{
  "name": "The set.mm Axiom Table",
  "description": "Every $a statement in Metamath's set.mm — logical axioms, definitions and syntax constructors kept apart — with the number of the library's 47,621 theorems whose proof closure reaches it, and that count as a share of the library.",
  "version": "2026-08-12",
  "sha256": "72e4a28dce786db0fff5de715447a23c965b3e0b59024653e995e2e05e92585d",
  "url": "https://f-keys.com/gonzalgo/setmm-axioms/",
  "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
  },
  "fields": [
    "label",
    "role",
    "dependents",
    "share_of_library_pct"
  ],
  "displayed": [
    {
      "key": "label",
      "label": "label"
    },
    {
      "key": "role",
      "label": "role"
    },
    {
      "key": "dependents",
      "label": "theorems depending"
    },
    {
      "key": "share_of_library_pct",
      "label": "share of library"
    }
  ],
  "rows": [
    {
      "label": "ax-mp",
      "role": "axiom",
      "dependents": 47562,
      "share_of_library_pct": 99.88
    },
    {
      "label": "wi",
      "role": "syntax constructor",
      "dependents": 47560,
      "share_of_library_pct": 99.87
    },
    {
      "label": "ax-1",
      "role": "axiom",
      "dependents": 47449,
      "share_of_library_pct": 99.64
    },
    {
      "label": "ax-2",
      "role": "axiom",
      "dependents": 47428,
      "share_of_library_pct": 99.59
    },
    {
      "label": "wn",
      "role": "syntax constructor",
      "dependents": 47276,
      "share_of_library_pct": 99.28
    },
    {
      "label": "ax-3",
      "role": "axiom",
      "dependents": 47221,
      "share_of_library_pct": 99.16
    },
    {
      "label": "wb",
      "role": "syntax constructor",
      "dependents": 47039,
      "share_of_library_pct": 98.78
    },
    {
      "label": "df-bi",
      "role": "definition",
      "dependents": 47036,
      "share_of_library_pct": 98.77
    },
    {
      "label": "wa",
      "role": "syntax constructor",
      "dependents": 45864,
      "share_of_library_pct": 96.31
    },
    {
      "label": "df-an",
      "role": "definition",
      "dependents": 45835,
      "share_of_library_pct": 96.25
    },
    {
      "label": "wal",
      "role": "syntax constructor",
      "dependents": 44820,
      "share_of_library_pct": 94.12
    },
    {
      "label": "ax-gen",
      "role": "axiom",
      "dependents": 44537,
      "share_of_library_pct": 93.52
    },
    {
      "label": "ax-4",
      "role": "axiom",
      "dependents": 44501,
      "share_of_library_pct": 93.45
    },
    {
      "label": "cv",
      "role": "syntax constructor",
      "dependents": 44410,
      "share_of_library_pct": 93.26
    },
    {
      "label": "wex",
      "role": "syntax constructor",
      "dependents": 44361,
      "share_of_library_pct": 93.15
    },
    {
      "label": "df-ex",
      "role": "definition",
      "dependents": 44315,
      "share_of_library_pct": 93.06
    },
    {
      "label": "wceq",
      "role": "syntax constructor",
      "dependents": 44262,
      "share_of_library_pct": 92.95
    },
    {
      "label": "ax-5",
      "role": "axiom",
      "dependents": 44067,
      "share_of_library_pct": 92.54
    },
    {
      "label": "ax-6",
      "role": "axiom",
      "dependents": 43859,
      "share_of_library_pct": 92.1
    },
    {
      "label": "ax-7",
      "role": "axiom",
      "dependents": 43787,
      "share_of_library_pct": 91.95
    },
    {
      "label": "wcel",
      "role": "syntax constructor",
      "dependents": 43165,
      "share_of_library_pct": 90.64
    },
    {
      "label": "ax-9",
      "role": "axiom",
      "dependents": 42398,
      "share_of_library_pct": 89.03
    },
    {
      "label": "ax-ext",
      "role": "axiom",
      "dependents": 42325,
      "share_of_library_pct": 88.88
    },
    {
      "label": "df-cleq",
      "role": "definition",
      "dependents": 42311,
      "share_of_library_pct": 88.85
    },
    {
      "label": "wsb",
      "role": "syntax constructor",
      "dependents": 42204,
      "share_of_library_pct": 88.62
    },
    {
      "label": "df-sb",
      "role": "definition",
      "dependents": 42191,
      "share_of_library_pct": 88.6
    },
    {
      "label": "ax-8",
      "role": "axiom",
      "dependents": 42120,
      "share_of_library_pct": 88.45
    },
    {
      "label": "df-clel",
      "role": "definition",
      "dependents": 42045,
      "share_of_library_pct": 88.29
    },
    {
      "label": "wtru",
      "role": "syntax constructor",
      "dependents": 42000,
      "share_of_library_pct": 88.2
    },
    {
      "label": "df-tru",
      "role": "definition",
      "dependents": 41970,
      "share_of_library_pct": 88.13
    },
    {
      "label": "cab",
      "role": "syntax constructor",
      "dependents": 41847,
      "share_of_library_pct": 87.88
    },
    {
      "label": "df-clab",
      "role": "definition",
      "dependents": 41839,
      "share_of_library_pct": 87.86
    },
    {
      "label": "wo",
      "role": "syntax constructor",
      "dependents": 41216,
      "share_of_library_pct": 86.55
    },
    {
      "label": "df-or",
      "role": "definition",
      "dependents": 41207,
      "share_of_library_pct": 86.53
    },
    {
      "label": "cvv",
      "role": "syntax constructor",
      "dependents": 40979,
      "share_of_library_pct": 86.05
    },
    {
      "label": "df-v",
      "role": "definition",
      "dependents": 40704,
      "share_of_library_pct": 85.47
    },
    {
      "label": "wss",
      "role": "syntax constructor",
      "dependents": 40114,
      "share_of_library_pct": 84.24
    },
    {
      "label": "df-ss",
      "role": "definition",
      "dependents": 40047,
      "share_of_library_pct": 84.1
    },
    {
      "label": "w3a",
      "role": "syntax constructor",
      "dependents": 39676,
      "share_of_library_pct": 83.32
    },
    {
      "label": "df-3an",
      "role": "definition",
      "dependents": 39652,
      "share_of_library_pct": 83.27
    },
    {
      "label": "csn",
      "role": "syntax constructor",
      "dependents": 39283,
      "share_of_library_pct": 82.49
    },
    {
      "label": "cdif",
      "role": "syntax constructor",
      "dependents": 39245,
      "share_of_library_pct": 82.41
    },
    {
      "label": "df-dif",
      "role": "definition",
      "dependents": 39225,
      "share_of_library_pct": 82.37
    },
    {
      "label": "wfal",
      "role": "syntax constructor",
      "dependents": 39200,
      "share_of_library_pct": 82.32
    },
    {
      "label": "cun",
      "role": "syntax constructor",
      "dependents": 39172,
      "share_of_library_pct": 82.26
    },
    {
      "label": "df-un",
      "role": "definition",
      "dependents": 39166,
      "share_of_library_pct": 82.25
    },
    {
      "label": "df-fal",
      "role": "definition",
      "dependents": 39163,
      "share_of_library_pct": 82.24
    },
    {
      "label": "c0",
      "role": "syntax constructor",
      "dependents": 39151,
      "share_of_library_pct": 82.21
    },
    {
      "label": "crab",
      "role": "syntax constructor",
      "dependents": 39150,
      "share_of_library_pct": 82.21
    },
    {
      "label": "df-rab",
      "role": "definition",
      "dependents": 39149,
      "share_of_library_pct": 82.21
    },
    {
      "label": "df-sn",
      "role": "definition",
      "dependents": 39139,
      "share_of_library_pct": 82.19
    },
    {
      "label": "df-nul",
      "role": "definition",
      "dependents": 39024,
      "share_of_library_pct": 81.95
    },
    {
      "label": "cpr",
      "role": "syntax constructor",
      "dependents": 38865,
      "share_of_library_pct": 81.61
    },
    {
      "label": "df-pr",
      "role": "definition",
      "dependents": 38827,
      "share_of_library_pct": 81.53
    },
    {
      "label": "cop",
      "role": "syntax constructor",
      "dependents": 38687,
      "share_of_library_pct": 81.24
    },
    {
      "label": "wral",
      "role": "syntax constructor",
      "dependents": 38541,
      "share_of_library_pct": 80.93
    },
    {
      "label": "df-ral",
      "role": "definition",
      "dependents": 38458,
      "share_of_library_pct": 80.76
    },
    {
      "label": "cif",
      "role": "syntax constructor",
      "dependents": 38421,
      "share_of_library_pct": 80.68
    },
    {
      "label": "df-if",
      "role": "definition",
      "dependents": 38415,
      "share_of_library_pct": 80.67
    },
    {
      "label": "df-op",
      "role": "definition",
      "dependents": 38324,
      "share_of_library_pct": 80.48
    },
    {
      "label": "wbr",
      "role": "syntax constructor",
      "dependents": 38288,
      "share_of_library_pct": 80.4
    },
    {
      "label": "df-br",
      "role": "definition",
      "dependents": 38043,
      "share_of_library_pct": 79.89
    },
    {
      "label": "cin",
      "role": "syntax constructor",
      "dependents": 37840,
      "share_of_library_pct": 79.46
    },
    {
      "label": "df-in",
      "role": "definition",
      "dependents": 37813,
      "share_of_library_pct": 79.4
    },
    {
      "label": "wrex",
      "role": "syntax constructor",
      "dependents": 37798,
      "share_of_library_pct": 79.37
    },
    {
      "label": "df-rex",
      "role": "definition",
      "dependents": 37723,
      "share_of_library_pct": 79.22
    },
    {
      "label": "copab",
      "role": "syntax constructor",
      "dependents": 37257,
      "share_of_library_pct": 78.24
    },
    {
      "label": "df-opab",
      "role": "definition",
      "dependents": 37256,
      "share_of_library_pct": 78.23
    },
    {
      "label": "ax-sep",
      "role": "axiom",
      "dependents": 37162,
      "share_of_library_pct": 78.04
    },
    {
      "label": "ax-12",
      "role": "axiom",
      "dependents": 36916,
      "share_of_library_pct": 77.52
    },
    {
      "label": "ax-pr",
      "role": "axiom",
      "dependents": 36828,
      "share_of_library_pct": 77.34
    },
    {
      "label": "cuni",
      "role": "syntax constructor",
      "dependents": 36618,
      "share_of_library_pct": 76.89
    },
    {
      "label": "df-uni",
      "role": "definition",
      "dependents": 36576,
      "share_of_library_pct": 76.81
    },
    {
      "label": "wnf",
      "role": "syntax constructor",
      "dependents": 36560,
      "share_of_library_pct": 76.77
    },
    {
      "label": "df-nf",
      "role": "definition",
      "dependents": 36558,
      "share_of_library_pct": 76.77
    },
    {
      "label": "cxp",
      "role": "syntax constructor",
      "dependents": 36351,
      "share_of_library_pct": 76.33
    },
    {
      "label": "ax-10",
      "role": "axiom",
      "dependents": 36165,
      "share_of_library_pct": 75.94
    },
    {
      "label": "df-xp",
      "role": "definition",
      "dependents": 36115,
      "share_of_library_pct": 75.84
    },
    {
      "label": "ax-11",
      "role": "axiom",
      "dependents": 35916,
      "share_of_library_pct": 75.42
    },
    {
      "label": "cdm",
      "role": "syntax constructor",
      "dependents": 35830,
      "share_of_library_pct": 75.24
    },
    {
      "label": "ccnv",
      "role": "syntax constructor",
      "dependents": 35765,
      "share_of_library_pct": 75.1
    },
    {
      "label": "df-dm",
      "role": "definition",
      "dependents": 35660,
      "share_of_library_pct": 74.88
    },
    {
      "label": "df-cnv",
      "role": "definition",
      "dependents": 35654,
      "share_of_library_pct": 74.87
    },
    {
      "label": "wne",
      "role": "syntax constructor",
      "dependents": 35543,
      "share_of_library_pct": 74.64
    },
    {
      "label": "wrel",
      "role": "syntax constructor",
      "dependents": 35451,
      "share_of_library_pct": 74.44
    },
    {
      "label": "df-rel",
      "role": "definition",
      "dependents": 35439,
      "share_of_library_pct": 74.42
    },
    {
      "label": "df-ne",
      "role": "definition",
      "dependents": 35399,
      "share_of_library_pct": 74.33
    },
    {
      "label": "cfv",
      "role": "syntax constructor",
      "dependents": 35278,
      "share_of_library_pct": 74.08
    },
    {
      "label": "cio",
      "role": "syntax constructor",
      "dependents": 35243,
      "share_of_library_pct": 74.01
    },
    {
      "label": "df-iota",
      "role": "definition",
      "dependents": 35231,
      "share_of_library_pct": 73.98
    },
    {
      "label": "wmo",
      "role": "syntax constructor",
      "dependents": 35091,
      "share_of_library_pct": 73.69
    },
    {
      "label": "df-mo",
      "role": "definition",
      "dependents": 35065,
      "share_of_library_pct": 73.63
    },
    {
      "label": "df-fv",
      "role": "definition",
      "dependents": 35023,
      "share_of_library_pct": 73.55
    },
    {
      "label": "ccom",
      "role": "syntax constructor",
      "dependents": 34763,
      "share_of_library_pct": 73.0
    },
    {
      "label": "df-co",
      "role": "definition",
      "dependents": 34689,
      "share_of_library_pct": 72.84
    },
    {
      "label": "df-nfc",
      "role": "definition",
      "dependents": 34674,
      "share_of_library_pct": 72.81
    },
    {
      "label": "wnfc",
      "role": "syntax constructor",
      "dependents": 34674,
      "share_of_library_pct": 72.81
    },
    {
      "label": "weu",
      "role": "syntax constructor",
      "dependents": 34649,
      "share_of_library_pct": 72.76
    },
    {
      "label": "df-eu",
      "role": "definition",
      "dependents": 34643,
      "share_of_library_pct": 72.75
    },
    {
      "label": "cid",
      "role": "syntax constructor",
      "dependents": 34579,
      "share_of_library_pct": 72.61
    },
    {
      "label": "wfun",
      "role": "syntax constructor",
      "dependents": 34345,
      "share_of_library_pct": 72.12
    },
    {
      "label": "df-id",
      "role": "definition",
      "dependents": 34260,
      "share_of_library_pct": 71.94
    },
    {
      "label": "df-fun",
      "role": "definition",
      "dependents": 34174,
      "share_of_library_pct": 71.76
    },
    {
      "label": "ax-nul",
      "role": "axiom",
      "dependents": 34035,
      "share_of_library_pct": 71.47
    },
    {
      "label": "crn",
      "role": "syntax constructor",
      "dependents": 34016,
      "share_of_library_pct": 71.43
    },
    {
      "label": "df-rn",
      "role": "definition",
      "dependents": 33809,
      "share_of_library_pct": 71.0
    },
    {
      "label": "cres",
      "role": "syntax constructor",
      "dependents": 33054,
      "share_of_library_pct": 69.41
    },
    {
      "label": "df-res",
      "role": "definition",
      "dependents": 33015,
      "share_of_library_pct": 69.33
    },
    {
      "label": "cmpt",
      "role": "syntax constructor",
      "dependents": 32675,
      "share_of_library_pct": 68.61
    },
    {
      "label": "df-mpt",
      "role": "definition",
      "dependents": 32638,
      "share_of_library_pct": 68.54
    },
    {
      "label": "cpw",
      "role": "syntax constructor",
      "dependents": 32596,
      "share_of_library_pct": 68.45
    },
    {
      "label": "wsbc",
      "role": "syntax constructor",
      "dependents": 32592,
      "share_of_library_pct": 68.44
    },
    {
      "label": "df-sbc",
      "role": "definition",
      "dependents": 32571,
      "share_of_library_pct": 68.4
    },
    {
      "label": "df-pw",
      "role": "definition",
      "dependents": 32497,
      "share_of_library_pct": 68.24
    },
    {
      "label": "wfn",
      "role": "syntax constructor",
      "dependents": 32453,
      "share_of_library_pct": 68.15
    },
    {
      "label": "df-fn",
      "role": "definition",
      "dependents": 32236,
      "share_of_library_pct": 67.69
    },
    {
      "label": "cima",
      "role": "syntax constructor",
      "dependents": 32010,
      "share_of_library_pct": 67.22
    },
    {
      "label": "df-ima",
      "role": "definition",
      "dependents": 31954,
      "share_of_library_pct": 67.1
    },
    {
      "label": "wf",
      "role": "syntax constructor",
      "dependents": 31737,
      "share_of_library_pct": 66.64
    },
    {
      "label": "df-f",
      "role": "definition",
      "dependents": 31667,
      "share_of_library_pct": 66.5
    },
    {
      "label": "ax-un",
      "role": "axiom",
      "dependents": 31580,
      "share_of_library_pct": 66.32
    },
    {
      "label": "co",
      "role": "syntax constructor",
      "dependents": 31261,
      "share_of_library_pct": 65.65
    },
    {
      "label": "df-ov",
      "role": "definition",
      "dependents": 30838,
      "share_of_library_pct": 64.76
    },
    {
      "label": "csb",
      "role": "syntax constructor",
      "dependents": 30644,
      "share_of_library_pct": 64.35
    },
    {
      "label": "df-csb",
      "role": "definition",
      "dependents": 30604,
      "share_of_library_pct": 64.27
    },
    {
      "label": "wfo",
      "role": "syntax constructor",
      "dependents": 29969,
      "share_of_library_pct": 62.93
    },
    {
      "label": "df-fo",
      "role": "definition",
      "dependents": 29918,
      "share_of_library_pct": 62.83
    },
    {
      "label": "wf1",
      "role": "syntax constructor",
      "dependents": 29688,
      "share_of_library_pct": 62.34
    },
    {
      "label": "df-f1",
      "role": "definition",
      "dependents": 29670,
      "share_of_library_pct": 62.3
    },
    {
      "label": "wf1o",
      "role": "syntax constructor",
      "dependents": 29620,
      "share_of_library_pct": 62.2
    },
    {
      "label": "ax-pow",
      "role": "axiom",
      "dependents": 29579,
      "share_of_library_pct": 62.11
    },
    {
      "label": "df-f1o",
      "role": "definition",
      "dependents": 29545,
      "share_of_library_pct": 62.04
    },
    {
      "label": "wreu",
      "role": "syntax constructor",
      "dependents": 28516,
      "share_of_library_pct": 59.88
    },
    {
      "label": "df-reu",
      "role": "definition",
      "dependents": 28510,
      "share_of_library_pct": 59.87
    },
    {
      "label": "ciun",
      "role": "syntax constructor",
      "dependents": 27768,
      "share_of_library_pct": 58.31
    },
    {
      "label": "df-iun",
      "role": "definition",
      "dependents": 27756,
      "share_of_library_pct": 58.29
    },
    {
      "label": "coprab",
      "role": "syntax constructor",
      "dependents": 27686,
      "share_of_library_pct": 58.14
    },
    {
      "label": "df-oprab",
      "role": "definition",
      "dependents": 27683,
      "share_of_library_pct": 58.13
    },
    {
      "label": "cmpo",
      "role": "syntax constructor",
      "dependents": 26875,
      "share_of_library_pct": 56.44
    },
    {
      "label": "df-mpo",
      "role": "definition",
      "dependents": 26805,
      "share_of_library_pct": 56.29
    },
    {
      "label": "w3o",
      "role": "syntax constructor",
      "dependents": 26790,
      "share_of_library_pct": 56.26
    },
    {
      "label": "df-3or",
      "role": "definition",
      "dependents": 26604,
      "share_of_library_pct": 55.87
    },
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