier2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierclim", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierclimd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fouriercn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fouriercnp", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem100", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem101", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem103", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem104", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem105", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem107", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem108", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem109", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem110", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem111", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem112", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem113", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem114", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem115", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem16", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem21", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem22", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem30", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem39", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem47", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem69", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem73", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem81", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem82", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem83", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem84", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem87", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem88", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem92", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem93", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fourierdlem95", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "fouriersw", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc1a", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc1cn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc1lem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc1lem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc1lem6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc2ditg", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc2ditglem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ftc2re", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "goeqi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "golem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "golem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "h1da", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "h1datom", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "h1datomi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "h1de2ci", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "h1de2ctlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hatomic", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hatomici", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hatomistici", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "heibor", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "heibor1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "heiborlem10", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "heiborlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hgt750lem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hh0oi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hhcms", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hhhl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hhssbnOLD", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hhsscms", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hilcms", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hilhl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hmopbdoptHIL", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "hmopd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hmopidmch", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "hmopidmchi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "hmopidmpj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "hmopidmpji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "ho01i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ho02i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ho0coi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ho0f", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ho0sub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ho0subi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ho0val", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoadd32", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoadd4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoaddass", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoaddcom", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoaddrid", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoaddridi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoaddsub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoaddsubass", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoaddsubassi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoaddsubi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hocsubdir", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoddi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hodid", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hodidi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoico1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoico2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoid1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoid1ri", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoif", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hoival", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "honegsub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "honegsubdi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "honegsubdi2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "honpcani", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "honpncani", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hopncani", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosd1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosd2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosub4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubadd4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubcl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubdi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubeq0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubid1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubneg", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubsub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubsub2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hosubsub4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hst0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstle", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstles", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstoh", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstrbi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstri", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstrlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstrlem3a", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstrlem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstrlem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hstrlem6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "hsupval2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iblabs", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iblabslem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ibladd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ibladdlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iblcncfioo", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ibliccsinexp", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iblidicc", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iblioosinexp", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iblmulc2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iblsub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iblulm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "icccmpALT", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "idhmop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "idleop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "intlewftc", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iscmet2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iscmet3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iscmet3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ismbf3d", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg1climres", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2add", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2addlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2cn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2cnlem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2const2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2gt0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2i1fseq", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2i1fseq2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2i1fseq3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2mono", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2monolem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2monolem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itg2seq", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgabs", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgadd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgaddlem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgaddlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgcn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgcoscmulx", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgexpif", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgfsum", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itggt0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgiccshift", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgioocnicc", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgmulc2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgmulc2lem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgmulc2lem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgparts", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgperiod", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgpowd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsbtaddcnst", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsin0pi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsin0pilem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsincmulx", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsinexp", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsinexplem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsubst", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsubsticc", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsubsticclem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgsubstlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgulm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "itgulm2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iunctb", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iunctb2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iunmbl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "iunmbl2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "jpi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "jplem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "jplem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "kbass6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "kbpj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "largei", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem10", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem12", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem13", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem15", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem16", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem19", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem20", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem21", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem22", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem23", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lcmineqlem6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lecm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lecmi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lecmii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ledi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ledii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lediri", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lejdii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lejdiri", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leop2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leop3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leopadd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leopmul", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leopmul2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leopmuli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leopnmid", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leoppos", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leoprf", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leoprf2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leopsq", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leoptr", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "leoptri", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lhe4.4ex1a", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lncnbd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lncnopbd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lnopcnbd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lnopcnre", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lnopco0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lnopeq", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lnopeq0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "lnopeqi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "logdivsqrle", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mayete3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mayetes3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfadd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfaddlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfimaopn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfimaopn2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfimaopnlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfinf", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbflim", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbflimlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbflimsup", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfmul", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfmulc2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfmullem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfmullem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfmvolf", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfresmf", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfsub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfsup", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mbfulm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdbr2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdbr4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdcompli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mddmd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mddmdin0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdexchi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdoc1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdoc2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsl0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsl1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsl2bi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsl2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsl3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsldmd1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslj1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslj2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslle1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslle2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslmd1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslmd1lem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslmd1lem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslmd1lem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslmd1lem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslmd2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslmd3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdslmd4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsym", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymlem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymlem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymlem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymlem6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymlem7", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "mdsymlem8", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "met2ndc", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "metcld", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "metcld2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "metcn4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "metcnp4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "metelcls", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "minveco", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "minvecolem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "minvecolem7", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmbdoplb", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmlnop0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmlnop0iALT", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmlnop0iHIL", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmlnopgt0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmlnopne0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmobndseqi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmop0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmop0h", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmopadji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmopadjlei", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmopadjlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmopcoadj0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmopcoadj2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmopcoadji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmopcoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmopleid", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmoptri2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nmounbseqi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "nonbooli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "normcan", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ococ", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ococi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ococin", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "omlsi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "omlsii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "opnmblALT", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "opsqrlem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "opsqrlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "opsqrlem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "opsqrlem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "opsqrlem6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "osum", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "osumcor2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "osumcori", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "osumi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ovoliun", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ovoliun2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ovoliunlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ovoliunnul", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pj0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pj11", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pj11i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pj2cocli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pj3cor1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pj3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pj3lem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pj3si", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjaddi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjaddii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjadj2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "pjadj2coi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjadj3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "pjadjcoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjadji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjadjii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjbdlni", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjch", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjch1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjchi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjci", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcjt2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjclem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjclem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjclem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjclem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjclem4a", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjclii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcmul1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "pjcmul2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "pjcocli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcofni", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcohcli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcohocli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjcompi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjddii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjdifnormi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjdifnormii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjds3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjdsi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjeq", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjfi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjfn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjfoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhcl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhcli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhclii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhf", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhfo", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhmop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhmopi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhmopidm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": true, "axioms_reached": "ax-cc ax-dc" }, { "theorem": "pjhth", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhtheu", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhtheu2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjhthlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjid", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjidmco", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjidmcoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjidmi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjige0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjige0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjimai", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjin1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjin2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjin3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjini", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjinormi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjinormii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjinvari", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjjsi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjlnopi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjmf1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjmuli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjmulii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjnel", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjneli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjnmopi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjnorm", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjnorm2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjnormi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjnormssi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjo", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoc1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoc1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoc2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoc2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoccl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoccoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoci", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjocini", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjococi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjocvec", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoi0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoi0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml5i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjoml6i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjomli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjop", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjopi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjopyth", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjopythi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjordi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjorthcoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjorthi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjpj0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjpjhth", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjpjhthi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjpji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjpo", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjpoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjpyth", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjpythi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjrn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjrni", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjs14i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjscji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjsdi2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjsdii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjspansn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjss1coi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjss2coi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjss2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjssdif1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjssdif2i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjssge0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjssge0ii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjsslem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjssmi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjssmii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjssposi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjssumi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjsubi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjsubii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjsumi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjtoi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjvec", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "pjvi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "qlax1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "qlax3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "qlax5i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "qlaxr3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "reheibor", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "riesz1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "riesz2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "riesz3i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "riesz4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "riesz4i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "rnbra", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "rrncms", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "rrnheibor", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "salexct", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "salexct3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "salgencntex", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "salgensscntex", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sh1dle", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "shatomici", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "shatomistici", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "shjshseli", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "shjshsi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "shlub", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "shlubi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spanpr", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansna", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnch", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnchi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansncol", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansncv", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansncv2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansncvi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansneleq", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansneleqi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansni", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnid", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnj", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnm0i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnmul", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnpji", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnscl", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnsh", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnss", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spansnss2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "spanunsni", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sqwvfoura", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sqwvfourb", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ssdmd1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ssdmd2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sshhococi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sshjval2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ssmd1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "ssmd2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "stcltrlem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "stcltrlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "stcltrthi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "stirling", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "stirlinglem15", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "stirlingr", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "stji1i", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "strb", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "stri", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "strlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "strlem3a", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "strlem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "strlem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "strlem6", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sumdmdi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sumdmdii", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sumdmdlem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sumdmdlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "sumspansn", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "supcvg", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "superpos", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "unictb", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "unierri", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "unopbd", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "vitali", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "vitalilem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "vitalilem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "voliun", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "voliune", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "voliunlem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "voliunlem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "voliunsge0", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "voliunsge0lem", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "volivth", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "volmea", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "volmeas", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "volsup", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "volsup2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "wallispi", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "wallispi2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "wallispilem1", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "wallispilem2", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "wallispilem3", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "wallispilem4", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "wallispilem5", "strongest": "countable choice", "reaches_full_choice": false, "reaches_countable_choice": true, "reaches_dependent_choice": false, "axioms_reached": "ax-cc" }, { "theorem": "0reno", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "1reno", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "addhalfcut", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axcc", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axcclem", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc2", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc2lem", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc3", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc3lem", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc3lem2", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc3lem4", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc4", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc4lem", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc4uz", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "axdc4uzlem", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "bcth", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "bcth2", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "bcth3", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "bcthlem5", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "bdaypw2bnd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "bdaypw2n0bnd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "bdaypw2n0bndlem", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "dcomex", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divdivs1d", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divmuldivsd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divmuls", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divmulsd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divsassd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divscan1d", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divscan2d", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divscan3d", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divscl", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divscld", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divsdird", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "divsrecd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "elreno2", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "htth", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "htthlem", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "ltdivmuls2d", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "ltdivmulsd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "ltmuldivs2d", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "ltmuldivsd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "nnsrecgt0d", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "noseqex", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "om2noseqoi", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "precsex", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "precsexlem10", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "precsexlem11", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "pw2cut", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "pw2cutp1", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "readdscl", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "recsex", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "recsexd", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "recut", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "remulscl", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "remulscllem1", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "renegscl", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "sdc", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "sdclem1", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "seqsex", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "ubth", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "ubthlem1", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "ubthlem3", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "z12bday", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "z12bdaylem", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" }, { "theorem": "z12bdaylem2", "strongest": "dependent choice", "reaches_full_choice": false, "reaches_countable_choice": false, "reaches_dependent_choice": true, "axioms_reached": "ax-dc" } ] }