%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU818^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n031.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:37:28 AM UTC 2026
% Result : Theorem 8.23s 2.75s
% Output : Refutation 8.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 7
% Syntax : Number of formulae : 36 ( 15 unt; 0 typ; 0 def)
% Number of atoms : 5357 (1430 equ; 0 cnn)
% Maximal formula atoms : 337 ( 148 avg)
% Number of connectives : 2622 ( 43 ~; 25 |;1035 &; 163 @)
% ( 2 <=>;1354 =>; 0 <=; 0 <~>)
% Maximal formula depth : 342 ( 69 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 684 ( 681 usr; 340 con; 0-6 aty)
% Number of variables : 49 ( 5 ^; 28 !; 16 ?; 49 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_360,type,
image1: $i > ( $i > $i ) > $i ).
thf(func_def_364,type,
injective: $i > $i > $i > $o ).
thf(func_def_365,type,
injFuncSet: $i > $i > $i ).
thf(func_def_369,type,
surjective: $i > $i > $i > $o ).
thf(func_def_370,type,
surjFuncSet: $i > $i > $i ).
thf(func_def_378,type,
omegaS: $i > $i ).
thf(func_def_382,type,
peano3: $i > $i > $i > $o ).
thf(func_def_383,type,
peano4: $i > $i > $i > $o ).
thf(func_def_384,type,
peano5: $i > $i > $i > $o ).
thf(func_def_386,type,
peano: $i > $i > $i > $o ).
thf(func_def_387,type,
transitiveset: $i > $o ).
thf(func_def_392,type,
stricttotalorderedByIn: $i > $o ).
thf(func_def_393,type,
wellorderedByIn: $i > $o ).
thf(func_def_394,type,
ordinal: $i > $o ).
thf(func_def_395,type,
limitOrdinal: $i > $o ).
thf(func_def_413,type,
sP0: $i > $i > $i > $i > $o ).
thf(func_def_414,type,
sP1: $i > $i > $i > $o ).
thf(func_def_415,type,
sP2: $i > $i > $i > $o ).
thf(func_def_417,type,
sP4: $i > $i > $i > $o ).
thf(func_def_418,type,
sP5: $i > $o > $i > $o ).
thf(func_def_420,type,
sP7: $i > $i > $i > $i > $o ).
thf(func_def_421,type,
sP8: $i > $i > $i > $i > $o ).
thf(func_def_423,type,
sP10: ( $i > $o ) > $i > $o ).
thf(func_def_424,type,
sP11: ( $i > $o ) > $i > $o ).
thf(func_def_427,type,
sP14: $i > $i > $i > $o ).
thf(func_def_428,type,
sP15: $i > ( $i > $o ) > $o ).
thf(func_def_430,type,
sP17: ( $i > $o ) > $o ).
thf(func_def_431,type,
sP18: $i > ( $i > $i ) > $i > $o ).
thf(func_def_433,type,
sP20: ( $i > $o ) > $o ).
thf(func_def_434,type,
sP21: $i > $i > $i > $i > $o ).
thf(func_def_435,type,
sP22: $i > $i > $i > $o ).
thf(func_def_436,type,
sP23: $i > $i > ( $i > $i ) > $o ).
thf(func_def_438,type,
sP25: $i > $i > $i > $i > $o ).
thf(func_def_439,type,
sP26: ( $i > $i ) > $i > $i > $o ).
thf(func_def_442,type,
sP29: ( $i > $o ) > $i > $o ).
thf(func_def_445,type,
sP32: $i > $i > $o ).
thf(func_def_447,type,
sP34: ( $i > $o ) > ( $i > $o ) > $i > $i > $o ).
thf(func_def_448,type,
sP35: $o > $i > $i > $i > $o ).
thf(func_def_449,type,
sP36: $i > $o > $i > $i > $o ).
thf(func_def_451,type,
sP38: ( $i > $o ) > $i > $o ).
thf(func_def_454,type,
sP41: $i > $o ).
thf(func_def_455,type,
sP42: $i > $i > $o ).
thf(func_def_457,type,
sP44: $i > $i > $o ).
thf(func_def_459,type,
sP46: $i > ( $i > $o ) > $o ).
thf(func_def_460,type,
sP47: $i > ( $i > $o ) > $i > $o ).
thf(func_def_461,type,
sP48: $i > $i > $o > $o ).
thf(func_def_462,type,
sP49: $i > $i > $i > $o ).
thf(func_def_463,type,
sP50: $i > $o ).
thf(func_def_464,type,
sP51: $i > $o ).
thf(func_def_465,type,
sP52: ( $i > $o ) > $o ).
thf(func_def_466,type,
sP53: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_467,type,
sP54: ( $i > $i > $o ) > $i > $o ).
thf(func_def_470,type,
sP57: $o > $i > $i > $o ).
thf(func_def_472,type,
sP59: ( $i > $o ) > $o ).
thf(func_def_473,type,
sP60: ( $i > $i ) > $i > $i > $o ).
thf(func_def_475,type,
sP62: ( $i > $o ) > $i > $o ).
thf(func_def_476,type,
sP63: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_477,type,
sP64: $i > $i > $i > $o ).
thf(func_def_478,type,
sP65: $i > ( $i > $i ) > $i > $o ).
thf(func_def_480,type,
sP67: ( $i > $o ) > $i > $o ).
thf(func_def_481,type,
sP68: $i > ( $i > $o ) > $o ).
thf(func_def_484,type,
sP71: ( $i > $o ) > $i > $o ).
thf(func_def_485,type,
sP72: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_486,type,
sP73: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_487,type,
sP74: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_488,type,
sP75: $i > $i > $i > $i > $i > $o > $o ).
thf(func_def_489,type,
sP76: ( $i > $o ) > $i > $o ).
thf(func_def_491,type,
sP78: ( $i > $o ) > $o ).
thf(func_def_492,type,
sP79: $o > $i > $i > $o ).
thf(func_def_493,type,
sP80: $i > $i > $i > $o ).
thf(func_def_494,type,
sP81: $i > $i > $i > $o ).
thf(func_def_495,type,
sP82: $i > $i > $i > $o ).
thf(func_def_496,type,
sP83: $i > $i > $i > $o ).
thf(func_def_497,type,
sP84: $i > $i > $i > $i > $i > $o ).
thf(func_def_500,type,
sP87: $i > $i > $i > $o ).
thf(func_def_501,type,
sP88: $i > $i > $o ).
thf(func_def_502,type,
sP89: $i > $i > $o ).
thf(func_def_503,type,
sP90: $i > $o ).
thf(func_def_504,type,
sP91: $i > $i > $o ).
thf(func_def_505,type,
sP92: $i > $i > $o ).
thf(func_def_506,type,
sP93: $i > $i > $i > $o ).
thf(func_def_507,type,
sP94: $i > $i > ( $i > $o ) > $o ).
thf(func_def_509,type,
sP96: $o > $i > $i > $i > $o ).
thf(func_def_510,type,
sP97: $i > $i > $i > $i > $o ).
thf(func_def_511,type,
sP98: ( $i > $o ) > $i > $o ).
thf(func_def_512,type,
sP99: $i > ( $i > $o ) > $i > $o ).
thf(func_def_513,type,
sP100: $i > $i > $i > $o ).
thf(func_def_515,type,
sP102: $i > ( $i > $o ) > $o ).
thf(func_def_517,type,
sP104: ( $i > $o ) > $i > $o ).
thf(func_def_519,type,
sP106: $i > $i > $i > $o ).
thf(func_def_521,type,
sK108: $i > $o ).
thf(func_def_535,type,
sK122: $i > $i > $i > $i > $i ).
thf(func_def_556,type,
sK143: $i > $i ).
thf(func_def_558,type,
sK145: $i > $i > $i > $i ).
thf(func_def_559,type,
sK146: $i > $i > $i > $i ).
thf(func_def_560,type,
sK147: $i > $i > $i > $i ).
thf(func_def_561,type,
sK148: $i > $i > $i > $i ).
thf(func_def_569,type,
sK156: $i > $i ).
thf(func_def_570,type,
sK157: ( $i > $i ) > $i > $i > $i ).
thf(func_def_580,type,
sK167: $i > $i > $o ).
thf(func_def_581,type,
sK168: $i > $i > $i > $i ).
thf(func_def_589,type,
sK176: $i > $o > $i > $i ).
thf(func_def_612,type,
sK199: $i > $o ).
thf(func_def_614,type,
sK201: $i > ( $i > $o ) > $i ).
thf(func_def_623,type,
sK210: $i > $i > $i > $i > $i ).
thf(func_def_624,type,
sK211: $i > $i > $i > $i > $i ).
thf(func_def_625,type,
sK212: $i > $i > $i > $i > $i ).
thf(func_def_626,type,
sK213: $i > $i > $i > $i > $i ).
thf(func_def_637,type,
sK224: $i > $i ).
thf(func_def_638,type,
sK225: ( $i > $i ) > $i > $i > $i ).
thf(func_def_644,type,
sK231: $i > $i > $o ).
thf(func_def_656,type,
sK243: $i > $o ).
thf(func_def_662,type,
sK249: $i > $o ).
thf(func_def_666,type,
sK253: ( $i > $o ) > $i > $i ).
thf(func_def_667,type,
sK254: ( $i > $o ) > $i > $i ).
thf(func_def_668,type,
sK255: ( $i > $o ) > $i > $i ).
thf(func_def_670,type,
sK257: $i > $o ).
thf(func_def_679,type,
sK266: $i > $o ).
thf(func_def_680,type,
sK267: $i > $o ).
thf(func_def_683,type,
sK270: $i > $i > $i ).
thf(func_def_693,type,
sK280: $i > $i > $i ).
thf(func_def_696,type,
sK283: $i > $i ).
thf(func_def_703,type,
sK290: $i > $i > $o ).
thf(func_def_707,type,
sK294: $i > $i > $i > $i ).
thf(func_def_708,type,
sK295: $i > $i > $i > $i > $i ).
thf(func_def_716,type,
sK303: ( $i > $o ) > $i ).
thf(func_def_717,type,
sK304: $i > $o ).
thf(func_def_719,type,
sK306: $i > $o ).
thf(func_def_720,type,
sK307: ( $i > $o ) > $i > $i ).
thf(func_def_721,type,
sK308: $i > ( $i > $o ) > $i ).
thf(func_def_746,type,
sK333: $i > ( $i > $o ) > $i ).
thf(func_def_747,type,
sK334: ( $i > $o ) > $i ).
thf(func_def_748,type,
sK335: $i > $o ).
thf(func_def_751,type,
sK338: $i > $i ).
thf(func_def_754,type,
sK341: $i > ( $i > $i ) > $i > $i ).
thf(func_def_760,type,
sK347: ( $i > $o ) > $i ).
thf(func_def_761,type,
sK348: ( $i > $o ) > $i ).
thf(func_def_762,type,
sK349: $i > $o ).
thf(func_def_764,type,
sK351: $i > $i > $i > $i ).
thf(func_def_765,type,
sK352: $i > $i > $i > $i > $i ).
thf(func_def_766,type,
sK353: $i > $i > $i > $i > $i ).
thf(func_def_767,type,
sK354: $i > $i > $i > $i > $i ).
thf(func_def_783,type,
sK370: $i > $i ).
thf(func_def_784,type,
sK371: ( $i > $i ) > $i > $i > $i ).
thf(func_def_785,type,
sK372: $i > $i > ( $i > $i ) > $i ).
thf(func_def_794,type,
sK381: $i > $i > $i > $i > $i ).
thf(func_def_795,type,
sK382: $i > $i > $i > $i > $i ).
thf(func_def_809,type,
sK396: $i > $o ).
thf(func_def_814,type,
sK401: $i > $i > $o ).
thf(func_def_831,type,
sK418: $i > $i ).
thf(func_def_832,type,
sK419: ( $i > $i ) > $i > $i > $i ).
thf(func_def_833,type,
sK420: ( $i > $i ) > $i > $i > $i ).
thf(func_def_834,type,
sK421: $i > ( $i > $i ) > $i > $i > $i ).
thf(func_def_841,type,
sK428: $i > $i > $i ).
thf(func_def_850,type,
sK437: $i > $o ).
thf(func_def_852,type,
sK439: $i > ( $i > $o ) > $i ).
thf(func_def_853,type,
sK440: ( $i > $o ) > $i > $i ).
thf(func_def_858,type,
sK445: $i > $i ).
thf(func_def_859,type,
sK446: ( $i > $i ) > $i > $i > $i ).
thf(func_def_862,type,
sK449: $i > $i > $i ).
thf(func_def_863,type,
sK450: $i > $i > $i ).
thf(func_def_869,type,
sK456: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_870,type,
sK457: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_873,type,
sK460: $i > $o ).
thf(func_def_874,type,
sK461: $i > $o ).
thf(func_def_888,type,
sK475: $i > $i ).
thf(func_def_890,type,
sK477: $i > ( $i > $i ) > $i > $i ).
thf(func_def_892,type,
sK479: $i > $o ).
thf(func_def_893,type,
sK480: ( $i > $o ) > $i > $i ).
thf(func_def_894,type,
sK481: ( $i > $o ) > $i > $i ).
thf(func_def_909,type,
sK496: $i > $i ).
thf(func_def_927,type,
sK514: $i > $i > $i ).
thf(func_def_928,type,
sK515: $i > $i ).
thf(func_def_945,type,
sK532: $i > $i > $i ).
thf(func_def_948,type,
sK535: $i > $i > $i ).
thf(func_def_956,type,
sK543: $i > ( $i > $o ) > $i > $i ).
thf(func_def_957,type,
sK544: $i > ( $i > $o ) > $i > $i ).
thf(func_def_958,type,
sK545: $i > ( $i > $o ) > $i ).
thf(func_def_961,type,
sK548: $i > $o ).
thf(func_def_965,type,
sK552: $i > $i > $i > $o ).
thf(func_def_991,type,
sK578: $i > $o ).
thf(func_def_1007,type,
sK594: $i > $i ).
thf(func_def_1008,type,
sK595: $i > $i ).
thf(func_def_1014,type,
sK601: ( $i > $o ) > $i ).
thf(func_def_1015,type,
sK602: $i > ( $i > $o ) > $i ).
thf(func_def_1016,type,
sK603: $i > $o ).
thf(func_def_1032,type,
sK619: $i > $i > $o ).
thf(func_def_1053,type,
sK640: $i > $o ).
thf(func_def_1057,type,
sK644: $i > $i ).
thf(func_def_1059,type,
sK646: $i > $i > $o ).
thf(func_def_1060,type,
sK647: ( $i > $i > $o ) > $i > $i ).
thf(func_def_1061,type,
sK648: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1062,type,
sK649: ( $i > $i > $o ) > $i > $i ).
thf(func_def_1063,type,
sK650: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_1084,type,
sK671: $i > $i > $i ).
thf(func_def_1118,type,
sK705: ( $i > $o ) > $i ).
thf(func_def_1119,type,
sK706: $i > ( $i > $o ) > $i ).
thf(func_def_1120,type,
sK707: $i > $o ).
thf(func_def_1146,type,
sK733: $i > $o ).
thf(func_def_1148,type,
sK735: $i > $i ).
thf(func_def_1149,type,
sK736: $i > $i ).
thf(func_def_1150,type,
sK737: ( $i > $i ) > $i > $i ).
thf(func_def_1151,type,
sK738: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1159,type,
sK746: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1160,type,
sK747: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1161,type,
sK748: ( $i > $o ) > $i > $i ).
thf(func_def_1165,type,
sK752: $i > $o ).
thf(func_def_1172,type,
sK759: $i > $i > $i > $i ).
thf(func_def_1173,type,
sK760: $i > $i > $i > $i > $i ).
thf(func_def_1182,type,
sK769: $i > $i ).
thf(func_def_1183,type,
sK770: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1184,type,
sK771: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1192,type,
sK779: $i > ( $i > $o ) > $i ).
thf(func_def_1193,type,
sK780: $i > ( $i > $o ) > $i ).
thf(func_def_1194,type,
sK781: ( $i > $o ) > $i > $i ).
thf(func_def_1196,type,
sK783: $i > $o ).
thf(func_def_1220,type,
sK807: $i > $i > $i ).
thf(func_def_1242,type,
sK829: $i > $i ).
thf(func_def_1244,type,
sK831: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1245,type,
sK832: $i > $i > $o ).
thf(func_def_1273,type,
sK860: $i > $i ).
thf(func_def_1303,type,
sK890: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1304,type,
sK891: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1305,type,
sK892: ( $i > $o ) > $i > $i ).
thf(func_def_1309,type,
sK896: $i > $o ).
thf(func_def_1317,type,
sK904: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1318,type,
sK905: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1319,type,
sK906: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1320,type,
sK907: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1321,type,
sK908: $i > $i > $o ).
thf(func_def_1331,type,
sK918: $i > $i > $o ).
thf(func_def_1336,type,
sK923: $i > $i > $i > $i > $i > $o > $i ).
thf(func_def_1351,type,
sK938: $i > $o ).
thf(func_def_1353,type,
sK940: $i > ( $i > $o ) > $i ).
thf(func_def_1354,type,
sK941: ( $i > $o ) > $i > $i ).
thf(func_def_1359,type,
sK946: $i > $o ).
thf(func_def_1368,type,
sK955: $i > ( $i > $o ) > $i ).
thf(func_def_1369,type,
sK956: ( $i > $o ) > $i ).
thf(func_def_1370,type,
sK957: $i > $o ).
thf(func_def_1371,type,
sK958: $i > $i > $i > $o ).
thf(func_def_1372,type,
sK959: $i > $i > $i > $i ).
thf(func_def_1376,type,
sK963: $i > $i > $i > $i ).
thf(func_def_1377,type,
sK964: $i > $i > $i > $i ).
thf(func_def_1392,type,
sK979: $i > $i ).
thf(func_def_1395,type,
sK982: $i > $i ).
thf(func_def_1409,type,
sK996: $i > $o ).
thf(func_def_1414,type,
sK1001: $i > $i > $i > $i ).
thf(func_def_1415,type,
sK1002: $i > $i > $i > $i ).
thf(func_def_1435,type,
sK1022: $i > $i > $i > $i > $i > $i ).
thf(func_def_1442,type,
sK1029: $i > $o ).
thf(func_def_1443,type,
sK1030: $i > $o ).
thf(func_def_1444,type,
sK1031: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1445,type,
sK1032: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1449,type,
sK1036: $i > $o ).
thf(func_def_1450,type,
sK1037: $i > $o ).
thf(func_def_1451,type,
sK1038: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1452,type,
sK1039: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1453,type,
sK1040: $i > $i > $i ).
thf(func_def_1454,type,
sK1041: $i > $i > $i ).
thf(func_def_1455,type,
sK1042: $i > $i > $i > $i ).
thf(func_def_1456,type,
sK1043: $i > $i > $i ).
thf(func_def_1457,type,
sK1044: $i > $i > $i ).
thf(func_def_1458,type,
sK1045: $i > $i ).
thf(func_def_1459,type,
sK1046: $i > $i ).
thf(func_def_1460,type,
sK1047: $i > $i ).
thf(func_def_1461,type,
sK1048: $i > $i ).
thf(func_def_1462,type,
sK1049: $i > $i > $i ).
thf(func_def_1463,type,
sK1050: $i > $i > $i ).
thf(func_def_1464,type,
sK1051: $i > $i > $i ).
thf(func_def_1465,type,
sK1052: $i > $i > $i ).
thf(func_def_1466,type,
sK1053: $i > $i > $i ).
thf(func_def_1467,type,
sK1054: $i > $i > $i ).
thf(func_def_1468,type,
sK1055: $i > $i > $i > $i ).
thf(func_def_1469,type,
sK1056: $i > $i > $i > $i ).
thf(func_def_1471,type,
sK1058: $i > $i ).
thf(func_def_1474,type,
sK1061: $i > $i > $i > $i ).
thf(func_def_1475,type,
sK1062: $i > $i > $i > $i ).
thf(func_def_1486,type,
sK1073: $i > $i > ( $i > $o ) > $i ).
thf(func_def_1488,type,
sK1075: $i > $o ).
thf(func_def_1494,type,
sK1081: $i > $i > $i ).
thf(func_def_1500,type,
sK1087: $i > $i > $i > $i > $o ).
thf(func_def_1511,type,
sK1098: $i > $i ).
thf(func_def_1512,type,
sK1099: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1513,type,
sK1100: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1514,type,
sK1101: ( $i > $o ) > $i > $i ).
thf(func_def_1517,type,
sK1104: $i > $o ).
thf(func_def_1522,type,
sK1109: $i > $o ).
thf(func_def_1525,type,
sK1112: $i > $o ).
thf(func_def_1526,type,
sK1113: ( $i > $o ) > $i > $i ).
thf(func_def_1527,type,
sK1114: $i > ( $i > $o ) > $i ).
thf(func_def_1532,type,
sK1119: $i > $o ).
thf(func_def_1533,type,
sK1120: ( $i > $o ) > $i > $i ).
thf(func_def_1534,type,
sK1121: ( $i > $o ) > $i > $i ).
thf(func_def_1542,type,
sK1129: $i > $o ).
thf(func_def_1559,type,
sK1146: $i > $i > $i > $i ).
thf(func_def_1584,type,
sK1171: $i > $i > $o ).
thf(func_def_1608,type,
ph1195:
!>[X0: $tType] : X0 ).
thf(f8599,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f8598,f3827]) ).
thf(f3827,plain,
( ( in @ sK142 @ sK141 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1716]) ).
thf(f1716,plain,
( ( notinself = $true )
& ( breln1unionIL = $true )
& ( powersetAx = $true )
& ( omegaSAx = $true )
& ( setukpairinjL = $true )
& ( setadjoinSub = $true )
& ( binintersectSubset1 = $true )
& ( ifSingleton = $true )
& ( setadjoinIL = $true )
& ( binintersectER = $true )
& ( image1Ex = $true )
& ( descr__Cong = $true )
& ( setunionsingleton2 = $true )
& ( powersetTE1 = $true )
& ( cartprodmempair1 = $true )
& ( omega0Ax = $true )
& ( setukpairinjR2 = $true )
& ( ordinalTransSet = $true )
& ( setunionE = $true )
& ( upairsetE = $true )
& ( theeq = $true )
& ( subsetTI = $true )
& ( funcext = $true )
& ( setminusERneg = $true )
& ( wellorderingAx = $true )
& ( setukpairIL = $true )
& ( setunionI = $true )
& ( ap2apEq2 = $true )
& ( choice2fnsingleton = $true )
& ( powersetTI1 = $true )
& ( setadjoinSub2 = $true )
& ( notinemptyset = $true )
& ( binunionRsub = $true )
& ( cartprodpairsurjEq = $true )
& ( emptysetAx = $true )
& ( cartprodmempair = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjL1 = $true )
& ( breln1unionI = $true )
& ( surjFuncSetFuncSurj = $true )
& ( singletonsuniq = $true )
& ( cartprodsndin = $true )
& ( complementUnionInPowersetComplement = $true )
& ( ubforcartprodlem1 = $true )
& ( ordinalMinLem1 = $true )
& ( breln1compprop = $true )
& ( subPowSU = $true )
& ( complementTnotintersectT = $true )
& ( singletonsubset = $true )
& ( woz13rule2 = $true )
& ( surjFuncSetFuncIn = $true )
& ( binintersectSubset5 = $true )
& ( iffalseProp1 = $true )
& ( binunionE = $true )
& ( exuI3 = $true )
& ( breln1invI = $true )
& ( doubleComplementSub1 = $true )
& ( foundation2 = $true )
& ( ex1I = $true )
& ( demorgan2a = $true )
& ( eqimpsubset2 = $true )
& ( setoftrueEq = $true )
& ( eqbreln = $true )
& ( vacuousDall = $true )
& ( cartprodpairmemER = $true )
& ( setminusLsub = $true )
& ( funcGraphProp2 = $true )
& ( inCongP = $true )
& ( contrasubsetT3 = $true )
& ( bs114d = $true )
& ( binintTransitive = $true )
& ( notdexE = $true )
& ( demorgan2 = $true )
& ( setukpairinjL2 = $true )
& ( dsetconstrI = $true )
& ( binunionIL = $true )
& ( setextAx = $true )
& ( upairequniteq = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setunionTransitive = $true )
& ( subbreln1 = $true )
& ( demorgan1a = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem2 = $true )
& ( upairsetIL = $true )
& ( dpsetconstrER = $true )
& ( eqimpsubset1 = $true )
& ( omegaSp = $true )
& ( binunionTILcontra = $true )
& ( binunionTIRcontra = $true )
& ( quantDeMorgan1 = $true )
& ( woz1_3 = $true )
& ( emptyinunitempty = $true )
& ( powersetI = $true )
& ( ksndsingleton = $true )
& ( setminusER = $true )
& ( quantDeMorgan3 = $true )
& ( brelnall1 = $true )
& ( setminusI = $true )
& ( sepInPowerset = $true )
& ( nonemptyE1 = $true )
& ( iffalseProp2 = $true )
& ( dsetconstrEL = $true )
& ( emptyI = $true )
& ( symdiffI2 = $true )
& ( binunionEcases = $true )
& ( kfstpairEq = $true )
& ( notequalI1 = $true )
& ( powersetE = $true )
& ( woz1_1 = $true )
& ( symdiffI1 = $true )
& ( iffalse = $true )
& ( nonemptyI1 = $true )
& ( lamProp = $true )
& ( setminusIRneg = $true )
& ( breln1invE = $true )
& ( singletonsswitch = $true )
& ( ex1I2 = $true )
& ( setextsub = $true )
& ( omega__Cong = $true )
& ( transitivesetOp2 = $true )
& ( singletoninpowunion = $true )
& ( binintersectEL = $true )
& ( $true
= ( ordinal @ sK141 ) )
& ( ( in @ sK142 @ sK141 )
= $true )
& ( ( subset @ sK142 @ sK141 )
!= $true )
& ( exu__Cong = $true )
& ( funcGraphProp3 = $true )
& ( funcextLem = $true )
& ( kpairp = $true )
& ( breln1compEex = $true )
& ( eqbreln1 = $true )
& ( setadjoinE = $true )
& ( binintersectSubset2 = $true )
& ( ex1E2 = $true )
& ( binunionT_lem = $true )
& ( beta1 = $true )
& ( breln1unionCommutes = $true )
& ( replAx = $true )
& ( sepSubset = $true )
& ( breln1invprop = $true )
& ( woz13rule4 = $true )
& ( demorgan1b = $true )
& ( setunionsingleton1 = $true )
& ( in__Cong = $true )
& ( binunionIR = $true )
& ( funcext2 = $true )
& ( prop2setI = $true )
& ( funcGraphProp4 = $true )
& ( powersetT_lem = $true )
& ( surjCantorThm = $true )
& ( apProp = $true )
& ( inIntersectImpInUnion2 = $true )
& ( binunionTE = $true )
& ( complementTcontraSubset = $true )
& ( binintersectSubset4 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setunionE2 = $true )
& ( powerset__Cong = $true )
& ( breln1unionIR = $true )
& ( injFuncSetFuncIn = $true )
& ( setbeta = $true )
& ( binunionLsub = $true )
& ( ap2p = $true )
& ( dpsetconstrI = $true )
& ( complementT_lem = $true )
& ( subbreln = $true )
& ( subsetemptysetimpeq = $true )
& ( breln1compE = $true )
& ( setukpairinjR12 = $true )
& ( subsetI1 = $true )
& ( lam2lamEq = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( omegaIndAx = $true )
& ( iftrueorfalse = $true )
& ( complementImpComplementIntersect = $true )
& ( breln1all2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( ap2apEq1 = $true )
& ( cartprodpairmemEL = $true )
& ( iftrue = $true )
& ( exuE1 = $true )
& ( breln1SetBreln1 = $true )
& ( setadjoinOr = $true )
& ( binintersectI = $true )
& ( dpsetconstrERa = $true )
& ( setadjoinAx = $true )
& ( exuE2 = $true )
& ( setukpairinjR11 = $true )
& ( demorgan2a1 = $true )
& ( contraSubsetComplement = $true )
& ( dsetconstrER = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( woz2A = $true )
& ( binintersectTERcontra = $true )
& ( setOfPairsIsBReln1 = $true )
& ( woz1_4 = $true )
& ( exuI1 = $true )
& ( iftrueProp2 = $true )
& ( subsetE2 = $true )
& ( woz2B = $true )
& ( funcGraphProp1 = $true )
& ( contrasubsetT = $true )
& ( setminusILneg = $true )
& ( upairinpowunion = $true )
& ( leftInvIsSurj = $true )
& ( doubleComplementE1 = $true )
& ( lamp = $true )
& ( emptysetE = $true )
& ( demorgan2a2 = $true )
& ( exuE3u = $true )
& ( upairset2E = $true )
& ( woz13rule0 = $true )
& ( binintersectT_lem = $true )
& ( beta2 = $true )
& ( breln1unionE = $true )
& ( symdiffIneg1 = $true )
& ( inPowerset = $true )
& ( image1I = $true )
& ( binintersectRsub = $true )
& ( breln1all1 = $true )
& ( notdallE = $true )
& ( demorgan2b = $true )
& ( theprop = $true )
& ( contrasubsetT2 = $true )
& ( setminusSubset2 = $true )
& ( setminusELneg = $true )
& ( descrp = $true )
& ( demorgan2b2 = $true )
& ( demorgan1 = $true )
& ( subset2powerset = $true )
& ( exuEu = $true )
& ( breln1unionEcases = $true )
& ( omegaSclos = $true )
& ( setminusEL = $true )
& ( woz13rule3 = $true )
& ( dsetconstr__Cong = $true )
& ( notequalI2 = $true )
& ( symdiffE = $true )
& ( setminusT_lem = $true )
& ( image1Ex1 = $true )
& ( dpsetconstrEL1 = $true )
& ( setOfPairsIsBReln = $true )
& ( funcinfuncset = $true )
& ( iftrueProp1 = $true )
& ( binintersectTELcontra = $true )
& ( exuI2 = $true )
& ( eta1 = $true )
& ( setext = $true )
& ( setunion__Cong = $true )
& ( transitivesetOp1 = $true )
& ( emptyInPowerset = $true )
& ( setadjoinIR = $true )
& ( inIntersectImpInUnion = $true )
& ( upairsetIR = $true )
& ( disjointsetsI1 = $true )
& ( nonemptyI = $true )
& ( kpairiskpair = $true )
& ( kfstsingleton = $true )
& ( injFuncInInjFuncSet = $true )
& ( woz1_5 = $true )
& ( setunionsingleton = $true )
& ( kpairsurjEq = $true )
& ( ubforcartprodlem3 = $true )
& ( ex1E1 = $true )
& ( image1E = $true )
& ( binunionTEcontra = $true )
& ( injFuncSetFuncInj = $true )
& ( peano0notS = $true )
& ( subsetTrans = $true )
& ( funcImageSingleton = $true )
& ( contrasubsetT1 = $true )
& ( dpsetconstrSub = $true )
& ( notinsingleton = $true )
& ( quantDeMorgan4 = $true )
& ( woz1_2 = $true )
& ( notsubsetI = $true )
& ( singletoninpowerset = $true )
& ( setukpairIR = $true )
& ( lam2p = $true )
& ( complementTI1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( noeltsimpempty = $true )
& ( foundationAx = $true )
& ( eqinunit = $true )
& ( emptyinPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( setukpairinjR1 = $true )
& ( setunionAx = $true )
& ( setukpairinjR = $true )
& ( cartprodfstin = $true )
& ( emptysetsubset = $true )
& ( peanoSinj = $true )
& ( woz13rule1 = $true )
& ( breln1unionprop = $true )
& ( cartprodsndpairEq = $true )
& ( singletonprop = $true )
& ( setminusSubset1 = $true )
& ( doubleComplementSub2 = $true )
& ( cartprodpairin = $true )
& ( woz2W = $true )
& ( doubleComplementEq = $true )
& ( upairsubunion = $true )
& ( prop2set2propI = $true )
& ( ifp = $true )
& ( upairset2IR = $true )
& ( eta2 = $true )
& ( subsetRefl = $true )
& ( setadjoin__Cong = $true )
& ( emptysetimpfalse = $true )
& ( binintersectSubset3 = $true )
& ( breln1compI = $true )
& ( powersetsubset = $true )
& ( uniqinunit = $true )
& ( complementTE1 = $true )
& ( emptyE1 = $true )
& ( nonemptyImpWitness = $true )
& ( subsetE = $true )
& ( app = $true )
& ( ksndpairEq = $true )
& ( infuncsetfunc = $true )
& ( woz2Ex = $true )
& ( setextT = $true )
& ( symdiffIneg2 = $true )
& ( quantDeMorgan2 = $true )
& ( brelnall2 = $true )
& ( subsetI2 = $true )
& ( dpsetconstrEL2 = $true )
& ( doubleComplementI1 = $true )
& ( image1Equiv = $true )
& ( notinself2 = $true )
& ( prop2setE = $true )
& ( secondinupair = $true )
& ( binintersectLsub = $true )
& ( emptyset__Cong = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK141,sK142])],[f1362,f1715,f1714]) ).
thf(f1714,plain,
( ? [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
& ? [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
& ( ( subset @ X1 @ X0 )
!= $true ) ) )
=> ( ( $true
= ( ordinal @ sK141 ) )
& ? [X1: $i] :
( ( ( in @ X1 @ sK141 )
= $true )
& ( ( subset @ X1 @ sK141 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1715,plain,
( ? [X1: $i] :
( ( ( in @ X1 @ sK141 )
= $true )
& ( ( subset @ X1 @ sK141 )
!= $true ) )
=> ( ( ( in @ sK142 @ sK141 )
= $true )
& ( ( subset @ sK142 @ sK141 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1362,plain,
( ( notinself = $true )
& ( breln1unionIL = $true )
& ( powersetAx = $true )
& ( omegaSAx = $true )
& ( setukpairinjL = $true )
& ( setadjoinSub = $true )
& ( binintersectSubset1 = $true )
& ( ifSingleton = $true )
& ( setadjoinIL = $true )
& ( binintersectER = $true )
& ( image1Ex = $true )
& ( descr__Cong = $true )
& ( setunionsingleton2 = $true )
& ( powersetTE1 = $true )
& ( cartprodmempair1 = $true )
& ( omega0Ax = $true )
& ( setukpairinjR2 = $true )
& ( ordinalTransSet = $true )
& ( setunionE = $true )
& ( upairsetE = $true )
& ( theeq = $true )
& ( subsetTI = $true )
& ( funcext = $true )
& ( setminusERneg = $true )
& ( wellorderingAx = $true )
& ( setukpairIL = $true )
& ( setunionI = $true )
& ( ap2apEq2 = $true )
& ( choice2fnsingleton = $true )
& ( powersetTI1 = $true )
& ( setadjoinSub2 = $true )
& ( notinemptyset = $true )
& ( binunionRsub = $true )
& ( cartprodpairsurjEq = $true )
& ( emptysetAx = $true )
& ( cartprodmempair = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjL1 = $true )
& ( breln1unionI = $true )
& ( surjFuncSetFuncSurj = $true )
& ( singletonsuniq = $true )
& ( cartprodsndin = $true )
& ( complementUnionInPowersetComplement = $true )
& ( ubforcartprodlem1 = $true )
& ( ordinalMinLem1 = $true )
& ( breln1compprop = $true )
& ( subPowSU = $true )
& ( complementTnotintersectT = $true )
& ( singletonsubset = $true )
& ( woz13rule2 = $true )
& ( surjFuncSetFuncIn = $true )
& ( binintersectSubset5 = $true )
& ( iffalseProp1 = $true )
& ( binunionE = $true )
& ( exuI3 = $true )
& ( breln1invI = $true )
& ( doubleComplementSub1 = $true )
& ( foundation2 = $true )
& ( ex1I = $true )
& ( demorgan2a = $true )
& ( eqimpsubset2 = $true )
& ( setoftrueEq = $true )
& ( eqbreln = $true )
& ( vacuousDall = $true )
& ( cartprodpairmemER = $true )
& ( setminusLsub = $true )
& ( funcGraphProp2 = $true )
& ( inCongP = $true )
& ( contrasubsetT3 = $true )
& ( bs114d = $true )
& ( binintTransitive = $true )
& ( notdexE = $true )
& ( demorgan2 = $true )
& ( setukpairinjL2 = $true )
& ( dsetconstrI = $true )
& ( binunionIL = $true )
& ( setextAx = $true )
& ( upairequniteq = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setunionTransitive = $true )
& ( subbreln1 = $true )
& ( demorgan1a = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem2 = $true )
& ( upairsetIL = $true )
& ( dpsetconstrER = $true )
& ( eqimpsubset1 = $true )
& ( omegaSp = $true )
& ( binunionTILcontra = $true )
& ( binunionTIRcontra = $true )
& ( quantDeMorgan1 = $true )
& ( woz1_3 = $true )
& ( emptyinunitempty = $true )
& ( powersetI = $true )
& ( ksndsingleton = $true )
& ( setminusER = $true )
& ( quantDeMorgan3 = $true )
& ( brelnall1 = $true )
& ( setminusI = $true )
& ( sepInPowerset = $true )
& ( nonemptyE1 = $true )
& ( iffalseProp2 = $true )
& ( dsetconstrEL = $true )
& ( emptyI = $true )
& ( symdiffI2 = $true )
& ( binunionEcases = $true )
& ( kfstpairEq = $true )
& ( notequalI1 = $true )
& ( powersetE = $true )
& ( woz1_1 = $true )
& ( symdiffI1 = $true )
& ( iffalse = $true )
& ( nonemptyI1 = $true )
& ( lamProp = $true )
& ( setminusIRneg = $true )
& ( breln1invE = $true )
& ( singletonsswitch = $true )
& ( ex1I2 = $true )
& ( setextsub = $true )
& ( omega__Cong = $true )
& ( transitivesetOp2 = $true )
& ( singletoninpowunion = $true )
& ( binintersectEL = $true )
& ? [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
& ? [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
& ( ( subset @ X1 @ X0 )
!= $true ) ) )
& ( exu__Cong = $true )
& ( funcGraphProp3 = $true )
& ( funcextLem = $true )
& ( kpairp = $true )
& ( breln1compEex = $true )
& ( eqbreln1 = $true )
& ( setadjoinE = $true )
& ( binintersectSubset2 = $true )
& ( ex1E2 = $true )
& ( binunionT_lem = $true )
& ( beta1 = $true )
& ( breln1unionCommutes = $true )
& ( replAx = $true )
& ( sepSubset = $true )
& ( breln1invprop = $true )
& ( woz13rule4 = $true )
& ( demorgan1b = $true )
& ( setunionsingleton1 = $true )
& ( in__Cong = $true )
& ( binunionIR = $true )
& ( funcext2 = $true )
& ( prop2setI = $true )
& ( funcGraphProp4 = $true )
& ( powersetT_lem = $true )
& ( surjCantorThm = $true )
& ( apProp = $true )
& ( inIntersectImpInUnion2 = $true )
& ( binunionTE = $true )
& ( complementTcontraSubset = $true )
& ( binintersectSubset4 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setunionE2 = $true )
& ( powerset__Cong = $true )
& ( breln1unionIR = $true )
& ( injFuncSetFuncIn = $true )
& ( setbeta = $true )
& ( binunionLsub = $true )
& ( ap2p = $true )
& ( dpsetconstrI = $true )
& ( complementT_lem = $true )
& ( subbreln = $true )
& ( subsetemptysetimpeq = $true )
& ( breln1compE = $true )
& ( setukpairinjR12 = $true )
& ( subsetI1 = $true )
& ( lam2lamEq = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( omegaIndAx = $true )
& ( iftrueorfalse = $true )
& ( complementImpComplementIntersect = $true )
& ( breln1all2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( ap2apEq1 = $true )
& ( cartprodpairmemEL = $true )
& ( iftrue = $true )
& ( exuE1 = $true )
& ( breln1SetBreln1 = $true )
& ( setadjoinOr = $true )
& ( binintersectI = $true )
& ( dpsetconstrERa = $true )
& ( setadjoinAx = $true )
& ( exuE2 = $true )
& ( setukpairinjR11 = $true )
& ( demorgan2a1 = $true )
& ( contraSubsetComplement = $true )
& ( dsetconstrER = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( woz2A = $true )
& ( binintersectTERcontra = $true )
& ( setOfPairsIsBReln1 = $true )
& ( woz1_4 = $true )
& ( exuI1 = $true )
& ( iftrueProp2 = $true )
& ( subsetE2 = $true )
& ( woz2B = $true )
& ( funcGraphProp1 = $true )
& ( contrasubsetT = $true )
& ( setminusILneg = $true )
& ( upairinpowunion = $true )
& ( leftInvIsSurj = $true )
& ( doubleComplementE1 = $true )
& ( lamp = $true )
& ( emptysetE = $true )
& ( demorgan2a2 = $true )
& ( exuE3u = $true )
& ( upairset2E = $true )
& ( woz13rule0 = $true )
& ( binintersectT_lem = $true )
& ( beta2 = $true )
& ( breln1unionE = $true )
& ( symdiffIneg1 = $true )
& ( inPowerset = $true )
& ( image1I = $true )
& ( binintersectRsub = $true )
& ( breln1all1 = $true )
& ( notdallE = $true )
& ( demorgan2b = $true )
& ( theprop = $true )
& ( contrasubsetT2 = $true )
& ( setminusSubset2 = $true )
& ( setminusELneg = $true )
& ( descrp = $true )
& ( demorgan2b2 = $true )
& ( demorgan1 = $true )
& ( subset2powerset = $true )
& ( exuEu = $true )
& ( breln1unionEcases = $true )
& ( omegaSclos = $true )
& ( setminusEL = $true )
& ( woz13rule3 = $true )
& ( dsetconstr__Cong = $true )
& ( notequalI2 = $true )
& ( symdiffE = $true )
& ( setminusT_lem = $true )
& ( image1Ex1 = $true )
& ( dpsetconstrEL1 = $true )
& ( setOfPairsIsBReln = $true )
& ( funcinfuncset = $true )
& ( iftrueProp1 = $true )
& ( binintersectTELcontra = $true )
& ( exuI2 = $true )
& ( eta1 = $true )
& ( setext = $true )
& ( setunion__Cong = $true )
& ( transitivesetOp1 = $true )
& ( emptyInPowerset = $true )
& ( setadjoinIR = $true )
& ( inIntersectImpInUnion = $true )
& ( upairsetIR = $true )
& ( disjointsetsI1 = $true )
& ( nonemptyI = $true )
& ( kpairiskpair = $true )
& ( kfstsingleton = $true )
& ( injFuncInInjFuncSet = $true )
& ( woz1_5 = $true )
& ( setunionsingleton = $true )
& ( kpairsurjEq = $true )
& ( ubforcartprodlem3 = $true )
& ( ex1E1 = $true )
& ( image1E = $true )
& ( binunionTEcontra = $true )
& ( injFuncSetFuncInj = $true )
& ( peano0notS = $true )
& ( subsetTrans = $true )
& ( funcImageSingleton = $true )
& ( contrasubsetT1 = $true )
& ( dpsetconstrSub = $true )
& ( notinsingleton = $true )
& ( quantDeMorgan4 = $true )
& ( woz1_2 = $true )
& ( notsubsetI = $true )
& ( singletoninpowerset = $true )
& ( setukpairIR = $true )
& ( lam2p = $true )
& ( complementTI1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( noeltsimpempty = $true )
& ( foundationAx = $true )
& ( eqinunit = $true )
& ( emptyinPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( setukpairinjR1 = $true )
& ( setunionAx = $true )
& ( setukpairinjR = $true )
& ( cartprodfstin = $true )
& ( emptysetsubset = $true )
& ( peanoSinj = $true )
& ( woz13rule1 = $true )
& ( breln1unionprop = $true )
& ( cartprodsndpairEq = $true )
& ( singletonprop = $true )
& ( setminusSubset1 = $true )
& ( doubleComplementSub2 = $true )
& ( cartprodpairin = $true )
& ( woz2W = $true )
& ( doubleComplementEq = $true )
& ( upairsubunion = $true )
& ( prop2set2propI = $true )
& ( ifp = $true )
& ( upairset2IR = $true )
& ( eta2 = $true )
& ( subsetRefl = $true )
& ( setadjoin__Cong = $true )
& ( emptysetimpfalse = $true )
& ( binintersectSubset3 = $true )
& ( breln1compI = $true )
& ( powersetsubset = $true )
& ( uniqinunit = $true )
& ( complementTE1 = $true )
& ( emptyE1 = $true )
& ( nonemptyImpWitness = $true )
& ( subsetE = $true )
& ( app = $true )
& ( ksndpairEq = $true )
& ( infuncsetfunc = $true )
& ( woz2Ex = $true )
& ( setextT = $true )
& ( symdiffIneg2 = $true )
& ( quantDeMorgan2 = $true )
& ( brelnall2 = $true )
& ( subsetI2 = $true )
& ( dpsetconstrEL2 = $true )
& ( doubleComplementI1 = $true )
& ( image1Equiv = $true )
& ( notinself2 = $true )
& ( prop2setE = $true )
& ( secondinupair = $true )
& ( binintersectLsub = $true )
& ( emptyset__Cong = $true ) ),
inference(flattening,[status(thm)],[f1361]) ).
thf(f1361,plain,
( ? [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
& ? [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
& ( ( subset @ X1 @ X0 )
!= $true ) ) )
& ( ordinalTransSet = $true )
& ( ordinalMinLem1 = $true )
& ( setunionTransitive = $true )
& ( transitivesetOp2 = $true )
& ( binintTransitive = $true )
& ( transitivesetOp1 = $true )
& ( peanoSinj = $true )
& ( peano0notS = $true )
& ( omegaSclos = $true )
& ( omegaSp = $true )
& ( notinself2 = $true )
& ( notinself = $true )
& ( foundation2 = $true )
& ( surjCantorThm = $true )
& ( leftInvIsSurj = $true )
& ( surjFuncSetFuncSurj = $true )
& ( surjFuncSetFuncIn = $true )
& ( injFuncSetFuncInj = $true )
& ( injFuncSetFuncIn = $true )
& ( injFuncInInjFuncSet = $true )
& ( image1I = $true )
& ( image1E = $true )
& ( image1Equiv = $true )
& ( image1Ex1 = $true )
& ( image1Ex = $true )
& ( woz2B = $true )
& ( woz2A = $true )
& ( woz2W = $true )
& ( woz2Ex = $true )
& ( breln1unionCommutes = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f943]) ).
thf(f943,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ( ( breln1unionE = $true )
=> ( ( breln1unionEcases = $true )
=> ( ( breln1unionCommutes = $true )
=> ( ( woz2Ex = $true )
=> ( ( woz2W = $true )
=> ( ( woz2A = $true )
=> ( ( woz2B = $true )
=> ( ( image1Ex = $true )
=> ( ( image1Ex1 = $true )
=> ( ( image1Equiv = $true )
=> ( ( image1E = $true )
=> ( ( image1I = $true )
=> ( ( injFuncInInjFuncSet = $true )
=> ( ( injFuncSetFuncIn = $true )
=> ( ( injFuncSetFuncInj = $true )
=> ( ( surjFuncSetFuncIn = $true )
=> ( ( surjFuncSetFuncSurj = $true )
=> ( ( leftInvIsSurj = $true )
=> ( ( surjCantorThm = $true )
=> ( ( foundation2 = $true )
=> ( ( notinself = $true )
=> ( ( notinself2 = $true )
=> ( ( omegaSp = $true )
=> ( ( omegaSclos = $true )
=> ( ( peano0notS = $true )
=> ( ( peanoSinj = $true )
=> ( ( transitivesetOp1 = $true )
=> ( ( binintTransitive = $true )
=> ( ( transitivesetOp2 = $true )
=> ( ( setunionTransitive = $true )
=> ( ( ordinalMinLem1 = $true )
=> ( ( ordinalTransSet = $true )
=> ! [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
=> ! [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
=> ( ( subset @ X1 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f942]) ).
thf(f942,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ( binintTransitive
=> ( transitivesetOp2
=> ( setunionTransitive
=> ( ordinalMinLem1
=> ( ordinalTransSet
=> ! [X0: $i] :
( ( ordinal @ X0 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( subset @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f342]) ).
thf(f342,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ( binintTransitive
=> ( transitivesetOp2
=> ( setunionTransitive
=> ( ordinalMinLem1
=> ( ordinalTransSet
=> ! [X11: $i] :
( ( ordinal @ X11 )
=> ! [X3: $i] :
( ( in @ X3 @ X11 )
=> ( subset @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f341]) ).
thf(f341,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ( binintTransitive
=> ( transitivesetOp2
=> ( setunionTransitive
=> ( ordinalMinLem1
=> ( ordinalTransSet
=> ! [X11: $i] :
( ( ordinal @ X11 )
=> ! [X3: $i] :
( ( in @ X3 @ X11 )
=> ( subset @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.PG6uUBnDB9/DTF2THF_9905.p',ordinalTransSet1) ).
thf(f8598,plain,
( ( in @ sK142 @ sK141 )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f8595,f7031]) ).
thf(f7031,plain,
( ( transitiveset @ sK141 )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f7030]) ).
thf(f7030,plain,
( $true
= ( ( wellorderedByIn @ sK141 )
& ( transitiveset @ sK141 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f5621]) ).
thf(f5621,plain,
( $true
= ( ^ [Y0: $i] :
( ( wellorderedByIn @ Y0 )
& ( transitiveset @ Y0 ) )
@ sK141 ) ),
inference(definition_unfolding,[status(thm)],[f3828,f4852]) ).
thf(f4852,plain,
( ordinal
= ( ^ [Y0: $i] :
( ( wellorderedByIn @ Y0 )
& ( transitiveset @ Y0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f883]) ).
thf(f883,plain,
( ordinal
= ( ^ [Y0: $i] :
( ( wellorderedByIn @ Y0 )
& ( transitiveset @ Y0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f882]) ).
thf(f882,plain,
( ordinal
= ( ^ [X0: $i] :
( ( transitiveset @ X0 )
& ( wellorderedByIn @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f338]) ).
thf(f338,axiom,
( ordinal
= ( ^ [X1: $i] :
( ( transitiveset @ X1 )
& ( wellorderedByIn @ X1 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.PG6uUBnDB9/DTF2THF_9905.p',ordinal) ).
thf(f3828,plain,
( $true
= ( ordinal @ sK141 ) ),
inference(cnf_transformation,[status(thm)],[f1716]) ).
thf(f8595,plain,
( ( ( transitiveset @ sK141 )
!= $true )
| ( ( in @ sK142 @ sK141 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f8592]) ).
thf(f8592,plain,
( ( $true != $true )
| ( ( transitiveset @ sK141 )
!= $true )
| ( ( in @ sK142 @ sK141 )
!= $true ) ),
inference(superposition,[status(thm)],[f3826,f6895]) ).
thf(f6895,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
| ( ( transitiveset @ X2 )
!= $true )
| ( ( in @ X3 @ X2 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f5842]) ).
thf(f5842,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
| ( ( transitiveset @ X2 )
!= $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f4250,f3701]) ).
thf(f3701,plain,
transitivesetOp1 = $true,
inference(cnf_transformation,[status(thm)],[f1716]) ).
thf(f4250,plain,
! [X2: $i,X3: $i] :
( ( ( transitiveset @ X2 )
!= $true )
| ( ( subset @ X3 @ X2 )
= $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( transitivesetOp1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f2062]) ).
thf(f2062,plain,
( ( ( transitivesetOp1 = $true )
| ( ( ( transitiveset @ sK345 )
= $true )
& ( ( subset @ sK346 @ sK345 )
!= $true )
& ( $true
= ( in @ sK346 @ sK345 ) ) ) )
& ( ! [X2: $i] :
( ( ( transitiveset @ X2 )
!= $true )
| ! [X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
| ( ( in @ X3 @ X2 )
!= $true ) ) )
| ( transitivesetOp1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK345,sK346])],[f2059,f2061,f2060]) ).
thf(f2060,plain,
( ? [X0: $i] :
( ( ( transitiveset @ X0 )
= $true )
& ? [X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
& ( ( in @ X1 @ X0 )
= $true ) ) )
=> ( ( ( transitiveset @ sK345 )
= $true )
& ? [X1: $i] :
( ( $true
!= ( subset @ X1 @ sK345 ) )
& ( ( in @ X1 @ sK345 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2061,plain,
( ? [X1: $i] :
( ( $true
!= ( subset @ X1 @ sK345 ) )
& ( ( in @ X1 @ sK345 )
= $true ) )
=> ( ( ( subset @ sK346 @ sK345 )
!= $true )
& ( $true
= ( in @ sK346 @ sK345 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2059,plain,
( ( ( transitivesetOp1 = $true )
| ? [X0: $i] :
( ( ( transitiveset @ X0 )
= $true )
& ? [X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
& ( ( in @ X1 @ X0 )
= $true ) ) ) )
& ( ! [X2: $i] :
( ( ( transitiveset @ X2 )
!= $true )
| ! [X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
| ( ( in @ X3 @ X2 )
!= $true ) ) )
| ( transitivesetOp1 != $true ) ) ),
inference(rectify,[status(thm)],[f2058]) ).
thf(f2058,plain,
( ( ( transitivesetOp1 = $true )
| ? [X0: $i] :
( ( ( transitiveset @ X0 )
= $true )
& ? [X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
& ( ( in @ X1 @ X0 )
= $true ) ) ) )
& ( ! [X0: $i] :
( ( ( transitiveset @ X0 )
!= $true )
| ! [X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X0 )
!= $true ) ) )
| ( transitivesetOp1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1306]) ).
thf(f1306,plain,
( ( transitivesetOp1 = $true )
<=> ! [X0: $i] :
( ( ( transitiveset @ X0 )
!= $true )
| ! [X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X0 )
!= $true ) ) ) ),
inference(ennf_transformation,[status(thm)],[f449]) ).
thf(f449,plain,
( ( transitivesetOp1 = $true )
<=> ! [X0: $i] :
( ( ( transitiveset @ X0 )
= $true )
=> ! [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
=> ( ( subset @ X1 @ X0 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f448]) ).
thf(f448,plain,
! [X0: $i] :
( ( ( transitiveset @ X0 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( subset @ X1 @ X0 ) ) )
= transitivesetOp1 ),
inference(rectify,[status(thm)],[f332]) ).
thf(f332,axiom,
! [X11: $i] :
( ( ( transitiveset @ X11 )
=> ! [X3: $i] :
( ( in @ X3 @ X11 )
=> ( subset @ X3 @ X11 ) ) )
= transitivesetOp1 ),
file('/export/starexec/sandbox/tmp/tmp.PG6uUBnDB9/DTF2THF_9905.p',transitivesetOp1) ).
thf(f3826,plain,
( ( subset @ sK142 @ sK141 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1716]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU818^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n031.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 12:36:03 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.33/0.49 ---- Original DTF file ---
% 0.33/0.49 thf(spec,logic,$$dhol).
% 0.33/0.49 %------------------------------------------------------------------------------
% 0.33/0.49 % File : SEU818^1 : TPTP v9.2.1. Released v3.7.0.
% 0.33/0.49 % Domain : Set Theory
% 0.33/0.49 % Problem : Ordinals
% 0.33/0.49 % Version : Especial.
% 0.33/0.49 % English : (! X:i.ordinal X -> (! A:i.in A X -> subset A X))
% 0.33/0.49
% 0.33/0.49 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.33/0.49 % Source : [Bro08]
% 0.33/0.49 % Names : ZFC320g [Bro08]
% 0.33/0.49
% 0.33/0.49 % Status : Theorem
% 0.33/0.49 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.33/0.49 % Syntax : Number of formulae : 738 ( 340 unt; 397 typ; 340 def)
% 0.33/0.49 % Number of atoms : 2101 ( 487 equ; 0 cnn)
% 0.33/0.49 % Maximal formula atoms : 337 ( 6 avg)
% 0.33/0.49 % Number of connectives : 4549 ( 78 ~; 17 |; 60 &;3298 @)
% 0.33/0.49 % ( 17 <=>;1079 =>; 0 <=; 0 <~>)
% 0.33/0.49 % Maximal formula depth : 341 ( 2 avg)
% 0.33/0.49 % Number of types : 2 ( 0 usr)
% 0.33/0.49 % Number of type conns : 194 ( 194 >; 0 *; 0 +; 0 <<)
% 0.33/0.49 % Number of symbols : 400 ( 397 usr; 338 con; 0-4 aty)
% 0.33/0.49 % Number of variables : 1181 ( 79 ^;1057 !; 45 ?;1181 :)
% 0.33/0.49 % SPC : TH0_THM_EQU_NAR
% 0.33/0.49
% 0.33/0.49 % Comments : http://mathgate.info/detsetitem.php?id=514
% 0.33/0.49 %------------------------------------------------------------------------------
% 0.33/0.49 thf(in_type,type,
% 0.33/0.49 in: $i > $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(exu_type,type,
% 0.33/0.49 exu: ( $i > $o ) > $o ).
% 0.33/0.49
% 0.33/0.49 thf(exu,definition,
% 0.33/0.49 ( exu
% 0.33/0.49 = ( ^ [Xphi: $i > $o] :
% 0.33/0.49 ? [Xx: $i] :
% 0.33/0.49 ( ( Xphi @ Xx )
% 0.33/0.49 & ! [Xy: $i] :
% 0.33/0.49 ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setextAx_type,type,
% 0.33/0.49 setextAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setextAx,definition,
% 0.33/0.49 ( setextAx
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 <=> ( in @ Xx @ B ) )
% 0.33/0.49 => ( A = B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptyset_type,type,
% 0.33/0.49 emptyset: $i ).
% 0.33/0.49
% 0.33/0.49 thf(emptysetAx_type,type,
% 0.33/0.49 emptysetAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptysetAx,definition,
% 0.33/0.49 ( emptysetAx
% 0.33/0.49 = ( ! [Xx: $i] :
% 0.33/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoin_type,type,
% 0.33/0.49 setadjoin: $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinAx_type,type,
% 0.33/0.49 setadjoinAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinAx,definition,
% 0.33/0.49 ( setadjoinAx
% 0.33/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.33/0.49 <=> ( ( Xy = Xx )
% 0.33/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(powerset_type,type,
% 0.33/0.49 powerset: $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(powersetAx_type,type,
% 0.33/0.49 powersetAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(powersetAx,definition,
% 0.33/0.49 ( powersetAx
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.33/0.49 <=> ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setunion_type,type,
% 0.33/0.49 setunion: $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(setunionAx_type,type,
% 0.33/0.49 setunionAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setunionAx,definition,
% 0.33/0.49 ( setunionAx
% 0.33/0.49 = ( ! [A: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.33/0.49 <=> ? [B: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 & ( in @ B @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(omega_type,type,
% 0.33/0.49 omega: $i ).
% 0.33/0.49
% 0.33/0.49 thf(omega0Ax_type,type,
% 0.33/0.49 omega0Ax: $o ).
% 0.33/0.49
% 0.33/0.49 thf(omega0Ax,definition,
% 0.33/0.49 ( omega0Ax
% 0.33/0.49 = ( in @ emptyset @ omega ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(omegaSAx_type,type,
% 0.33/0.49 omegaSAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(omegaSAx,definition,
% 0.33/0.49 ( omegaSAx
% 0.33/0.49 = ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ omega )
% 0.33/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(omegaIndAx_type,type,
% 0.33/0.49 omegaIndAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(omegaIndAx,definition,
% 0.33/0.49 ( omegaIndAx
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ( ( in @ emptyset @ A )
% 0.33/0.49 & ! [Xx: $i] :
% 0.33/0.49 ( ( ( in @ Xx @ omega )
% 0.33/0.49 & ( in @ Xx @ A ) )
% 0.33/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ omega )
% 0.33/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(replAx_type,type,
% 0.33/0.49 replAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(replAx,definition,
% 0.33/0.49 ( replAx
% 0.33/0.49 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( exu
% 0.33/0.49 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.33/0.49 => ? [B: $i] :
% 0.33/0.49 ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 <=> ? [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ A )
% 0.33/0.49 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(foundationAx_type,type,
% 0.33/0.49 foundationAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(foundationAx,definition,
% 0.33/0.49 ( foundationAx
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.33/0.49 => ? [B: $i] :
% 0.33/0.49 ( ( in @ B @ A )
% 0.33/0.49 & ~ ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(wellorderingAx_type,type,
% 0.33/0.49 wellorderingAx: $o ).
% 0.33/0.49
% 0.33/0.49 thf(wellorderingAx,definition,
% 0.33/0.49 ( wellorderingAx
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ? [B: $i] :
% 0.33/0.49 ( ! [C: $i] :
% 0.33/0.49 ( ( in @ C @ B )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ C )
% 0.33/0.49 => ( in @ Xx @ A ) ) )
% 0.33/0.49 & ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( ( in @ Xx @ A )
% 0.33/0.49 & ( in @ Xy @ A ) )
% 0.33/0.49 => ( ! [C: $i] :
% 0.33/0.49 ( ( in @ C @ B )
% 0.33/0.49 => ( ( in @ Xx @ C )
% 0.33/0.49 <=> ( in @ Xy @ C ) ) )
% 0.33/0.49 => ( Xx = Xy ) ) )
% 0.33/0.49 & ! [C: $i,D: $i] :
% 0.33/0.49 ( ( ( in @ C @ B )
% 0.33/0.49 & ( in @ D @ B ) )
% 0.33/0.49 => ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ C )
% 0.33/0.49 => ( in @ Xx @ D ) )
% 0.33/0.49 | ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ D )
% 0.33/0.49 => ( in @ Xx @ C ) ) ) )
% 0.33/0.49 & ! [C: $i] :
% 0.33/0.49 ( ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ C )
% 0.33/0.49 => ( in @ Xx @ A ) )
% 0.33/0.49 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.33/0.49 => ? [D: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ D @ B )
% 0.33/0.49 & ( in @ Xx @ C )
% 0.33/0.49 & ~ ? [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ D )
% 0.33/0.49 & ( in @ Xy @ C ) )
% 0.33/0.49 & ! [E: $i] :
% 0.33/0.49 ( ( in @ E @ B )
% 0.33/0.49 => ( ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ E )
% 0.33/0.49 => ( in @ Xy @ D ) )
% 0.33/0.49 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(descr_type,type,
% 0.33/0.49 descr: ( $i > $o ) > $i ).
% 0.33/0.49
% 0.33/0.49 thf(descrp_type,type,
% 0.33/0.49 descrp: $o ).
% 0.33/0.49
% 0.33/0.49 thf(descrp,definition,
% 0.33/0.49 ( descrp
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 => ( Xphi
% 0.33/0.49 @ ( descr
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstr_type,type,
% 0.33/0.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstrI_type,type,
% 0.33/0.49 dsetconstrI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstrI,definition,
% 0.33/0.49 ( dsetconstrI
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ( Xphi @ Xx )
% 0.33/0.49 => ( in @ Xx
% 0.33/0.49 @ ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstrEL_type,type,
% 0.33/0.49 dsetconstrEL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstrEL,definition,
% 0.33/0.49 ( dsetconstrEL
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx
% 0.33/0.49 @ ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.33/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstrER_type,type,
% 0.33/0.49 dsetconstrER: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstrER,definition,
% 0.33/0.49 ( dsetconstrER
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx
% 0.33/0.49 @ ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.33/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exuE1_type,type,
% 0.33/0.49 exuE1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exuE1,definition,
% 0.33/0.49 ( exuE1
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 => ? [Xx: $i] :
% 0.33/0.49 ( ( Xphi @ Xx )
% 0.33/0.49 & ! [Xy: $i] :
% 0.33/0.49 ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(prop2set_type,type,
% 0.33/0.49 prop2set: $o > $i ).
% 0.33/0.49
% 0.33/0.49 thf(prop2setE_type,type,
% 0.33/0.49 prop2setE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(prop2setE,definition,
% 0.33/0.49 ( prop2setE
% 0.33/0.49 = ( ! [Xphi: $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.33/0.49 => Xphi ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptysetE_type,type,
% 0.33/0.49 emptysetE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptysetE,definition,
% 0.33/0.49 ( emptysetE
% 0.33/0.49 = ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ emptyset )
% 0.33/0.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptysetimpfalse_type,type,
% 0.33/0.49 emptysetimpfalse: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptysetimpfalse,definition,
% 0.33/0.49 ( emptysetimpfalse
% 0.33/0.49 = ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ emptyset )
% 0.33/0.49 => $false ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(notinemptyset_type,type,
% 0.33/0.49 notinemptyset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(notinemptyset,definition,
% 0.33/0.49 ( notinemptyset
% 0.33/0.49 = ( ! [Xx: $i] :
% 0.33/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exuE3e_type,type,
% 0.33/0.49 exuE3e: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exuE3e,definition,
% 0.33/0.49 ( exuE3e
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setext_type,type,
% 0.33/0.49 setext: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setext,definition,
% 0.33/0.49 ( setext
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ Xx @ B ) )
% 0.33/0.49 => ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ A ) )
% 0.33/0.49 => ( A = B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptyI_type,type,
% 0.33/0.49 emptyI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptyI,definition,
% 0.33/0.49 ( emptyI
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ~ ( in @ Xx @ A )
% 0.33/0.49 => ( A = emptyset ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(noeltsimpempty_type,type,
% 0.33/0.49 noeltsimpempty: $o ).
% 0.33/0.49
% 0.33/0.49 thf(noeltsimpempty,definition,
% 0.33/0.49 ( noeltsimpempty
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ~ ( in @ Xx @ A )
% 0.33/0.49 => ( A = emptyset ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setbeta_type,type,
% 0.33/0.49 setbeta: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setbeta,definition,
% 0.33/0.49 ( setbeta
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ( in @ Xx
% 0.33/0.49 @ ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.33/0.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(nonempty_type,type,
% 0.33/0.49 nonempty: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(nonempty,definition,
% 0.33/0.49 ( nonempty
% 0.33/0.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(nonemptyE1_type,type,
% 0.33/0.49 nonemptyE1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(nonemptyE1,definition,
% 0.33/0.49 ( nonemptyE1
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ( nonempty @ A )
% 0.33/0.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(nonemptyI_type,type,
% 0.33/0.49 nonemptyI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(nonemptyI,definition,
% 0.33/0.49 ( nonemptyI
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ( Xphi @ Xx )
% 0.33/0.49 => ( nonempty
% 0.33/0.49 @ ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(nonemptyI1_type,type,
% 0.33/0.49 nonemptyI1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(nonemptyI1,definition,
% 0.33/0.49 ( nonemptyI1
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.33/0.49 => ( nonempty @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinIL_type,type,
% 0.33/0.49 setadjoinIL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinIL,definition,
% 0.33/0.49 ( setadjoinIL
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptyinunitempty_type,type,
% 0.33/0.49 emptyinunitempty: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptyinunitempty,definition,
% 0.33/0.49 ( emptyinunitempty
% 0.33/0.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinIR_type,type,
% 0.33/0.49 setadjoinIR: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinIR,definition,
% 0.33/0.49 ( setadjoinIR
% 0.33/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ A )
% 0.33/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinE_type,type,
% 0.33/0.49 setadjoinE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinE,definition,
% 0.33/0.49 ( setadjoinE
% 0.33/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.33/0.49 => ! [Xphi: $o] :
% 0.33/0.49 ( ( ( Xy = Xx )
% 0.33/0.49 => Xphi )
% 0.33/0.49 => ( ( ( in @ Xy @ A )
% 0.33/0.49 => Xphi )
% 0.33/0.49 => Xphi ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinOr_type,type,
% 0.33/0.49 setadjoinOr: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinOr,definition,
% 0.33/0.49 ( setadjoinOr
% 0.33/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.33/0.49 => ( ( Xy = Xx )
% 0.33/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setoftrueEq_type,type,
% 0.33/0.49 setoftrueEq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setoftrueEq,definition,
% 0.33/0.49 ( setoftrueEq
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xx: $i] : $true )
% 0.33/0.49 = A ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(powersetI_type,type,
% 0.33/0.49 powersetI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(powersetI,definition,
% 0.33/0.49 ( powersetI
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ A ) )
% 0.33/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptyinPowerset_type,type,
% 0.33/0.49 emptyinPowerset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptyinPowerset,definition,
% 0.33/0.49 ( emptyinPowerset
% 0.33/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptyInPowerset_type,type,
% 0.33/0.49 emptyInPowerset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptyInPowerset,definition,
% 0.33/0.49 ( emptyInPowerset
% 0.33/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(powersetE_type,type,
% 0.33/0.49 powersetE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(powersetE,definition,
% 0.33/0.49 ( powersetE
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.33/0.49 => ( ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setunionI_type,type,
% 0.33/0.49 setunionI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setunionI,definition,
% 0.33/0.49 ( setunionI
% 0.33/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 => ( ( in @ B @ A )
% 0.33/0.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setunionE_type,type,
% 0.33/0.49 setunionE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setunionE,definition,
% 0.33/0.49 ( setunionE
% 0.33/0.49 = ( ! [A: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.33/0.49 => ! [Xphi: $o] :
% 0.33/0.49 ( ! [B: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 => ( ( in @ B @ A )
% 0.33/0.49 => Xphi ) )
% 0.33/0.49 => Xphi ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subPowSU_type,type,
% 0.33/0.49 subPowSU: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subPowSU,definition,
% 0.33/0.49 ( subPowSU
% 0.33/0.49 = ( ! [A: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exuE2_type,type,
% 0.33/0.49 exuE2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exuE2,definition,
% 0.33/0.49 ( exuE2
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 => ? [Xx: $i] :
% 0.33/0.49 ! [Xy: $i] :
% 0.33/0.49 ( ( Xphi @ Xy )
% 0.33/0.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(nonemptyImpWitness_type,type,
% 0.33/0.49 nonemptyImpWitness: $o ).
% 0.33/0.49
% 0.33/0.49 thf(nonemptyImpWitness,definition,
% 0.33/0.49 ( nonemptyImpWitness
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ( nonempty @ A )
% 0.33/0.49 => ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & $true ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(uniqinunit_type,type,
% 0.33/0.49 uniqinunit: $o ).
% 0.33/0.49
% 0.33/0.49 thf(uniqinunit,definition,
% 0.33/0.49 ( uniqinunit
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(notinsingleton_type,type,
% 0.33/0.49 notinsingleton: $o ).
% 0.33/0.49
% 0.33/0.49 thf(notinsingleton,definition,
% 0.33/0.49 ( notinsingleton
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx != Xy )
% 0.33/0.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(eqinunit_type,type,
% 0.33/0.49 eqinunit: $o ).
% 0.33/0.49
% 0.33/0.49 thf(eqinunit,definition,
% 0.33/0.49 ( eqinunit
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx = Xy )
% 0.33/0.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(singletonsswitch_type,type,
% 0.33/0.49 singletonsswitch: $o ).
% 0.33/0.49
% 0.33/0.49 thf(singletonsswitch,definition,
% 0.33/0.49 ( singletonsswitch
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.33/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(upairsetE_type,type,
% 0.33/0.49 upairsetE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairsetE,definition,
% 0.33/0.49 ( upairsetE
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.33/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.33/0.49 => ( ( Xz = Xx )
% 0.33/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(upairsetIL_type,type,
% 0.33/0.49 upairsetIL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairsetIL,definition,
% 0.33/0.49 ( upairsetIL
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(upairsetIR_type,type,
% 0.33/0.49 upairsetIR: $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairsetIR,definition,
% 0.33/0.49 ( upairsetIR
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptyE1_type,type,
% 0.33/0.49 emptyE1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptyE1,definition,
% 0.33/0.49 ( emptyE1
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ( Xphi @ Xx ) )
% 0.33/0.49 => ( ( ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 = emptyset )
% 0.33/0.49 => $false ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(vacuousDall_type,type,
% 0.33/0.49 vacuousDall: $o ).
% 0.33/0.49
% 0.33/0.49 thf(vacuousDall,definition,
% 0.33/0.49 ( vacuousDall
% 0.33/0.49 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ emptyset )
% 0.33/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(quantDeMorgan1_type,type,
% 0.33/0.49 quantDeMorgan1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(quantDeMorgan1,definition,
% 0.33/0.49 ( quantDeMorgan1
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ~ ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( Xphi @ Xx ) )
% 0.33/0.49 => ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(quantDeMorgan2_type,type,
% 0.33/0.49 quantDeMorgan2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(quantDeMorgan2,definition,
% 0.33/0.49 ( quantDeMorgan2
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ~ ( Xphi @ Xx ) )
% 0.33/0.49 => ~ ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(quantDeMorgan3_type,type,
% 0.33/0.49 quantDeMorgan3: $o ).
% 0.33/0.49
% 0.33/0.49 thf(quantDeMorgan3,definition,
% 0.33/0.49 ( quantDeMorgan3
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ~ ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ( Xphi @ Xx ) )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(quantDeMorgan4_type,type,
% 0.33/0.49 quantDeMorgan4: $o ).
% 0.33/0.49
% 0.33/0.49 thf(quantDeMorgan4,definition,
% 0.33/0.49 ( quantDeMorgan4
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ~ ( Xphi @ Xx ) )
% 0.33/0.49 => ~ ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(prop2setI_type,type,
% 0.33/0.49 prop2setI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(prop2setI,definition,
% 0.33/0.49 ( prop2setI
% 0.33/0.49 = ( ! [Xphi: $o] :
% 0.33/0.49 ( Xphi
% 0.33/0.49 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(set2prop_type,type,
% 0.33/0.49 set2prop: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(prop2set2propI_type,type,
% 0.33/0.49 prop2set2propI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(prop2set2propI,definition,
% 0.33/0.49 ( prop2set2propI
% 0.33/0.49 = ( ! [Xphi: $o] :
% 0.33/0.49 ( Xphi
% 0.33/0.49 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(notdexE_type,type,
% 0.33/0.49 notdexE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(notdexE,definition,
% 0.33/0.49 ( notdexE
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ~ ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ( Xphi @ Xx ) )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(notdallE_type,type,
% 0.33/0.49 notdallE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(notdallE,definition,
% 0.33/0.49 ( notdallE
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ~ ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( Xphi @ Xx ) )
% 0.33/0.49 => ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exuI1_type,type,
% 0.33/0.49 exuI1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exuI1,definition,
% 0.33/0.49 ( exuI1
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ? [Xx: $i] :
% 0.33/0.49 ( ( Xphi @ Xx )
% 0.33/0.49 & ! [Xy: $i] :
% 0.33/0.49 ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xx = Xy ) ) )
% 0.33/0.49 => ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exuI3_type,type,
% 0.33/0.49 exuI3: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exuI3,definition,
% 0.33/0.49 ( exuI3
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.33/0.49 => ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xphi @ Xx )
% 0.33/0.49 => ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xx = Xy ) ) )
% 0.33/0.49 => ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exuI2_type,type,
% 0.33/0.49 exuI2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exuI2,definition,
% 0.33/0.49 ( exuI2
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ? [Xx: $i] :
% 0.33/0.49 ! [Xy: $i] :
% 0.33/0.49 ( ( Xphi @ Xy )
% 0.33/0.49 <=> ( Xy = Xx ) )
% 0.33/0.49 => ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(inCongP_type,type,
% 0.33/0.49 inCongP: $o ).
% 0.33/0.49
% 0.33/0.49 thf(inCongP,definition,
% 0.33/0.49 ( inCongP
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( A = B )
% 0.33/0.49 => ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx = Xy )
% 0.33/0.49 => ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(in__Cong_type,type,
% 0.33/0.49 in__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(in__Cong,definition,
% 0.33/0.49 ( in__Cong
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( A = B )
% 0.33/0.49 => ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx = Xy )
% 0.33/0.49 => ( ( in @ Xx @ A )
% 0.33/0.49 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exuE3u_type,type,
% 0.33/0.49 exuE3u: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exuE3u,definition,
% 0.33/0.49 ( exuE3u
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 => ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xphi @ Xx )
% 0.33/0.49 => ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exu__Cong_type,type,
% 0.33/0.49 exu__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exu__Cong,definition,
% 0.33/0.49 ( exu__Cong
% 0.33/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.33/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx = Xy )
% 0.33/0.49 => ( ( Xphi @ Xx )
% 0.33/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.33/0.49 => ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 <=> ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptyset__Cong_type,type,
% 0.33/0.49 emptyset__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptyset__Cong,definition,
% 0.33/0.49 ( emptyset__Cong
% 0.33/0.49 = ( emptyset = emptyset ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoin__Cong_type,type,
% 0.33/0.49 setadjoin__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoin__Cong,definition,
% 0.33/0.49 ( setadjoin__Cong
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx = Xy )
% 0.33/0.49 => ! [Xz: $i,Xu: $i] :
% 0.33/0.49 ( ( Xz = Xu )
% 0.33/0.49 => ( ( setadjoin @ Xx @ Xz )
% 0.33/0.49 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(powerset__Cong_type,type,
% 0.33/0.49 powerset__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(powerset__Cong,definition,
% 0.33/0.49 ( powerset__Cong
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( A = B )
% 0.33/0.49 => ( ( powerset @ A )
% 0.33/0.49 = ( powerset @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setunion__Cong_type,type,
% 0.33/0.49 setunion__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setunion__Cong,definition,
% 0.33/0.49 ( setunion__Cong
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( A = B )
% 0.33/0.49 => ( ( setunion @ A )
% 0.33/0.49 = ( setunion @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(omega__Cong_type,type,
% 0.33/0.49 omega__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(omega__Cong,definition,
% 0.33/0.49 ( omega__Cong
% 0.33/0.49 = ( omega = omega ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(exuEu_type,type,
% 0.33/0.49 exuEu: $o ).
% 0.33/0.49
% 0.33/0.49 thf(exuEu,definition,
% 0.33/0.49 ( exuEu
% 0.33/0.49 = ( ! [Xphi: $i > $o] :
% 0.33/0.49 ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 => ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xphi @ Xx )
% 0.33/0.49 => ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(descr__Cong_type,type,
% 0.33/0.49 descr__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(descr__Cong,definition,
% 0.33/0.49 ( descr__Cong
% 0.33/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.33/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx = Xy )
% 0.33/0.49 => ( ( Xphi @ Xx )
% 0.33/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.33/0.49 => ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 => ( ( exu
% 0.33/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.33/0.49 => ( ( descr
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 = ( descr
% 0.33/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstr__Cong_type,type,
% 0.33/0.49 dsetconstr__Cong: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dsetconstr__Cong,definition,
% 0.33/0.49 ( dsetconstr__Cong
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( A = B )
% 0.33/0.49 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( ( Xx = Xy )
% 0.33/0.49 => ( ( Xphi @ Xx )
% 0.33/0.49 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.33/0.49 => ( ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 = ( dsetconstr @ B
% 0.33/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subset_type,type,
% 0.33/0.49 subset: $i > $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(disjoint_type,type,
% 0.33/0.49 disjoint: $i > $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(setsmeet_type,type,
% 0.33/0.49 setsmeet: $i > $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(subsetI1_type,type,
% 0.33/0.49 subsetI1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subsetI1,definition,
% 0.33/0.49 ( subsetI1
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ Xx @ B ) )
% 0.33/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(eqimpsubset2_type,type,
% 0.33/0.49 eqimpsubset2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(eqimpsubset2,definition,
% 0.33/0.49 ( eqimpsubset2
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( A = B )
% 0.33/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(eqimpsubset1_type,type,
% 0.33/0.49 eqimpsubset1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(eqimpsubset1,definition,
% 0.33/0.49 ( eqimpsubset1
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( A = B )
% 0.33/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subsetI2_type,type,
% 0.33/0.49 subsetI2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subsetI2,definition,
% 0.33/0.49 ( subsetI2
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ Xx @ B ) )
% 0.33/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(emptysetsubset_type,type,
% 0.33/0.49 emptysetsubset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(emptysetsubset,definition,
% 0.33/0.49 ( emptysetsubset
% 0.33/0.49 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subsetE_type,type,
% 0.33/0.49 subsetE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subsetE,definition,
% 0.33/0.49 ( subsetE
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subsetE2_type,type,
% 0.33/0.49 subsetE2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subsetE2,definition,
% 0.33/0.49 ( subsetE2
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( ~ ( in @ Xx @ B )
% 0.33/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(notsubsetI_type,type,
% 0.33/0.49 notsubsetI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(notsubsetI,definition,
% 0.33/0.49 ( notsubsetI
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ~ ( in @ Xx @ B )
% 0.33/0.49 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(notequalI1_type,type,
% 0.33/0.49 notequalI1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(notequalI1,definition,
% 0.33/0.49 ( notequalI1
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ~ ( subset @ A @ B )
% 0.33/0.49 => ( A != B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(notequalI2_type,type,
% 0.33/0.49 notequalI2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(notequalI2,definition,
% 0.33/0.49 ( notequalI2
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ~ ( in @ Xx @ B )
% 0.33/0.49 => ( A != B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subsetRefl_type,type,
% 0.33/0.49 subsetRefl: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subsetRefl,definition,
% 0.33/0.49 ( subsetRefl
% 0.33/0.49 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subsetTrans_type,type,
% 0.33/0.49 subsetTrans: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subsetTrans,definition,
% 0.33/0.49 ( subsetTrans
% 0.33/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( ( subset @ B @ C )
% 0.33/0.49 => ( subset @ A @ C ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinSub_type,type,
% 0.33/0.49 setadjoinSub: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinSub,definition,
% 0.33/0.49 ( setadjoinSub
% 0.33/0.49 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinSub2_type,type,
% 0.33/0.49 setadjoinSub2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setadjoinSub2,definition,
% 0.33/0.49 ( setadjoinSub2
% 0.33/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subset2powerset_type,type,
% 0.33/0.49 subset2powerset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subset2powerset,definition,
% 0.33/0.49 ( subset2powerset
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setextsub_type,type,
% 0.33/0.49 setextsub: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setextsub,definition,
% 0.33/0.49 ( setextsub
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( ( subset @ B @ A )
% 0.33/0.49 => ( A = B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(subsetemptysetimpeq_type,type,
% 0.33/0.49 subsetemptysetimpeq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(subsetemptysetimpeq,definition,
% 0.33/0.49 ( subsetemptysetimpeq
% 0.33/0.49 = ( ! [A: $i] :
% 0.33/0.49 ( ( subset @ A @ emptyset )
% 0.33/0.49 => ( A = emptyset ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(powersetI1_type,type,
% 0.33/0.49 powersetI1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(powersetI1,definition,
% 0.33/0.49 ( powersetI1
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( subset @ B @ A )
% 0.33/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(powersetE1_type,type,
% 0.33/0.49 powersetE1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(powersetE1,definition,
% 0.33/0.49 ( powersetE1
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.33/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(inPowerset_type,type,
% 0.33/0.49 inPowerset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(inPowerset,definition,
% 0.33/0.49 ( inPowerset
% 0.33/0.49 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(powersetsubset_type,type,
% 0.33/0.49 powersetsubset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(powersetsubset,definition,
% 0.33/0.49 ( powersetsubset
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(sepInPowerset_type,type,
% 0.33/0.49 sepInPowerset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(sepInPowerset,definition,
% 0.33/0.49 ( sepInPowerset
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( in
% 0.33/0.49 @ ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 @ ( powerset @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(sepSubset_type,type,
% 0.33/0.49 sepSubset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(sepSubset,definition,
% 0.33/0.49 ( sepSubset
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( subset
% 0.33/0.49 @ ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 @ A ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binunion_type,type,
% 0.33/0.49 binunion: $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(binunionIL_type,type,
% 0.33/0.49 binunionIL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binunionIL,definition,
% 0.33/0.49 ( binunionIL
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(upairset2IR_type,type,
% 0.33/0.49 upairset2IR: $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairset2IR,definition,
% 0.33/0.49 ( upairset2IR
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binunionIR_type,type,
% 0.33/0.49 binunionIR: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binunionIR,definition,
% 0.33/0.49 ( binunionIR
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binunionEcases_type,type,
% 0.33/0.49 binunionEcases: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binunionEcases,definition,
% 0.33/0.49 ( binunionEcases
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.33/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.33/0.49 => ( ( ( in @ Xx @ A )
% 0.33/0.49 => Xphi )
% 0.33/0.49 => ( ( ( in @ Xx @ B )
% 0.33/0.49 => Xphi )
% 0.33/0.49 => Xphi ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binunionE_type,type,
% 0.33/0.49 binunionE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binunionE,definition,
% 0.33/0.49 ( binunionE
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.33/0.49 => ( ( in @ Xx @ A )
% 0.33/0.49 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binunionLsub_type,type,
% 0.33/0.49 binunionLsub: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binunionLsub,definition,
% 0.33/0.49 ( binunionLsub
% 0.33/0.49 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binunionRsub_type,type,
% 0.33/0.49 binunionRsub: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binunionRsub,definition,
% 0.33/0.49 ( binunionRsub
% 0.33/0.49 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersect_type,type,
% 0.33/0.49 binintersect: $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectI_type,type,
% 0.33/0.49 binintersectI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectI,definition,
% 0.33/0.49 ( binintersectI
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset5_type,type,
% 0.33/0.49 binintersectSubset5: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset5,definition,
% 0.33/0.49 ( binintersectSubset5
% 0.33/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.33/0.49 ( ( subset @ C @ A )
% 0.33/0.49 => ( ( subset @ C @ B )
% 0.33/0.49 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectEL_type,type,
% 0.33/0.49 binintersectEL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectEL,definition,
% 0.33/0.49 ( binintersectEL
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.33/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectLsub_type,type,
% 0.33/0.49 binintersectLsub: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectLsub,definition,
% 0.33/0.49 ( binintersectLsub
% 0.33/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset2_type,type,
% 0.33/0.49 binintersectSubset2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset2,definition,
% 0.33/0.49 ( binintersectSubset2
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( ( binintersect @ A @ B )
% 0.33/0.49 = A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset3_type,type,
% 0.33/0.49 binintersectSubset3: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset3,definition,
% 0.33/0.49 ( binintersectSubset3
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( ( binintersect @ A @ B )
% 0.33/0.49 = B )
% 0.33/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectER_type,type,
% 0.33/0.49 binintersectER: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectER,definition,
% 0.33/0.49 ( binintersectER
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.33/0.49 => ( in @ Xx @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(disjointsetsI1_type,type,
% 0.33/0.49 disjointsetsI1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(disjointsetsI1,definition,
% 0.33/0.49 ( disjointsetsI1
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ~ ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ( in @ Xx @ B ) )
% 0.33/0.49 => ( ( binintersect @ A @ B )
% 0.33/0.49 = emptyset ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectRsub_type,type,
% 0.33/0.49 binintersectRsub: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectRsub,definition,
% 0.33/0.49 ( binintersectRsub
% 0.33/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset4_type,type,
% 0.33/0.49 binintersectSubset4: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset4,definition,
% 0.33/0.49 ( binintersectSubset4
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( subset @ B @ A )
% 0.33/0.49 => ( ( binintersect @ A @ B )
% 0.33/0.49 = B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset1_type,type,
% 0.33/0.49 binintersectSubset1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(binintersectSubset1,definition,
% 0.33/0.49 ( binintersectSubset1
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( ( binintersect @ A @ B )
% 0.33/0.49 = A )
% 0.33/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(bs114d_type,type,
% 0.33/0.49 bs114d: $o ).
% 0.33/0.49
% 0.33/0.49 thf(bs114d,definition,
% 0.33/0.49 ( bs114d
% 0.33/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.33/0.49 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.33/0.49 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(regular_type,type,
% 0.33/0.49 regular: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminus_type,type,
% 0.33/0.49 setminus: $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(setminusI_type,type,
% 0.33/0.49 setminusI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusI,definition,
% 0.33/0.49 ( setminusI
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ~ ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusEL_type,type,
% 0.33/0.49 setminusEL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusEL,definition,
% 0.33/0.49 ( setminusEL
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.33/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusER_type,type,
% 0.33/0.49 setminusER: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusER,definition,
% 0.33/0.49 ( setminusER
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.33/0.49 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusSubset2_type,type,
% 0.33/0.49 setminusSubset2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusSubset2,definition,
% 0.33/0.49 ( setminusSubset2
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( subset @ A @ B )
% 0.33/0.49 => ( ( setminus @ A @ B )
% 0.33/0.49 = emptyset ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusERneg_type,type,
% 0.33/0.49 setminusERneg: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusERneg,definition,
% 0.33/0.49 ( setminusERneg
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.33/0.49 => ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusELneg_type,type,
% 0.33/0.49 setminusELneg: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusELneg,definition,
% 0.33/0.49 ( setminusELneg
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.33/0.49 => ( ~ ( in @ Xx @ B )
% 0.33/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusILneg_type,type,
% 0.33/0.49 setminusILneg: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusILneg,definition,
% 0.33/0.49 ( setminusILneg
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ~ ( in @ Xx @ A )
% 0.33/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusIRneg_type,type,
% 0.33/0.49 setminusIRneg: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusIRneg,definition,
% 0.33/0.49 ( setminusIRneg
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ B )
% 0.33/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusLsub_type,type,
% 0.33/0.49 setminusLsub: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusLsub,definition,
% 0.33/0.49 ( setminusLsub
% 0.33/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setminusSubset1_type,type,
% 0.33/0.49 setminusSubset1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setminusSubset1,definition,
% 0.33/0.49 ( setminusSubset1
% 0.33/0.49 = ( ! [A: $i,B: $i] :
% 0.33/0.49 ( ( ( setminus @ A @ B )
% 0.33/0.49 = emptyset )
% 0.33/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(symdiff_type,type,
% 0.33/0.49 symdiff: $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffE_type,type,
% 0.33/0.49 symdiffE: $o ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffE,definition,
% 0.33/0.49 ( symdiffE
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.33/0.49 => ! [Xphi: $o] :
% 0.33/0.49 ( ( ( in @ Xx @ A )
% 0.33/0.49 => ( ~ ( in @ Xx @ B )
% 0.33/0.49 => Xphi ) )
% 0.33/0.49 => ( ( ~ ( in @ Xx @ A )
% 0.33/0.49 => ( ( in @ Xx @ B )
% 0.33/0.49 => Xphi ) )
% 0.33/0.49 => Xphi ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffI1_type,type,
% 0.33/0.49 symdiffI1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffI1,definition,
% 0.33/0.49 ( symdiffI1
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ~ ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffI2_type,type,
% 0.33/0.49 symdiffI2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffI2,definition,
% 0.33/0.49 ( symdiffI2
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ~ ( in @ Xx @ A )
% 0.33/0.49 => ( ( in @ Xx @ B )
% 0.33/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffIneg1_type,type,
% 0.33/0.49 symdiffIneg1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffIneg1,definition,
% 0.33/0.49 ( symdiffIneg1
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ( in @ Xx @ B )
% 0.33/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffIneg2_type,type,
% 0.33/0.49 symdiffIneg2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(symdiffIneg2,definition,
% 0.33/0.49 ( symdiffIneg2
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ~ ( in @ Xx @ A )
% 0.33/0.49 => ( ~ ( in @ Xx @ B )
% 0.33/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(iskpair_type,type,
% 0.33/0.49 iskpair: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(secondinupair_type,type,
% 0.33/0.49 secondinupair: $o ).
% 0.33/0.49
% 0.33/0.49 thf(secondinupair,definition,
% 0.33/0.49 ( secondinupair
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairIL_type,type,
% 0.33/0.49 setukpairIL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairIL,definition,
% 0.33/0.49 ( setukpairIL
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairIR_type,type,
% 0.33/0.49 setukpairIR: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairIR,definition,
% 0.33/0.49 ( setukpairIR
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(kpairiskpair_type,type,
% 0.33/0.49 kpairiskpair: $o ).
% 0.33/0.49
% 0.33/0.49 thf(kpairiskpair,definition,
% 0.33/0.49 ( kpairiskpair
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(kpair_type,type,
% 0.33/0.49 kpair: $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(kpairp_type,type,
% 0.33/0.49 kpairp: $o ).
% 0.33/0.49
% 0.33/0.49 thf(kpairp,definition,
% 0.33/0.49 ( kpairp
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprod_type,type,
% 0.33/0.49 cartprod: $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(singletonsubset_type,type,
% 0.33/0.49 singletonsubset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(singletonsubset,definition,
% 0.33/0.49 ( singletonsubset
% 0.33/0.49 = ( ! [A: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(singletoninpowerset_type,type,
% 0.33/0.49 singletoninpowerset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(singletoninpowerset,definition,
% 0.33/0.49 ( singletoninpowerset
% 0.33/0.49 = ( ! [A: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(singletoninpowunion_type,type,
% 0.33/0.49 singletoninpowunion: $o ).
% 0.33/0.49
% 0.33/0.49 thf(singletoninpowunion,definition,
% 0.33/0.49 ( singletoninpowunion
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(upairset2E_type,type,
% 0.33/0.49 upairset2E: $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairset2E,definition,
% 0.33/0.49 ( upairset2E
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.33/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.33/0.49 => ( ( Xz = Xx )
% 0.33/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(upairsubunion_type,type,
% 0.33/0.49 upairsubunion: $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairsubunion,definition,
% 0.33/0.49 ( upairsubunion
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(upairinpowunion_type,type,
% 0.33/0.49 upairinpowunion: $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairinpowunion,definition,
% 0.33/0.49 ( upairinpowunion
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ubforcartprodlem1_type,type,
% 0.33/0.49 ubforcartprodlem1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ubforcartprodlem1,definition,
% 0.33/0.49 ( ubforcartprodlem1
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ubforcartprodlem2_type,type,
% 0.33/0.49 ubforcartprodlem2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ubforcartprodlem2,definition,
% 0.33/0.49 ( ubforcartprodlem2
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ubforcartprodlem3_type,type,
% 0.33/0.49 ubforcartprodlem3: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ubforcartprodlem3,definition,
% 0.33/0.49 ( ubforcartprodlem3
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodpairin_type,type,
% 0.33/0.49 cartprodpairin: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodpairin,definition,
% 0.33/0.49 ( cartprodpairin
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodmempair1_type,type,
% 0.33/0.49 cartprodmempair1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodmempair1,definition,
% 0.33/0.49 ( cartprodmempair1
% 0.33/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.33/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.33/0.49 => ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ? [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 & ( Xu
% 0.33/0.49 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodmempair_type,type,
% 0.33/0.49 cartprodmempair: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodmempair,definition,
% 0.33/0.49 ( cartprodmempair
% 0.33/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.33/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.33/0.49 => ( iskpair @ Xu ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setunionE2_type,type,
% 0.33/0.49 setunionE2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setunionE2,definition,
% 0.33/0.49 ( setunionE2
% 0.33/0.49 = ( ! [A: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.33/0.49 => ? [X: $i] :
% 0.33/0.49 ( ( in @ X @ A )
% 0.33/0.49 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setunionsingleton1_type,type,
% 0.33/0.49 setunionsingleton1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setunionsingleton1,definition,
% 0.33/0.49 ( setunionsingleton1
% 0.33/0.49 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setunionsingleton2_type,type,
% 0.33/0.49 setunionsingleton2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setunionsingleton2,definition,
% 0.33/0.49 ( setunionsingleton2
% 0.33/0.49 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setunionsingleton_type,type,
% 0.33/0.49 setunionsingleton: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setunionsingleton,definition,
% 0.33/0.49 ( setunionsingleton
% 0.33/0.49 = ( ! [Xx: $i] :
% 0.33/0.49 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.33/0.49 = Xx ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(singleton_type,type,
% 0.33/0.49 singleton: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(singletonprop_type,type,
% 0.33/0.49 singletonprop: $o ).
% 0.33/0.49
% 0.33/0.49 thf(singletonprop,definition,
% 0.33/0.49 ( singletonprop
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ A )
% 0.33/0.49 => ( ( Xphi @ Xx )
% 0.33/0.49 => ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) )
% 0.33/0.49 => ( ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ( Xphi @ Xx ) )
% 0.33/0.49 => ( singleton
% 0.33/0.49 @ ( dsetconstr @ A
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ex1_type,type,
% 0.33/0.49 ex1: $i > ( $i > $o ) > $o ).
% 0.33/0.49
% 0.33/0.49 thf(ex1E1_type,type,
% 0.33/0.49 ex1E1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ex1E1,definition,
% 0.33/0.49 ( ex1E1
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ( ex1 @ A
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.49 => ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ex1I_type,type,
% 0.33/0.49 ex1I: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ex1I,definition,
% 0.33/0.49 ( ex1I
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( ( Xphi @ Xx )
% 0.33/0.49 => ( ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ A )
% 0.33/0.49 => ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xy = Xx ) ) )
% 0.33/0.49 => ( ex1 @ A
% 0.33/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ex1I2_type,type,
% 0.33/0.49 ex1I2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ex1I2,definition,
% 0.33/0.49 ( ex1I2
% 0.33/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ A )
% 0.33/0.49 => ( ( Xphi @ Xx )
% 0.33/0.49 => ( ( Xphi @ Xy )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) )
% 0.33/0.49 => ( ? [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 & ( Xphi @ Xx ) )
% 0.33/0.49 => ( ex1 @ A
% 0.33/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(singletonsuniq_type,type,
% 0.33/0.49 singletonsuniq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(singletonsuniq,definition,
% 0.33/0.49 ( singletonsuniq
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( ( setadjoin @ Xx @ emptyset )
% 0.33/0.49 = ( setadjoin @ Xy @ emptyset ) )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(atmost1p_type,type,
% 0.33/0.49 atmost1p: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(atleast2p_type,type,
% 0.33/0.49 atleast2p: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(atmost2p_type,type,
% 0.33/0.49 atmost2p: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairsetp_type,type,
% 0.33/0.49 upairsetp: $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjL1_type,type,
% 0.33/0.49 setukpairinjL1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjL1,definition,
% 0.33/0.49 ( setukpairinjL1
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.33/0.49 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.33/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(kfstsingleton_type,type,
% 0.33/0.49 kfstsingleton: $o ).
% 0.33/0.49
% 0.33/0.49 thf(kfstsingleton,definition,
% 0.33/0.49 ( kfstsingleton
% 0.33/0.49 = ( ! [Xu: $i] :
% 0.33/0.49 ( ( iskpair @ Xu )
% 0.33/0.49 => ( singleton
% 0.33/0.49 @ ( dsetconstr @ ( setunion @ Xu )
% 0.33/0.49 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(theprop_type,type,
% 0.33/0.49 theprop: $o ).
% 0.33/0.49
% 0.33/0.49 thf(theprop,definition,
% 0.33/0.49 ( theprop
% 0.33/0.49 = ( ! [X: $i] :
% 0.33/0.49 ( ( singleton @ X )
% 0.33/0.49 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(kfst_type,type,
% 0.33/0.49 kfst: $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(kfstpairEq_type,type,
% 0.33/0.49 kfstpairEq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(kfstpairEq,definition,
% 0.33/0.49 ( kfstpairEq
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.33/0.49 = Xx ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodfstin_type,type,
% 0.33/0.49 cartprodfstin: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodfstin,definition,
% 0.33/0.49 ( cartprodfstin
% 0.33/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.33/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.33/0.49 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjL2_type,type,
% 0.33/0.49 setukpairinjL2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjL2,definition,
% 0.33/0.49 ( setukpairinjL2
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.33/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.33/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.33/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjL_type,type,
% 0.33/0.49 setukpairinjL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjL,definition,
% 0.33/0.49 ( setukpairinjL
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.33/0.49 ( ( ( kpair @ Xx @ Xy )
% 0.33/0.49 = ( kpair @ Xz @ Xu ) )
% 0.33/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR11_type,type,
% 0.33/0.49 setukpairinjR11: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR11,definition,
% 0.33/0.49 ( setukpairinjR11
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx = Xy )
% 0.33/0.49 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.33/0.49 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR12_type,type,
% 0.33/0.49 setukpairinjR12: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR12,definition,
% 0.33/0.49 ( setukpairinjR12
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xx = Xy )
% 0.33/0.49 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.33/0.49 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR1_type,type,
% 0.33/0.49 setukpairinjR1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR1,definition,
% 0.33/0.49 ( setukpairinjR1
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.33/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.33/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.33/0.49 => ( ( Xz = Xu )
% 0.33/0.49 => ( Xy = Xu ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(upairequniteq_type,type,
% 0.33/0.49 upairequniteq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(upairequniteq,definition,
% 0.33/0.49 ( upairequniteq
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.33/0.49 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.33/0.49 = ( setadjoin @ Xz @ emptyset ) )
% 0.33/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR2_type,type,
% 0.33/0.49 setukpairinjR2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR2,definition,
% 0.33/0.49 ( setukpairinjR2
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.33/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.33/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.33/0.49 => ( Xy = Xu ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR_type,type,
% 0.33/0.49 setukpairinjR: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setukpairinjR,definition,
% 0.33/0.49 ( setukpairinjR
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.33/0.49 ( ( ( kpair @ Xx @ Xy )
% 0.33/0.49 = ( kpair @ Xz @ Xu ) )
% 0.33/0.49 => ( Xy = Xu ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ksndsingleton_type,type,
% 0.33/0.49 ksndsingleton: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ksndsingleton,definition,
% 0.33/0.49 ( ksndsingleton
% 0.33/0.49 = ( ! [Xu: $i] :
% 0.33/0.49 ( ( iskpair @ Xu )
% 0.33/0.49 => ( singleton
% 0.33/0.49 @ ( dsetconstr @ ( setunion @ Xu )
% 0.33/0.49 @ ^ [Xx: $i] :
% 0.33/0.49 ( Xu
% 0.33/0.49 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ksnd_type,type,
% 0.33/0.49 ksnd: $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(ksndpairEq_type,type,
% 0.33/0.49 ksndpairEq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ksndpairEq,definition,
% 0.33/0.49 ( ksndpairEq
% 0.33/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.33/0.49 = Xy ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(kpairsurjEq_type,type,
% 0.33/0.49 kpairsurjEq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(kpairsurjEq,definition,
% 0.33/0.49 ( kpairsurjEq
% 0.33/0.49 = ( ! [Xu: $i] :
% 0.33/0.49 ( ( iskpair @ Xu )
% 0.33/0.49 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.33/0.49 = Xu ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodsndin_type,type,
% 0.33/0.49 cartprodsndin: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodsndin,definition,
% 0.33/0.49 ( cartprodsndin
% 0.33/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.33/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.33/0.49 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodpairmemEL_type,type,
% 0.33/0.49 cartprodpairmemEL: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodpairmemEL,definition,
% 0.33/0.49 ( cartprodpairmemEL
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.33/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodpairmemER_type,type,
% 0.33/0.49 cartprodpairmemER: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodpairmemER,definition,
% 0.33/0.49 ( cartprodpairmemER
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.33/0.49 => ( in @ Xy @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodmempaircEq_type,type,
% 0.33/0.49 cartprodmempaircEq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodmempaircEq,definition,
% 0.33/0.49 ( cartprodmempaircEq
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( ( kpair @ Xx @ Xy )
% 0.33/0.49 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodfstpairEq_type,type,
% 0.33/0.49 cartprodfstpairEq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodfstpairEq,definition,
% 0.33/0.49 ( cartprodfstpairEq
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.33/0.49 = Xx ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodsndpairEq_type,type,
% 0.33/0.49 cartprodsndpairEq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodsndpairEq,definition,
% 0.33/0.49 ( cartprodsndpairEq
% 0.33/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.33/0.49 = Xy ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodpairsurjEq_type,type,
% 0.33/0.49 cartprodpairsurjEq: $o ).
% 0.33/0.49
% 0.33/0.49 thf(cartprodpairsurjEq,definition,
% 0.33/0.49 ( cartprodpairsurjEq
% 0.33/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.33/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.33/0.49 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.33/0.49 = Xu ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(breln_type,type,
% 0.33/0.49 breln: $i > $i > $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstr_type,type,
% 0.33/0.49 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrI_type,type,
% 0.33/0.49 dpsetconstrI: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrI,definition,
% 0.33/0.49 ( dpsetconstrI
% 0.33/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( ( Xphi @ Xx @ Xy )
% 0.33/0.49 => ( in @ ( kpair @ Xx @ Xy )
% 0.33/0.49 @ ( dpsetconstr @ A @ B
% 0.33/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrSub_type,type,
% 0.33/0.49 dpsetconstrSub: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrSub,definition,
% 0.33/0.49 ( dpsetconstrSub
% 0.33/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.33/0.49 ( subset
% 0.33/0.49 @ ( dpsetconstr @ A @ B
% 0.33/0.49 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.33/0.49 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(setOfPairsIsBReln_type,type,
% 0.33/0.49 setOfPairsIsBReln: $o ).
% 0.33/0.49
% 0.33/0.49 thf(setOfPairsIsBReln,definition,
% 0.33/0.49 ( setOfPairsIsBReln
% 0.33/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.33/0.49 ( breln @ A @ B
% 0.33/0.49 @ ( dpsetconstr @ A @ B
% 0.33/0.49 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrERa_type,type,
% 0.33/0.49 dpsetconstrERa: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrERa,definition,
% 0.33/0.49 ( dpsetconstrERa
% 0.33/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.33/0.49 @ ( dpsetconstr @ A @ B
% 0.33/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.33/0.49 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrEL1_type,type,
% 0.33/0.49 dpsetconstrEL1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrEL1,definition,
% 0.33/0.49 ( dpsetconstrEL1
% 0.33/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.33/0.49 @ ( dpsetconstr @ A @ B
% 0.33/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.33/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrEL2_type,type,
% 0.33/0.49 dpsetconstrEL2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrEL2,definition,
% 0.33/0.49 ( dpsetconstrEL2
% 0.33/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.33/0.49 @ ( dpsetconstr @ A @ B
% 0.33/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.33/0.49 => ( in @ Xy @ B ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrER_type,type,
% 0.33/0.49 dpsetconstrER: $o ).
% 0.33/0.49
% 0.33/0.49 thf(dpsetconstrER,definition,
% 0.33/0.49 ( dpsetconstrER
% 0.33/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.33/0.49 @ ( dpsetconstr @ A @ B
% 0.33/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.33/0.49 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(func_type,type,
% 0.33/0.49 func: $i > $i > $i > $o ).
% 0.33/0.49
% 0.33/0.49 thf(funcSet_type,type,
% 0.33/0.49 funcSet: $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(funcImageSingleton_type,type,
% 0.33/0.49 funcImageSingleton: $o ).
% 0.33/0.49
% 0.33/0.49 thf(funcImageSingleton,definition,
% 0.33/0.49 ( funcImageSingleton
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.49 ( ( func @ A @ B @ Xf )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( singleton
% 0.33/0.49 @ ( dsetconstr @ B
% 0.33/0.49 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(apProp_type,type,
% 0.33/0.49 apProp: $o ).
% 0.33/0.49
% 0.33/0.49 thf(apProp,definition,
% 0.33/0.49 ( apProp
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.49 ( ( func @ A @ B @ Xf )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in
% 0.33/0.49 @ ( setunion
% 0.33/0.49 @ ( dsetconstr @ B
% 0.33/0.49 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.33/0.49 @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ap_type,type,
% 0.33/0.49 ap: $i > $i > $i > $i > $i ).
% 0.33/0.49
% 0.33/0.49 thf(app_type,type,
% 0.33/0.49 app: $o ).
% 0.33/0.49
% 0.33/0.49 thf(app,definition,
% 0.33/0.49 ( app
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.49 ( ( func @ A @ B @ Xf )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(infuncsetfunc_type,type,
% 0.33/0.49 infuncsetfunc: $o ).
% 0.33/0.49
% 0.33/0.49 thf(infuncsetfunc,definition,
% 0.33/0.49 ( infuncsetfunc
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.33/0.49 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(ap2p_type,type,
% 0.33/0.49 ap2p: $o ).
% 0.33/0.49
% 0.33/0.49 thf(ap2p,definition,
% 0.33/0.49 ( ap2p
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(funcinfuncset_type,type,
% 0.33/0.49 funcinfuncset: $o ).
% 0.33/0.49
% 0.33/0.49 thf(funcinfuncset,definition,
% 0.33/0.49 ( funcinfuncset
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.49 ( ( func @ A @ B @ Xf )
% 0.33/0.49 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(lamProp_type,type,
% 0.33/0.49 lamProp: $o ).
% 0.33/0.49
% 0.33/0.49 thf(lamProp,definition,
% 0.33/0.49 ( lamProp
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.33/0.49 => ( func @ A @ B
% 0.33/0.49 @ ( dpsetconstr @ A @ B
% 0.33/0.49 @ ^ [Xx: $i,Xy: $i] :
% 0.33/0.49 ( ( Xf @ Xx )
% 0.33/0.49 = Xy ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(lam_type,type,
% 0.33/0.49 lam: $i > $i > ( $i > $i ) > $i ).
% 0.33/0.49
% 0.33/0.49 thf(lamp_type,type,
% 0.33/0.49 lamp: $o ).
% 0.33/0.49
% 0.33/0.49 thf(lamp,definition,
% 0.33/0.49 ( lamp
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.33/0.49 => ( func @ A @ B
% 0.33/0.49 @ ( lam @ A @ B
% 0.33/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(lam2p_type,type,
% 0.33/0.49 lam2p: $o ).
% 0.33/0.49
% 0.33/0.49 thf(lam2p,definition,
% 0.33/0.49 ( lam2p
% 0.33/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.33/0.49 => ( in
% 0.33/0.49 @ ( lam @ A @ B
% 0.33/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.33/0.49 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(brelnall1_type,type,
% 0.33/0.49 brelnall1: $o ).
% 0.33/0.49
% 0.33/0.49 thf(brelnall1,definition,
% 0.33/0.49 ( brelnall1
% 0.33/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.33/0.49 ( ( breln @ A @ B @ R )
% 0.33/0.49 => ! [Xphi: $i > $o] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.49 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.33/0.49 => ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ R )
% 0.33/0.49 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf(brelnall2_type,type,
% 0.33/0.49 brelnall2: $o ).
% 0.33/0.49
% 0.33/0.49 thf(brelnall2,definition,
% 0.33/0.49 ( brelnall2
% 0.33/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.33/0.49 ( ( breln @ A @ B @ R )
% 0.33/0.49 => ! [Xphi: $i > $o] :
% 0.33/0.49 ( ! [Xx: $i] :
% 0.33/0.49 ( ( in @ Xx @ A )
% 0.33/0.49 => ! [Xy: $i] :
% 0.33/0.49 ( ( in @ Xy @ B )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ R )
% 0.33/0.50 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(ex1E2_type,type,
% 0.33/0.50 ex1E2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(ex1E2,definition,
% 0.33/0.50 ( ex1E2
% 0.33/0.50 = ( ! [A: $i,Xphi: $i > $o] :
% 0.33/0.50 ( ( ex1 @ A
% 0.33/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( Xphi @ Xx )
% 0.33/0.50 => ( ( Xphi @ Xy )
% 0.33/0.50 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(funcGraphProp1_type,type,
% 0.33/0.50 funcGraphProp1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(funcGraphProp1,definition,
% 0.33/0.50 ( funcGraphProp1
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( func @ A @ B @ Xf )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(funcGraphProp3_type,type,
% 0.33/0.50 funcGraphProp3: $o ).
% 0.33/0.50
% 0.33/0.50 thf(funcGraphProp3,definition,
% 0.33/0.50 ( funcGraphProp3
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(funcGraphProp2_type,type,
% 0.33/0.50 funcGraphProp2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(funcGraphProp2,definition,
% 0.33/0.50 ( funcGraphProp2
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( func @ A @ B @ Xf )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ B )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.33/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.33/0.50 = Xy ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(funcextLem_type,type,
% 0.33/0.50 funcextLem: $o ).
% 0.33/0.50
% 0.33/0.50 thf(funcextLem,definition,
% 0.33/0.50 ( funcextLem
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( func @ A @ B @ Xf )
% 0.33/0.50 => ! [Xg: $i] :
% 0.33/0.50 ( ( func @ A @ B @ Xg )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.33/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ B )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(funcGraphProp4_type,type,
% 0.33/0.50 funcGraphProp4: $o ).
% 0.33/0.50
% 0.33/0.50 thf(funcGraphProp4,definition,
% 0.33/0.50 ( funcGraphProp4
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ B )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.33/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.33/0.50 = Xy ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(subbreln_type,type,
% 0.33/0.50 subbreln: $o ).
% 0.33/0.50
% 0.33/0.50 thf(subbreln,definition,
% 0.33/0.50 ( subbreln
% 0.33/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.33/0.50 ( ( breln @ A @ B @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln @ A @ B @ S )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ B )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.33/0.50 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(eqbreln_type,type,
% 0.33/0.50 eqbreln: $o ).
% 0.33/0.50
% 0.33/0.50 thf(eqbreln,definition,
% 0.33/0.50 ( eqbreln
% 0.33/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.33/0.50 ( ( breln @ A @ B @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln @ A @ B @ S )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ B )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ B )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.33/0.50 => ( R = S ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(funcext_type,type,
% 0.33/0.50 funcext: $o ).
% 0.33/0.50
% 0.33/0.50 thf(funcext,definition,
% 0.33/0.50 ( funcext
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( func @ A @ B @ Xf )
% 0.33/0.50 => ! [Xg: $i] :
% 0.33/0.50 ( ( func @ A @ B @ Xg )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.33/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.33/0.50 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(funcext2_type,type,
% 0.33/0.50 funcext2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(funcext2,definition,
% 0.33/0.50 ( funcext2
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.33/0.50 => ! [Xg: $i] :
% 0.33/0.50 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.33/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.33/0.50 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(ap2apEq1_type,type,
% 0.33/0.50 ap2apEq1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(ap2apEq1,definition,
% 0.33/0.50 ( ap2apEq1
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.33/0.50 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(ap2apEq2_type,type,
% 0.33/0.50 ap2apEq2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(ap2apEq2,definition,
% 0.33/0.50 ( ap2apEq2
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( func @ A @ B @ Xf )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.33/0.50 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(beta1_type,type,
% 0.33/0.50 beta1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(beta1,definition,
% 0.33/0.50 ( beta1
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( ap @ A @ B
% 0.33/0.50 @ ( lam @ A @ B
% 0.33/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.33/0.50 @ Xx )
% 0.33/0.50 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(eta1_type,type,
% 0.33/0.50 eta1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(eta1,definition,
% 0.33/0.50 ( eta1
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( func @ A @ B @ Xf )
% 0.33/0.50 => ( ( lam @ A @ B
% 0.33/0.50 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.33/0.50 = Xf ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(lam2lamEq_type,type,
% 0.33/0.50 lam2lamEq: $o ).
% 0.33/0.50
% 0.33/0.50 thf(lam2lamEq,definition,
% 0.33/0.50 ( lam2lamEq
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.33/0.50 => ( ( lam @ A @ B
% 0.33/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.33/0.50 = ( lam @ A @ B
% 0.33/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(beta2_type,type,
% 0.33/0.50 beta2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(beta2,definition,
% 0.33/0.50 ( beta2
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( ap @ A @ B
% 0.33/0.50 @ ( lam @ A @ B
% 0.33/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.33/0.50 @ Xx )
% 0.33/0.50 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(eta2_type,type,
% 0.33/0.50 eta2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(eta2,definition,
% 0.33/0.50 ( eta2
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.33/0.50 => ( ( lam @ A @ B
% 0.33/0.50 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.33/0.50 = Xf ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(iffalseProp1_type,type,
% 0.33/0.50 iffalseProp1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(iffalseProp1,definition,
% 0.33/0.50 ( iffalseProp1
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ~ Xphi
% 0.33/0.50 => ( in @ Xy
% 0.33/0.50 @ ( dsetconstr @ A
% 0.33/0.50 @ ^ [Xz: $i] :
% 0.33/0.50 ( ( Xphi
% 0.33/0.50 & ( Xz = Xx ) )
% 0.33/0.50 | ( ~ Xphi
% 0.33/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(iffalseProp2_type,type,
% 0.33/0.50 iffalseProp2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(iffalseProp2,definition,
% 0.33/0.50 ( iffalseProp2
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ~ Xphi
% 0.33/0.50 => ( ( dsetconstr @ A
% 0.33/0.50 @ ^ [Xz: $i] :
% 0.33/0.50 ( ( Xphi
% 0.33/0.50 & ( Xz = Xx ) )
% 0.33/0.50 | ( ~ Xphi
% 0.33/0.50 & ( Xz = Xy ) ) ) )
% 0.33/0.50 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(iftrueProp1_type,type,
% 0.33/0.50 iftrueProp1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(iftrueProp1,definition,
% 0.33/0.50 ( iftrueProp1
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( Xphi
% 0.33/0.50 => ( in @ Xx
% 0.33/0.50 @ ( dsetconstr @ A
% 0.33/0.50 @ ^ [Xz: $i] :
% 0.33/0.50 ( ( Xphi
% 0.33/0.50 & ( Xz = Xx ) )
% 0.33/0.50 | ( ~ Xphi
% 0.33/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(iftrueProp2_type,type,
% 0.33/0.50 iftrueProp2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(iftrueProp2,definition,
% 0.33/0.50 ( iftrueProp2
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( Xphi
% 0.33/0.50 => ( ( dsetconstr @ A
% 0.33/0.50 @ ^ [Xz: $i] :
% 0.33/0.50 ( ( Xphi
% 0.33/0.50 & ( Xz = Xx ) )
% 0.33/0.50 | ( ~ Xphi
% 0.33/0.50 & ( Xz = Xy ) ) ) )
% 0.33/0.50 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(ifSingleton_type,type,
% 0.33/0.50 ifSingleton: $o ).
% 0.33/0.50
% 0.33/0.50 thf(ifSingleton,definition,
% 0.33/0.50 ( ifSingleton
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( singleton
% 0.33/0.50 @ ( dsetconstr @ A
% 0.33/0.50 @ ^ [Xz: $i] :
% 0.33/0.50 ( ( Xphi
% 0.33/0.50 & ( Xz = Xx ) )
% 0.33/0.50 | ( ~ Xphi
% 0.33/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(if_type,type,
% 0.33/0.50 if: $i > $o > $i > $i > $i ).
% 0.33/0.50
% 0.33/0.50 thf(ifp_type,type,
% 0.33/0.50 ifp: $o ).
% 0.33/0.50
% 0.33/0.50 thf(ifp,definition,
% 0.33/0.50 ( ifp
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(theeq_type,type,
% 0.33/0.50 theeq: $o ).
% 0.33/0.50
% 0.33/0.50 thf(theeq,definition,
% 0.33/0.50 ( theeq
% 0.33/0.50 = ( ! [X: $i] :
% 0.33/0.50 ( ( singleton @ X )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ X )
% 0.33/0.50 => ( ( setunion @ X )
% 0.33/0.50 = Xx ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(iftrue_type,type,
% 0.33/0.50 iftrue: $o ).
% 0.33/0.50
% 0.33/0.50 thf(iftrue,definition,
% 0.33/0.50 ( iftrue
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( Xphi
% 0.33/0.50 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.33/0.50 = Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(iffalse_type,type,
% 0.33/0.50 iffalse: $o ).
% 0.33/0.50
% 0.33/0.50 thf(iffalse,definition,
% 0.33/0.50 ( iffalse
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ~ Xphi
% 0.33/0.50 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.33/0.50 = Xy ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(iftrueorfalse_type,type,
% 0.33/0.50 iftrueorfalse: $o ).
% 0.33/0.50
% 0.33/0.50 thf(iftrueorfalse,definition,
% 0.33/0.50 ( iftrueorfalse
% 0.33/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binintersectT_lem_type,type,
% 0.33/0.50 binintersectT_lem: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binintersectT_lem,definition,
% 0.33/0.50 ( binintersectT_lem
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binunionT_lem_type,type,
% 0.33/0.50 binunionT_lem: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binunionT_lem,definition,
% 0.33/0.50 ( binunionT_lem
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(powersetT_lem_type,type,
% 0.33/0.50 powersetT_lem: $o ).
% 0.33/0.50
% 0.33/0.50 thf(powersetT_lem,definition,
% 0.33/0.50 ( powersetT_lem
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(setminusT_lem_type,type,
% 0.33/0.50 setminusT_lem: $o ).
% 0.33/0.50
% 0.33/0.50 thf(setminusT_lem,definition,
% 0.33/0.50 ( setminusT_lem
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementT_lem_type,type,
% 0.33/0.50 complementT_lem: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementT_lem,definition,
% 0.33/0.50 ( complementT_lem
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(setextT_type,type,
% 0.33/0.50 setextT: $o ).
% 0.33/0.50
% 0.33/0.50 thf(setextT,definition,
% 0.33/0.50 ( setextT
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ X )
% 0.33/0.50 => ( in @ Xx @ Y ) ) )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ Y )
% 0.33/0.50 => ( in @ Xx @ X ) ) )
% 0.33/0.50 => ( X = Y ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(subsetTI_type,type,
% 0.33/0.50 subsetTI: $o ).
% 0.33/0.50
% 0.33/0.50 thf(subsetTI,definition,
% 0.33/0.50 ( subsetTI
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ X )
% 0.33/0.50 => ( in @ Xx @ Y ) ) )
% 0.33/0.50 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(powersetTI1_type,type,
% 0.33/0.50 powersetTI1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(powersetTI1,definition,
% 0.33/0.50 ( powersetTI1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ X )
% 0.33/0.50 => ( in @ Xx @ Y ) ) )
% 0.33/0.50 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(powersetTE1_type,type,
% 0.33/0.50 powersetTE1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(powersetTE1,definition,
% 0.33/0.50 ( powersetTE1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ X @ ( powerset @ Y ) )
% 0.33/0.50 => ( ( in @ Xx @ X )
% 0.33/0.50 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementTI1_type,type,
% 0.33/0.50 complementTI1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementTI1,definition,
% 0.33/0.50 ( complementTI1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ X )
% 0.33/0.50 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementTE1_type,type,
% 0.33/0.50 complementTE1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementTE1,definition,
% 0.33/0.50 ( complementTE1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.33/0.50 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binintersectTELcontra_type,type,
% 0.33/0.50 binintersectTELcontra: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binintersectTELcontra,definition,
% 0.33/0.50 ( binintersectTELcontra
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ~ ( in @ Xx @ X )
% 0.33/0.50 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binintersectTERcontra_type,type,
% 0.33/0.50 binintersectTERcontra: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binintersectTERcontra,definition,
% 0.33/0.50 ( binintersectTERcontra
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ~ ( in @ Xx @ Y )
% 0.33/0.50 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(contrasubsetT_type,type,
% 0.33/0.50 contrasubsetT: $o ).
% 0.33/0.50
% 0.33/0.50 thf(contrasubsetT,definition,
% 0.33/0.50 ( contrasubsetT
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.33/0.50 => ( ( in @ Xx @ Y )
% 0.33/0.50 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(contrasubsetT1_type,type,
% 0.33/0.50 contrasubsetT1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(contrasubsetT1,definition,
% 0.33/0.50 ( contrasubsetT1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( subset @ X @ Y )
% 0.33/0.50 => ( ~ ( in @ Xx @ Y )
% 0.33/0.50 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(contrasubsetT2_type,type,
% 0.33/0.50 contrasubsetT2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(contrasubsetT2,definition,
% 0.33/0.50 ( contrasubsetT2
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ X @ Y )
% 0.33/0.50 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(contrasubsetT3_type,type,
% 0.33/0.50 contrasubsetT3: $o ).
% 0.33/0.50
% 0.33/0.50 thf(contrasubsetT3,definition,
% 0.33/0.50 ( contrasubsetT3
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.33/0.50 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementI1_type,type,
% 0.33/0.50 doubleComplementI1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementI1,definition,
% 0.33/0.50 ( doubleComplementI1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ X )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementE1_type,type,
% 0.33/0.50 doubleComplementE1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementE1,definition,
% 0.33/0.50 ( doubleComplementE1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.33/0.50 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementSub1_type,type,
% 0.33/0.50 doubleComplementSub1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementSub1,definition,
% 0.33/0.50 ( doubleComplementSub1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementSub2_type,type,
% 0.33/0.50 doubleComplementSub2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementSub2,definition,
% 0.33/0.50 ( doubleComplementSub2
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementEq_type,type,
% 0.33/0.50 doubleComplementEq: $o ).
% 0.33/0.50
% 0.33/0.50 thf(doubleComplementEq,definition,
% 0.33/0.50 ( doubleComplementEq
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ( X
% 0.33/0.50 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementTnotintersectT_type,type,
% 0.33/0.50 complementTnotintersectT: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementTnotintersectT,definition,
% 0.33/0.50 ( complementTnotintersectT
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.33/0.50 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementImpComplementIntersect_type,type,
% 0.33/0.50 complementImpComplementIntersect: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementImpComplementIntersect,definition,
% 0.33/0.50 ( complementImpComplementIntersect
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementSubsetComplementIntersect_type,type,
% 0.33/0.50 complementSubsetComplementIntersect: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementSubsetComplementIntersect,definition,
% 0.33/0.50 ( complementSubsetComplementIntersect
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementInPowersetComplementIntersect_type,type,
% 0.33/0.50 complementInPowersetComplementIntersect: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementInPowersetComplementIntersect,definition,
% 0.33/0.50 ( complementInPowersetComplementIntersect
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(contraSubsetComplement_type,type,
% 0.33/0.50 contraSubsetComplement: $o ).
% 0.33/0.50
% 0.33/0.50 thf(contraSubsetComplement,definition,
% 0.33/0.50 ( contraSubsetComplement
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ Y )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementTcontraSubset_type,type,
% 0.33/0.50 complementTcontraSubset: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementTcontraSubset,definition,
% 0.33/0.50 ( complementTcontraSubset
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.33/0.50 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binunionTILcontra_type,type,
% 0.33/0.50 binunionTILcontra: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binunionTILcontra,definition,
% 0.33/0.50 ( binunionTILcontra
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.33/0.50 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binunionTIRcontra_type,type,
% 0.33/0.50 binunionTIRcontra: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binunionTIRcontra,definition,
% 0.33/0.50 ( binunionTIRcontra
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.33/0.50 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(inIntersectImpInUnion_type,type,
% 0.33/0.50 inIntersectImpInUnion: $o ).
% 0.33/0.50
% 0.33/0.50 thf(inIntersectImpInUnion,definition,
% 0.33/0.50 ( inIntersectImpInUnion
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.33/0.50 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(inIntersectImpInUnion2_type,type,
% 0.33/0.50 inIntersectImpInUnion2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(inIntersectImpInUnion2,definition,
% 0.33/0.50 ( inIntersectImpInUnion2
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.33/0.50 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(inIntersectImpInIntersectUnions_type,type,
% 0.33/0.50 inIntersectImpInIntersectUnions: $o ).
% 0.33/0.50
% 0.33/0.50 thf(inIntersectImpInIntersectUnions,definition,
% 0.33/0.50 ( inIntersectImpInIntersectUnions
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.33/0.50 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(intersectInPowersetIntersectUnions_type,type,
% 0.33/0.50 intersectInPowersetIntersectUnions: $o ).
% 0.33/0.50
% 0.33/0.50 thf(intersectInPowersetIntersectUnions,definition,
% 0.33/0.50 ( intersectInPowersetIntersectUnions
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(inComplementUnionImpNotIn1_type,type,
% 0.33/0.50 inComplementUnionImpNotIn1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(inComplementUnionImpNotIn1,definition,
% 0.33/0.50 ( inComplementUnionImpNotIn1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.33/0.50 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(inComplementUnionImpInComplement1_type,type,
% 0.33/0.50 inComplementUnionImpInComplement1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(inComplementUnionImpInComplement1,definition,
% 0.33/0.50 ( inComplementUnionImpInComplement1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binunionTE_type,type,
% 0.33/0.50 binunionTE: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binunionTE,definition,
% 0.33/0.50 ( binunionTE
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xphi: $o,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.33/0.50 => ( ( ( in @ Xx @ X )
% 0.33/0.50 => Xphi )
% 0.33/0.50 => ( ( ( in @ Xx @ Y )
% 0.33/0.50 => Xphi )
% 0.33/0.50 => Xphi ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binunionTEcontra_type,type,
% 0.33/0.50 binunionTEcontra: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binunionTEcontra,definition,
% 0.33/0.50 ( binunionTEcontra
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ~ ( in @ Xx @ X )
% 0.33/0.50 => ( ~ ( in @ Xx @ Y )
% 0.33/0.50 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2a1_type,type,
% 0.33/0.50 demorgan2a1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2a1,definition,
% 0.33/0.50 ( demorgan2a1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(complementUnionInPowersetComplement_type,type,
% 0.33/0.50 complementUnionInPowersetComplement: $o ).
% 0.33/0.50
% 0.33/0.50 thf(complementUnionInPowersetComplement,definition,
% 0.33/0.50 ( complementUnionInPowersetComplement
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2a2_type,type,
% 0.33/0.50 demorgan2a2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2a2,definition,
% 0.33/0.50 ( demorgan2a2
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan1a_type,type,
% 0.33/0.50 demorgan1a: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan1a,definition,
% 0.33/0.50 ( demorgan1a
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.33/0.50 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan1b_type,type,
% 0.33/0.50 demorgan1b: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan1b,definition,
% 0.33/0.50 ( demorgan1b
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan1_type,type,
% 0.33/0.50 demorgan1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan1,definition,
% 0.33/0.50 ( demorgan1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.33/0.50 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2a_type,type,
% 0.33/0.50 demorgan2a: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2a,definition,
% 0.33/0.50 ( demorgan2a
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.33/0.50 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2b2_type,type,
% 0.33/0.50 demorgan2b2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2b2,definition,
% 0.33/0.50 ( demorgan2b2
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.33/0.50 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2b_type,type,
% 0.33/0.50 demorgan2b: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2b,definition,
% 0.33/0.50 ( demorgan2b
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.33/0.50 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2_type,type,
% 0.33/0.50 demorgan2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(demorgan2,definition,
% 0.33/0.50 ( demorgan2
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.33/0.50 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule0_type,type,
% 0.33/0.50 woz13rule0: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule0,definition,
% 0.33/0.50 ( woz13rule0
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.33/0.50 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule1_type,type,
% 0.33/0.50 woz13rule1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule1,definition,
% 0.33/0.50 ( woz13rule1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ X @ Z )
% 0.33/0.50 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule2_type,type,
% 0.33/0.50 woz13rule2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule2,definition,
% 0.33/0.50 ( woz13rule2
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ Y @ Z )
% 0.33/0.50 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule3_type,type,
% 0.33/0.50 woz13rule3: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule3,definition,
% 0.33/0.50 ( woz13rule3
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ X @ Y )
% 0.33/0.50 => ( ( subset @ X @ Z )
% 0.33/0.50 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule4_type,type,
% 0.33/0.50 woz13rule4: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz13rule4,definition,
% 0.33/0.50 ( woz13rule4
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ! [W: $i] :
% 0.33/0.50 ( ( in @ W @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ X @ Z )
% 0.33/0.50 => ( ( subset @ Y @ W )
% 0.33/0.50 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_1_type,type,
% 0.33/0.50 woz1_1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_1,definition,
% 0.33/0.50 ( woz1_1
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_2_type,type,
% 0.33/0.50 woz1_2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_2,definition,
% 0.33/0.50 ( woz1_2
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ! [W: $i] :
% 0.33/0.50 ( ( in @ W @ ( powerset @ A ) )
% 0.33/0.50 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.33/0.50 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_3_type,type,
% 0.33/0.50 woz1_3: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_3,definition,
% 0.33/0.50 ( woz1_3
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ! [Z: $i] :
% 0.33/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_4_type,type,
% 0.33/0.50 woz1_4: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_4,definition,
% 0.33/0.50 ( woz1_4
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.33/0.50 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_5_type,type,
% 0.33/0.50 woz1_5: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz1_5,definition,
% 0.33/0.50 ( woz1_5
% 0.33/0.50 = ( ! [A: $i,X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.33/0.50 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1_type,type,
% 0.33/0.50 breln1: $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1all2_type,type,
% 0.33/0.50 breln1all2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1all2,definition,
% 0.33/0.50 ( breln1all2
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [Xphi: $i > $o] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ R )
% 0.33/0.50 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1Set_type,type,
% 0.33/0.50 breln1Set: $i > $i ).
% 0.33/0.50
% 0.33/0.50 thf(breln1SetBreln1_type,type,
% 0.33/0.50 breln1SetBreln1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1SetBreln1,definition,
% 0.33/0.50 ( breln1SetBreln1
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( in @ R @ ( breln1Set @ A ) )
% 0.33/0.50 => ( breln1 @ A @ R ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(transitive_type,type,
% 0.33/0.50 transitive: $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(antisymmetric_type,type,
% 0.33/0.50 antisymmetric: $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(reflexive_type,type,
% 0.33/0.50 reflexive: $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(refltransitive_type,type,
% 0.33/0.50 refltransitive: $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(refllinearorder_type,type,
% 0.33/0.50 refllinearorder: $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(reflwellordering_type,type,
% 0.33/0.50 reflwellordering: $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(choice2fnsingleton_type,type,
% 0.33/0.50 choice2fnsingleton: $o ).
% 0.33/0.50
% 0.33/0.50 thf(choice2fnsingleton,definition,
% 0.33/0.50 ( choice2fnsingleton
% 0.33/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ? [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ B )
% 0.33/0.50 & ( Xphi @ Xx @ Xy ) ) )
% 0.33/0.50 => ! [R: $i] :
% 0.33/0.50 ( ( in @ R @ ( breln1Set @ B ) )
% 0.33/0.50 => ( ( reflwellordering @ B @ R )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( singleton
% 0.33/0.50 @ ( dsetconstr @ B
% 0.33/0.50 @ ^ [Xy: $i] :
% 0.33/0.50 ( ( Xphi @ Xx @ Xy )
% 0.33/0.50 & ! [Xz: $i] :
% 0.33/0.50 ( ( in @ Xz @ B )
% 0.33/0.50 => ( ( Xphi @ Xx @ Xz )
% 0.33/0.50 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(setOfPairsIsBReln1_type,type,
% 0.33/0.50 setOfPairsIsBReln1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(setOfPairsIsBReln1,definition,
% 0.33/0.50 ( setOfPairsIsBReln1
% 0.33/0.50 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.33/0.50 ( breln1 @ A
% 0.33/0.50 @ ( dpsetconstr @ A @ A
% 0.33/0.50 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1all1_type,type,
% 0.33/0.50 breln1all1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1all1,definition,
% 0.33/0.50 ( breln1all1
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [Xphi: $i > $o] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ R )
% 0.33/0.50 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(subbreln1_type,type,
% 0.33/0.50 subbreln1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(subbreln1,definition,
% 0.33/0.50 ( subbreln1
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.33/0.50 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(eqbreln1_type,type,
% 0.33/0.50 eqbreln1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(eqbreln1,definition,
% 0.33/0.50 ( eqbreln1
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.33/0.50 => ( R = S ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1invset_type,type,
% 0.33/0.50 breln1invset: $i > $i > $i ).
% 0.33/0.50
% 0.33/0.50 thf(breln1invprop_type,type,
% 0.33/0.50 breln1invprop: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1invprop,definition,
% 0.33/0.50 ( breln1invprop
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1invI_type,type,
% 0.33/0.50 breln1invI: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1invI,definition,
% 0.33/0.50 ( breln1invI
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1invE_type,type,
% 0.33/0.50 breln1invE: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1invE,definition,
% 0.33/0.50 ( breln1invE
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compset_type,type,
% 0.33/0.50 breln1compset: $i > $i > $i > $i ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compprop_type,type,
% 0.33/0.50 breln1compprop: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compprop,definition,
% 0.33/0.50 ( breln1compprop
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compI_type,type,
% 0.33/0.50 breln1compI: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compI,definition,
% 0.33/0.50 ( breln1compI
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ! [Xz: $i] :
% 0.33/0.50 ( ( in @ Xz @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.33/0.50 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compE_type,type,
% 0.33/0.50 breln1compE: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compE,definition,
% 0.33/0.50 ( breln1compE
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.33/0.50 => ? [Xz: $i] :
% 0.33/0.50 ( ( in @ Xz @ A )
% 0.33/0.50 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.33/0.50 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compEex_type,type,
% 0.33/0.50 breln1compEex: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1compEex,definition,
% 0.33/0.50 ( breln1compEex
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.33/0.50 => ! [Xphi: $o] :
% 0.33/0.50 ( ! [Xz: $i] :
% 0.33/0.50 ( ( in @ Xz @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.33/0.50 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.33/0.50 => Xphi ) ) )
% 0.33/0.50 => Xphi ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionprop_type,type,
% 0.33/0.50 breln1unionprop: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionprop,definition,
% 0.33/0.50 ( breln1unionprop
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionIL_type,type,
% 0.33/0.50 breln1unionIL: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionIL,definition,
% 0.33/0.50 ( breln1unionIL
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionIR_type,type,
% 0.33/0.50 breln1unionIR: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionIR,definition,
% 0.33/0.50 ( breln1unionIR
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionI_type,type,
% 0.33/0.50 breln1unionI: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionI,definition,
% 0.33/0.50 ( breln1unionI
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.33/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionE_type,type,
% 0.33/0.50 breln1unionE: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionE,definition,
% 0.33/0.50 ( breln1unionE
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionEcases_type,type,
% 0.33/0.50 breln1unionEcases: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionEcases,definition,
% 0.33/0.50 ( breln1unionEcases
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.33/0.50 => ! [Xphi: $o] :
% 0.33/0.50 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.33/0.50 => Xphi )
% 0.33/0.50 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.33/0.50 => Xphi )
% 0.33/0.50 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionCommutes_type,type,
% 0.33/0.50 breln1unionCommutes: $o ).
% 0.33/0.50
% 0.33/0.50 thf(breln1unionCommutes,definition,
% 0.33/0.50 ( breln1unionCommutes
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ( ( binunion @ R @ S )
% 0.33/0.50 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz2Ex_type,type,
% 0.33/0.50 woz2Ex: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz2Ex,definition,
% 0.33/0.50 ( woz2Ex
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ( R
% 0.33/0.50 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz2W_type,type,
% 0.33/0.50 woz2W: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz2W,definition,
% 0.33/0.50 ( woz2W
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.33/0.50 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz2A_type,type,
% 0.33/0.50 woz2A: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz2A,definition,
% 0.33/0.50 ( woz2A
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [T: $i] :
% 0.33/0.50 ( ( breln1 @ A @ T )
% 0.33/0.50 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.33/0.50 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(woz2B_type,type,
% 0.33/0.50 woz2B: $o ).
% 0.33/0.50
% 0.33/0.50 thf(woz2B,definition,
% 0.33/0.50 ( woz2B
% 0.33/0.50 = ( ! [A: $i,R: $i] :
% 0.33/0.50 ( ( breln1 @ A @ R )
% 0.33/0.50 => ! [S: $i] :
% 0.33/0.50 ( ( breln1 @ A @ S )
% 0.33/0.50 => ! [T: $i] :
% 0.33/0.50 ( ( breln1 @ A @ T )
% 0.33/0.50 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.33/0.50 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(image1Ex_type,type,
% 0.33/0.50 image1Ex: $o ).
% 0.33/0.50
% 0.33/0.50 thf(image1Ex,definition,
% 0.33/0.50 ( image1Ex
% 0.33/0.50 = ( ! [A: $i,Xf: $i > $i] :
% 0.33/0.50 ? [B: $i] :
% 0.33/0.50 ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ B )
% 0.33/0.50 <=> ? [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 & ( Xx
% 0.33/0.50 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(image1Ex1_type,type,
% 0.33/0.50 image1Ex1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(image1Ex1,definition,
% 0.33/0.50 ( image1Ex1
% 0.33/0.50 = ( ! [A: $i,Xf: $i > $i] :
% 0.33/0.50 ( exu
% 0.33/0.50 @ ^ [B: $i] :
% 0.33/0.50 ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ B )
% 0.33/0.50 <=> ? [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 & ( Xx
% 0.33/0.50 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(image1_type,type,
% 0.33/0.50 image1: $i > ( $i > $i ) > $i ).
% 0.33/0.50
% 0.33/0.50 thf(image1Equiv_type,type,
% 0.33/0.50 image1Equiv: $o ).
% 0.33/0.50
% 0.33/0.50 thf(image1Equiv,definition,
% 0.33/0.50 ( image1Equiv
% 0.33/0.50 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx
% 0.33/0.50 @ ( image1 @ A
% 0.33/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.33/0.50 <=> ? [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 & ( Xx
% 0.33/0.50 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(image1E_type,type,
% 0.33/0.50 image1E: $o ).
% 0.33/0.50
% 0.33/0.50 thf(image1E,definition,
% 0.33/0.50 ( image1E
% 0.33/0.50 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.33/0.50 ( ( in @ Xx
% 0.33/0.50 @ ( image1 @ A
% 0.33/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.33/0.50 => ? [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 & ( Xx
% 0.33/0.50 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(image1I_type,type,
% 0.33/0.50 image1I: $o ).
% 0.33/0.50
% 0.33/0.50 thf(image1I,definition,
% 0.33/0.50 ( image1I
% 0.33/0.50 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.33/0.50 ( ? [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ A )
% 0.33/0.50 & ( Xx
% 0.33/0.50 = ( Xf @ Xy ) ) )
% 0.33/0.50 => ( in @ Xx
% 0.33/0.50 @ ( image1 @ A
% 0.33/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(injective_type,type,
% 0.33/0.50 injective: $i > $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(injFuncSet_type,type,
% 0.33/0.50 injFuncSet: $i > $i > $i ).
% 0.33/0.50
% 0.33/0.50 thf(injFuncInInjFuncSet_type,type,
% 0.33/0.50 injFuncInInjFuncSet: $o ).
% 0.33/0.50
% 0.33/0.50 thf(injFuncInInjFuncSet,definition,
% 0.33/0.50 ( injFuncInInjFuncSet
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.33/0.50 => ( ( injective @ A @ B @ Xf )
% 0.33/0.50 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(injFuncSetFuncIn_type,type,
% 0.33/0.50 injFuncSetFuncIn: $o ).
% 0.33/0.50
% 0.33/0.50 thf(injFuncSetFuncIn,definition,
% 0.33/0.50 ( injFuncSetFuncIn
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.33/0.50 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(injFuncSetFuncInj_type,type,
% 0.33/0.50 injFuncSetFuncInj: $o ).
% 0.33/0.50
% 0.33/0.50 thf(injFuncSetFuncInj,definition,
% 0.33/0.50 ( injFuncSetFuncInj
% 0.33/0.50 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.33/0.50 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(surjective_type,type,
% 0.33/0.50 surjective: $i > $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(surjFuncSet_type,type,
% 0.33/0.50 surjFuncSet: $i > $i > $i ).
% 0.33/0.50
% 0.33/0.50 thf(surjFuncSetFuncIn_type,type,
% 0.33/0.50 surjFuncSetFuncIn: $o ).
% 0.33/0.50
% 0.33/0.50 thf(surjFuncSetFuncIn,definition,
% 0.33/0.50 ( surjFuncSetFuncIn
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.33/0.50 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(surjFuncSetFuncSurj_type,type,
% 0.33/0.50 surjFuncSetFuncSurj: $o ).
% 0.33/0.50
% 0.33/0.50 thf(surjFuncSetFuncSurj,definition,
% 0.33/0.50 ( surjFuncSetFuncSurj
% 0.33/0.50 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.33/0.50 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(leftInvIsSurj_type,type,
% 0.33/0.50 leftInvIsSurj: $o ).
% 0.33/0.50
% 0.33/0.50 thf(leftInvIsSurj,definition,
% 0.33/0.50 ( leftInvIsSurj
% 0.33/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.33/0.50 => ! [Xg: $i] :
% 0.33/0.50 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.33/0.50 => ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.33/0.50 = Xx ) )
% 0.33/0.50 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(surjCantorThm_type,type,
% 0.33/0.50 surjCantorThm: $o ).
% 0.33/0.50
% 0.33/0.50 thf(surjCantorThm,definition,
% 0.33/0.50 ( surjCantorThm
% 0.33/0.50 = ( ! [A: $i,Xf: $i] :
% 0.33/0.50 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.33/0.50 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(foundation2_type,type,
% 0.33/0.50 foundation2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(foundation2,definition,
% 0.33/0.50 ( foundation2
% 0.33/0.50 = ( ! [A: $i] :
% 0.33/0.50 ( ( nonempty @ A )
% 0.33/0.50 => ? [X: $i] :
% 0.33/0.50 ( ( in @ X @ A )
% 0.33/0.50 & ( ( binintersect @ X @ A )
% 0.33/0.50 = emptyset ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(notinself_type,type,
% 0.33/0.50 notinself: $o ).
% 0.33/0.50
% 0.33/0.50 thf(notinself,definition,
% 0.33/0.50 ( notinself
% 0.33/0.50 = ( ! [A: $i] :
% 0.33/0.50 ~ ( in @ A @ A ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(notinself2_type,type,
% 0.33/0.50 notinself2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(notinself2,definition,
% 0.33/0.50 ( notinself2
% 0.33/0.50 = ( ! [A: $i,B: $i] :
% 0.33/0.50 ( ( in @ A @ B )
% 0.33/0.50 => ~ ( in @ B @ A ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(omegaS_type,type,
% 0.33/0.50 omegaS: $i > $i ).
% 0.33/0.50
% 0.33/0.50 thf(omegaSp_type,type,
% 0.33/0.50 omegaSp: $o ).
% 0.33/0.50
% 0.33/0.50 thf(omegaSp,definition,
% 0.33/0.50 ( omegaSp
% 0.33/0.50 = ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ omega )
% 0.33/0.50 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(omegaSclos_type,type,
% 0.33/0.50 omegaSclos: $o ).
% 0.33/0.50
% 0.33/0.50 thf(omegaSclos,definition,
% 0.33/0.50 ( omegaSclos
% 0.33/0.50 = ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ omega )
% 0.33/0.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(peano0notS_type,type,
% 0.33/0.50 peano0notS: $o ).
% 0.33/0.50
% 0.33/0.50 thf(peano0notS,definition,
% 0.33/0.50 ( peano0notS
% 0.33/0.50 = ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ omega )
% 0.33/0.50 => ( ( omegaS @ Xx )
% 0.33/0.50 != emptyset ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(peano3_type,type,
% 0.33/0.50 peano3: $i > $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(peano4_type,type,
% 0.33/0.50 peano4: $i > $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(peano5_type,type,
% 0.33/0.50 peano5: $i > $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(peanoSinj_type,type,
% 0.33/0.50 peanoSinj: $o ).
% 0.33/0.50
% 0.33/0.50 thf(peanoSinj,definition,
% 0.33/0.50 ( peanoSinj
% 0.33/0.50 = ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ omega )
% 0.33/0.50 => ! [Xy: $i] :
% 0.33/0.50 ( ( in @ Xy @ omega )
% 0.33/0.50 => ( ( ( omegaS @ Xx )
% 0.33/0.50 = ( omegaS @ Xy ) )
% 0.33/0.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(peano_type,type,
% 0.33/0.50 peano: $i > $i > $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(transitiveset_type,type,
% 0.33/0.50 transitiveset: $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(transitiveset,definition,
% 0.33/0.50 ( transitiveset
% 0.33/0.50 = ( ^ [A: $i] :
% 0.33/0.50 ! [X: $i] :
% 0.33/0.50 ( ( in @ X @ A )
% 0.33/0.50 => ( subset @ X @ A ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(transitivesetOp1_type,type,
% 0.33/0.50 transitivesetOp1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(transitivesetOp1,definition,
% 0.33/0.50 ( transitivesetOp1
% 0.33/0.50 = ( ! [X: $i] :
% 0.33/0.50 ( ( transitiveset @ X )
% 0.33/0.50 => ! [A: $i] :
% 0.33/0.50 ( ( in @ A @ X )
% 0.33/0.50 => ( subset @ A @ X ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(binintTransitive_type,type,
% 0.33/0.50 binintTransitive: $o ).
% 0.33/0.50
% 0.33/0.50 thf(binintTransitive,definition,
% 0.33/0.50 ( binintTransitive
% 0.33/0.50 = ( ! [X: $i] :
% 0.33/0.50 ( ( transitiveset @ X )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( transitiveset @ Y )
% 0.33/0.50 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(transitivesetOp2_type,type,
% 0.33/0.50 transitivesetOp2: $o ).
% 0.33/0.50
% 0.33/0.50 thf(transitivesetOp2,definition,
% 0.33/0.50 ( transitivesetOp2
% 0.33/0.50 = ( ! [X: $i] :
% 0.33/0.50 ( ( transitiveset @ X )
% 0.33/0.50 => ! [A: $i,Xx: $i] :
% 0.33/0.50 ( ( in @ A @ X )
% 0.33/0.50 => ( ( in @ Xx @ A )
% 0.33/0.50 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(setunionTransitive_type,type,
% 0.33/0.50 setunionTransitive: $o ).
% 0.33/0.50
% 0.33/0.50 thf(setunionTransitive,definition,
% 0.33/0.50 ( setunionTransitive
% 0.33/0.50 = ( ! [X: $i] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ X )
% 0.33/0.50 => ( transitiveset @ Xx ) )
% 0.33/0.50 => ( transitiveset @ ( setunion @ X ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(stricttotalorderedByIn_type,type,
% 0.33/0.50 stricttotalorderedByIn: $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(stricttotalorderedByIn,definition,
% 0.33/0.50 ( stricttotalorderedByIn
% 0.33/0.50 = ( ^ [A: $i] :
% 0.33/0.50 ( ! [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ A )
% 0.33/0.50 => ! [X: $i] :
% 0.33/0.50 ( ( in @ X @ A )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ A )
% 0.33/0.50 => ( ( ( in @ Xx @ X )
% 0.33/0.50 & ( in @ X @ Y ) )
% 0.33/0.50 => ( in @ Xx @ Y ) ) ) ) )
% 0.33/0.50 & ! [X: $i] :
% 0.33/0.50 ( ( in @ X @ A )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ A )
% 0.33/0.50 => ( ( X = Y )
% 0.33/0.50 | ( in @ X @ Y )
% 0.33/0.50 | ( in @ Y @ X ) ) ) )
% 0.33/0.50 & ! [X: $i] :
% 0.33/0.50 ( ( in @ X @ A )
% 0.33/0.50 => ~ ( in @ X @ X ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(wellorderedByIn_type,type,
% 0.33/0.50 wellorderedByIn: $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(wellorderedByIn,definition,
% 0.33/0.50 ( wellorderedByIn
% 0.33/0.50 = ( ^ [A: $i] :
% 0.33/0.50 ( ( stricttotalorderedByIn @ A )
% 0.33/0.50 & ! [X: $i] :
% 0.33/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.33/0.50 => ( ( nonempty @ X )
% 0.33/0.50 => ? [Xx: $i] :
% 0.33/0.50 ( ( in @ Xx @ X )
% 0.33/0.50 & ! [Y: $i] :
% 0.33/0.50 ( ( in @ Y @ X )
% 0.33/0.50 => ( ( Xx = Y )
% 0.33/0.50 | ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(ordinal_type,type,
% 0.33/0.50 ordinal: $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(ordinal,definition,
% 0.33/0.50 ( ordinal
% 0.33/0.50 = ( ^ [Xx: $i] :
% 0.33/0.50 ( ( transitiveset @ Xx )
% 0.33/0.50 & ( wellorderedByIn @ Xx ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(limitOrdinal_type,type,
% 0.33/0.50 limitOrdinal: $i > $o ).
% 0.33/0.50
% 0.33/0.50 thf(ordinalMinLem1_type,type,
% 0.33/0.50 ordinalMinLem1: $o ).
% 0.33/0.50
% 0.33/0.50 thf(ordinalMinLem1,definition,
% 0.33/0.50 ( ordinalMinLem1
% 0.33/0.50 = ( ! [X: $i] :
% 0.33/0.50 ( ( ordinal @ X )
% 0.33/0.50 => ! [Y: $i] :
% 0.33/0.50 ( ( ordinal @ Y )
% 0.33/0.50 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(ordinalTransSet_type,type,
% 0.33/0.50 ordinalTransSet: $o ).
% 0.33/0.50
% 0.33/0.50 thf(ordinalTransSet,definition,
% 0.33/0.50 ( ordinalTransSet
% 0.33/0.50 = ( ! [X: $i] :
% 0.33/0.50 ( ( ordinal @ X )
% 0.33/0.50 => ! [Xx: $i,A: $i] :
% 0.33/0.50 ( ( in @ A @ X )
% 0.33/0.50 => ( ( in @ Xx @ A )
% 0.33/0.50 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 thf(ordinalTransSet1,conjecture,
% 0.33/0.50 ( setextAx
% 0.33/0.50 => ( emptysetAx
% 0.33/0.50 => ( setadjoinAx
% 0.33/0.50 => ( powersetAx
% 0.33/0.50 => ( setunionAx
% 0.33/0.50 => ( omega0Ax
% 0.33/0.50 => ( omegaSAx
% 0.33/0.50 => ( omegaIndAx
% 0.33/0.50 => ( replAx
% 0.33/0.50 => ( foundationAx
% 0.33/0.50 => ( wellorderingAx
% 0.33/0.50 => ( descrp
% 0.33/0.50 => ( dsetconstrI
% 0.33/0.50 => ( dsetconstrEL
% 0.33/0.50 => ( dsetconstrER
% 0.33/0.50 => ( exuE1
% 0.33/0.50 => ( prop2setE
% 0.33/0.50 => ( emptysetE
% 0.33/0.50 => ( emptysetimpfalse
% 0.33/0.50 => ( notinemptyset
% 0.33/0.50 => ( exuE3e
% 0.33/0.50 => ( setext
% 0.33/0.50 => ( emptyI
% 0.33/0.50 => ( noeltsimpempty
% 0.33/0.50 => ( setbeta
% 0.33/0.50 => ( nonemptyE1
% 0.33/0.50 => ( nonemptyI
% 0.33/0.50 => ( nonemptyI1
% 0.33/0.50 => ( setadjoinIL
% 0.33/0.50 => ( emptyinunitempty
% 0.33/0.50 => ( setadjoinIR
% 0.33/0.50 => ( setadjoinE
% 0.33/0.50 => ( setadjoinOr
% 0.33/0.50 => ( setoftrueEq
% 0.33/0.50 => ( powersetI
% 0.33/0.50 => ( emptyinPowerset
% 0.33/0.50 => ( emptyInPowerset
% 0.33/0.50 => ( powersetE
% 0.33/0.50 => ( setunionI
% 0.33/0.50 => ( setunionE
% 0.33/0.50 => ( subPowSU
% 0.33/0.50 => ( exuE2
% 0.33/0.50 => ( nonemptyImpWitness
% 0.33/0.50 => ( uniqinunit
% 0.33/0.50 => ( notinsingleton
% 0.33/0.50 => ( eqinunit
% 0.33/0.50 => ( singletonsswitch
% 0.33/0.50 => ( upairsetE
% 0.33/0.50 => ( upairsetIL
% 0.33/0.50 => ( upairsetIR
% 0.33/0.50 => ( emptyE1
% 0.33/0.50 => ( vacuousDall
% 0.33/0.50 => ( quantDeMorgan1
% 0.33/0.50 => ( quantDeMorgan2
% 0.33/0.50 => ( quantDeMorgan3
% 0.33/0.50 => ( quantDeMorgan4
% 0.33/0.50 => ( prop2setI
% 0.33/0.50 => ( prop2set2propI
% 0.33/0.50 => ( notdexE
% 0.33/0.50 => ( notdallE
% 0.33/0.50 => ( exuI1
% 0.33/0.50 => ( exuI3
% 0.33/0.50 => ( exuI2
% 0.33/0.50 => ( inCongP
% 0.33/0.50 => ( in__Cong
% 0.33/0.50 => ( exuE3u
% 0.33/0.50 => ( exu__Cong
% 0.33/0.50 => ( emptyset__Cong
% 0.33/0.50 => ( setadjoin__Cong
% 0.33/0.50 => ( powerset__Cong
% 0.33/0.50 => ( setunion__Cong
% 0.33/0.50 => ( omega__Cong
% 0.33/0.50 => ( exuEu
% 0.33/0.50 => ( descr__Cong
% 0.33/0.50 => ( dsetconstr__Cong
% 0.33/0.50 => ( subsetI1
% 0.33/0.50 => ( eqimpsubset2
% 0.33/0.50 => ( eqimpsubset1
% 0.33/0.50 => ( subsetI2
% 0.33/0.50 => ( emptysetsubset
% 0.33/0.50 => ( subsetE
% 0.33/0.50 => ( subsetE2
% 0.33/0.50 => ( notsubsetI
% 0.33/0.50 => ( notequalI1
% 0.33/0.50 => ( notequalI2
% 0.33/0.50 => ( subsetRefl
% 0.33/0.50 => ( subsetTrans
% 0.33/0.50 => ( setadjoinSub
% 0.33/0.50 => ( setadjoinSub2
% 0.33/0.50 => ( subset2powerset
% 0.33/0.50 => ( setextsub
% 0.33/0.50 => ( subsetemptysetimpeq
% 0.33/0.50 => ( powersetI1
% 0.33/0.50 => ( powersetE1
% 0.33/0.50 => ( inPowerset
% 0.33/0.50 => ( powersetsubset
% 0.33/0.50 => ( sepInPowerset
% 0.33/0.50 => ( sepSubset
% 0.33/0.50 => ( binunionIL
% 0.33/0.50 => ( upairset2IR
% 0.33/0.50 => ( binunionIR
% 0.33/0.50 => ( binunionEcases
% 0.33/0.50 => ( binunionE
% 0.33/0.50 => ( binunionLsub
% 0.33/0.50 => ( binunionRsub
% 0.33/0.50 => ( binintersectI
% 0.33/0.50 => ( binintersectSubset5
% 0.33/0.50 => ( binintersectEL
% 0.33/0.50 => ( binintersectLsub
% 0.33/0.50 => ( binintersectSubset2
% 0.33/0.50 => ( binintersectSubset3
% 0.33/0.50 => ( binintersectER
% 0.33/0.50 => ( disjointsetsI1
% 0.33/0.50 => ( binintersectRsub
% 0.33/0.50 => ( binintersectSubset4
% 0.33/0.50 => ( binintersectSubset1
% 0.33/0.50 => ( bs114d
% 0.33/0.50 => ( setminusI
% 0.33/0.50 => ( setminusEL
% 0.33/0.50 => ( setminusER
% 0.33/0.50 => ( setminusSubset2
% 0.33/0.50 => ( setminusERneg
% 0.33/0.50 => ( setminusELneg
% 0.33/0.50 => ( setminusILneg
% 0.33/0.50 => ( setminusIRneg
% 0.33/0.50 => ( setminusLsub
% 0.33/0.50 => ( setminusSubset1
% 0.33/0.50 => ( symdiffE
% 0.33/0.50 => ( symdiffI1
% 0.33/0.50 => ( symdiffI2
% 0.33/0.50 => ( symdiffIneg1
% 0.33/0.50 => ( symdiffIneg2
% 0.33/0.50 => ( secondinupair
% 0.33/0.50 => ( setukpairIL
% 0.33/0.50 => ( setukpairIR
% 0.33/0.50 => ( kpairiskpair
% 0.33/0.50 => ( kpairp
% 0.33/0.50 => ( singletonsubset
% 0.33/0.50 => ( singletoninpowerset
% 0.33/0.50 => ( singletoninpowunion
% 0.33/0.50 => ( upairset2E
% 0.33/0.50 => ( upairsubunion
% 0.33/0.50 => ( upairinpowunion
% 0.33/0.50 => ( ubforcartprodlem1
% 0.33/0.50 => ( ubforcartprodlem2
% 0.33/0.50 => ( ubforcartprodlem3
% 0.33/0.50 => ( cartprodpairin
% 0.33/0.50 => ( cartprodmempair1
% 0.33/0.50 => ( cartprodmempair
% 0.33/0.50 => ( setunionE2
% 0.33/0.50 => ( setunionsingleton1
% 0.33/0.50 => ( setunionsingleton2
% 0.33/0.50 => ( setunionsingleton
% 0.33/0.50 => ( singletonprop
% 0.33/0.50 => ( ex1E1
% 0.33/0.50 => ( ex1I
% 0.33/0.50 => ( ex1I2
% 0.33/0.50 => ( singletonsuniq
% 0.33/0.50 => ( setukpairinjL1
% 0.33/0.50 => ( kfstsingleton
% 0.33/0.50 => ( theprop
% 0.33/0.50 => ( kfstpairEq
% 0.33/0.50 => ( cartprodfstin
% 0.33/0.50 => ( setukpairinjL2
% 0.33/0.50 => ( setukpairinjL
% 0.33/0.50 => ( setukpairinjR11
% 0.33/0.50 => ( setukpairinjR12
% 0.33/0.50 => ( setukpairinjR1
% 0.33/0.50 => ( upairequniteq
% 0.33/0.50 => ( setukpairinjR2
% 0.33/0.50 => ( setukpairinjR
% 0.33/0.50 => ( ksndsingleton
% 0.33/0.50 => ( ksndpairEq
% 0.33/0.50 => ( kpairsurjEq
% 0.33/0.50 => ( cartprodsndin
% 0.33/0.50 => ( cartprodpairmemEL
% 0.33/0.50 => ( cartprodpairmemER
% 0.33/0.50 => ( cartprodmempaircEq
% 0.33/0.50 => ( cartprodfstpairEq
% 0.33/0.50 => ( cartprodsndpairEq
% 0.33/0.50 => ( cartprodpairsurjEq
% 0.33/0.50 => ( dpsetconstrI
% 0.33/0.50 => ( dpsetconstrSub
% 0.33/0.50 => ( setOfPairsIsBReln
% 0.33/0.50 => ( dpsetconstrERa
% 0.33/0.50 => ( dpsetconstrEL1
% 0.33/0.50 => ( dpsetconstrEL2
% 0.33/0.50 => ( dpsetconstrER
% 0.33/0.50 => ( funcImageSingleton
% 0.33/0.50 => ( apProp
% 0.33/0.50 => ( app
% 0.33/0.50 => ( infuncsetfunc
% 0.33/0.50 => ( ap2p
% 0.33/0.50 => ( funcinfuncset
% 0.33/0.50 => ( lamProp
% 0.33/0.50 => ( lamp
% 0.33/0.50 => ( lam2p
% 0.33/0.50 => ( brelnall1
% 0.33/0.50 => ( brelnall2
% 0.33/0.50 => ( ex1E2
% 0.33/0.50 => ( funcGraphProp1
% 0.33/0.50 => ( funcGraphProp3
% 0.33/0.50 => ( funcGraphProp2
% 0.33/0.50 => ( funcextLem
% 0.33/0.50 => ( funcGraphProp4
% 0.33/0.50 => ( subbreln
% 0.33/0.50 => ( eqbreln
% 0.33/0.50 => ( funcext
% 0.33/0.50 => ( funcext2
% 0.33/0.50 => ( ap2apEq1
% 0.33/0.50 => ( ap2apEq2
% 0.33/0.50 => ( beta1
% 0.33/0.50 => ( eta1
% 0.33/0.50 => ( lam2lamEq
% 0.33/0.50 => ( beta2
% 0.33/0.50 => ( eta2
% 0.33/0.50 => ( iffalseProp1
% 0.33/0.50 => ( iffalseProp2
% 0.33/0.50 => ( iftrueProp1
% 0.33/0.50 => ( iftrueProp2
% 0.33/0.50 => ( ifSingleton
% 0.33/0.50 => ( ifp
% 0.33/0.50 => ( theeq
% 0.33/0.50 => ( iftrue
% 0.33/0.50 => ( iffalse
% 0.33/0.50 => ( iftrueorfalse
% 0.33/0.50 => ( binintersectT_lem
% 0.33/0.50 => ( binunionT_lem
% 0.33/0.50 => ( powersetT_lem
% 0.33/0.50 => ( setminusT_lem
% 0.33/0.50 => ( complementT_lem
% 0.33/0.50 => ( setextT
% 0.33/0.50 => ( subsetTI
% 0.33/0.50 => ( powersetTI1
% 0.33/0.50 => ( powersetTE1
% 0.33/0.50 => ( complementTI1
% 0.33/0.50 => ( complementTE1
% 0.33/0.50 => ( binintersectTELcontra
% 0.33/0.50 => ( binintersectTERcontra
% 0.33/0.50 => ( contrasubsetT
% 0.33/0.50 => ( contrasubsetT1
% 0.33/0.50 => ( contrasubsetT2
% 0.33/0.50 => ( contrasubsetT3
% 0.33/0.50 => ( doubleComplementI1
% 0.33/0.50 => ( doubleComplementE1
% 0.33/0.50 => ( doubleComplementSub1
% 0.33/0.50 => ( doubleComplementSub2
% 0.33/0.50 => ( doubleComplementEq
% 0.33/0.50 => ( complementTnotintersectT
% 0.33/0.50 => ( complementImpComplementIntersect
% 0.33/0.50 => ( complementSubsetComplementIntersect
% 0.33/0.50 => ( complementInPowersetComplementIntersect
% 0.33/0.50 => ( contraSubsetComplement
% 0.33/0.50 => ( complementTcontraSubset
% 0.33/0.50 => ( binunionTILcontra
% 0.33/0.50 => ( binunionTIRcontra
% 0.33/0.50 => ( inIntersectImpInUnion
% 0.33/0.50 => ( inIntersectImpInUnion2
% 0.33/0.50 => ( inIntersectImpInIntersectUnions
% 0.33/0.50 => ( intersectInPowersetIntersectUnions
% 0.33/0.50 => ( inComplementUnionImpNotIn1
% 0.33/0.50 => ( inComplementUnionImpInComplement1
% 0.33/0.50 => ( binunionTE
% 0.33/0.50 => ( binunionTEcontra
% 0.33/0.50 => ( demorgan2a1
% 0.33/0.50 => ( complementUnionInPowersetComplement
% 0.33/0.50 => ( demorgan2a2
% 0.33/0.50 => ( demorgan1a
% 0.33/0.50 => ( demorgan1b
% 0.33/0.50 => ( demorgan1
% 0.33/0.50 => ( demorgan2a
% 0.33/0.50 => ( demorgan2b2
% 0.33/0.50 => ( demorgan2b
% 0.33/0.50 => ( demorgan2
% 0.33/0.50 => ( woz13rule0
% 0.33/0.50 => ( woz13rule1
% 0.33/0.50 => ( woz13rule2
% 0.33/0.50 => ( woz13rule3
% 0.33/0.50 => ( woz13rule4
% 0.33/0.50 => ( woz1_1
% 0.33/0.50 => ( woz1_2
% 0.33/0.50 => ( woz1_3
% 0.33/0.50 => ( woz1_4
% 0.33/0.50 => ( woz1_5
% 0.33/0.50 => ( breln1all2
% 0.33/0.50 => ( breln1SetBreln1
% 0.33/0.50 => ( choice2fnsingleton
% 0.33/0.50 => ( setOfPairsIsBReln1
% 0.33/0.50 => ( breln1all1
% 0.33/0.50 => ( subbreln1
% 0.33/0.50 => ( eqbreln1
% 0.33/0.50 => ( breln1invprop
% 0.33/0.50 => ( breln1invI
% 0.33/0.50 => ( breln1invE
% 0.33/0.50 => ( breln1compprop
% 0.33/0.50 => ( breln1compI
% 0.33/0.50 => ( breln1compE
% 0.33/0.50 => ( breln1compEex
% 0.33/0.50 => ( breln1unionprop
% 0.33/0.50 => ( breln1unionIL
% 0.33/0.50 => ( breln1unionIR
% 0.33/0.50 => ( breln1unionI
% 0.33/0.50 => ( breln1unionE
% 0.33/0.50 => ( breln1unionEcases
% 0.33/0.50 => ( breln1unionCommutes
% 0.33/0.50 => ( woz2Ex
% 0.33/0.50 => ( woz2W
% 0.33/0.50 => ( woz2A
% 0.33/0.50 => ( woz2B
% 0.33/0.50 => ( image1Ex
% 0.33/0.50 => ( image1Ex1
% 0.33/0.50 => ( image1Equiv
% 0.33/0.50 => ( image1E
% 0.33/0.50 => ( image1I
% 0.33/0.50 => ( injFuncInInjFuncSet
% 0.33/0.50 => ( injFuncSetFuncIn
% 0.33/0.50 => ( injFuncSetFuncInj
% 0.33/0.50 => ( surjFuncSetFuncIn
% 0.33/0.50 => ( surjFuncSetFuncSurj
% 0.33/0.50 => ( leftInvIsSurj
% 0.33/0.50 => ( surjCantorThm
% 0.33/0.50 => ( foundation2
% 0.33/0.50 => ( notinself
% 0.33/0.50 => ( notinself2
% 0.33/0.50 => ( omegaSp
% 0.33/0.50 => ( omegaSclos
% 0.33/0.50 => ( peano0notS
% 0.33/0.50 => ( peanoSinj
% 0.33/0.50 => ( transitivesetOp1
% 0.33/0.50 => ( binintTransitive
% 0.33/0.50 => ( transitivesetOp2
% 0.33/0.50 => ( setunionTransitive
% 0.33/0.50 => ( ordinalMinLem1
% 0.33/0.50 => ( ordinalTransSet
% 0.33/0.50 => ! [X: $i] :
% 0.33/0.50 ( ( ordinal @ X )
% 0.33/0.50 => ! [A: $i] :
% 0.33/0.50 ( ( in @ A @ X )
% 0.33/0.50 => ( subset @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.50
% 0.33/0.50 %------------------------------------------------------------------------------
% 0.33/0.50 ------------------------
% 3.56/1.85 ---- Embedded in THF ---
% 3.56/1.85 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.56/1.85 %%% Generated on Tue Feb 24 12:36:04 EST 2026
% 3.56/1.85 %%% using '$$dhol' embedding, version 1.3.0.
% 3.56/1.85 %%% Logic specification used:
% 3.56/1.85 %%% thf(spec, logic, $$dhol).
% 3.56/1.85
% 3.56/1.85 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.PG6uUBnDB9/DTF2DTF9905.p
% See solution above
% 3.56/1.85 ------------------------
% 3.71/1.93 ---- Cleaned THF ---
% 3.71/1.93 thf(in_type,type,
% 3.71/1.93 in: $i > $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(exu_type,type,
% 3.71/1.93 exu: ( $i > $o ) > $o ).
% 3.71/1.93
% 3.71/1.93 thf(exu,definition,
% 3.71/1.93 ( exu
% 3.71/1.93 = ( ^ [Xphi: $i > $o] :
% 3.71/1.93 ? [Xx: $i] :
% 3.71/1.93 ( ( Xphi @ Xx )
% 3.71/1.93 & ! [Xy: $i] :
% 3.71/1.93 ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setextAx_type,type,
% 3.71/1.93 setextAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setextAx,definition,
% 3.71/1.93 ( setextAx
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 <=> ( in @ Xx @ B ) )
% 3.71/1.93 => ( A = B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptyset_type,type,
% 3.71/1.93 emptyset: $i ).
% 3.71/1.93
% 3.71/1.93 thf(emptysetAx_type,type,
% 3.71/1.93 emptysetAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptysetAx,definition,
% 3.71/1.93 ( emptysetAx
% 3.71/1.93 = ( ! [Xx: $i] :
% 3.71/1.93 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoin_type,type,
% 3.71/1.93 setadjoin: $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinAx_type,type,
% 3.71/1.93 setadjoinAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinAx,definition,
% 3.71/1.93 ( setadjoinAx
% 3.71/1.93 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.71/1.93 <=> ( ( Xy = Xx )
% 3.71/1.93 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powerset_type,type,
% 3.71/1.93 powerset: $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(powersetAx_type,type,
% 3.71/1.93 powersetAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetAx,definition,
% 3.71/1.93 ( powersetAx
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( in @ B @ ( powerset @ A ) )
% 3.71/1.93 <=> ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setunion_type,type,
% 3.71/1.93 setunion: $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(setunionAx_type,type,
% 3.71/1.93 setunionAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setunionAx,definition,
% 3.71/1.93 ( setunionAx
% 3.71/1.93 = ( ! [A: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( setunion @ A ) )
% 3.71/1.93 <=> ? [B: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 & ( in @ B @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(omega_type,type,
% 3.71/1.93 omega: $i ).
% 3.71/1.93
% 3.71/1.93 thf(omega0Ax_type,type,
% 3.71/1.93 omega0Ax: $o ).
% 3.71/1.93
% 3.71/1.93 thf(omega0Ax,definition,
% 3.71/1.93 ( omega0Ax
% 3.71/1.93 = ( in @ emptyset @ omega ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(omegaSAx_type,type,
% 3.71/1.93 omegaSAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(omegaSAx,definition,
% 3.71/1.93 ( omegaSAx
% 3.71/1.93 = ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ omega )
% 3.71/1.93 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(omegaIndAx_type,type,
% 3.71/1.93 omegaIndAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(omegaIndAx,definition,
% 3.71/1.93 ( omegaIndAx
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ( ( in @ emptyset @ A )
% 3.71/1.93 & ! [Xx: $i] :
% 3.71/1.93 ( ( ( in @ Xx @ omega )
% 3.71/1.93 & ( in @ Xx @ A ) )
% 3.71/1.93 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ omega )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(replAx_type,type,
% 3.71/1.93 replAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(replAx,definition,
% 3.71/1.93 ( replAx
% 3.71/1.93 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( exu
% 3.71/1.93 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.71/1.93 => ? [B: $i] :
% 3.71/1.93 ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 <=> ? [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(foundationAx_type,type,
% 3.71/1.93 foundationAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(foundationAx,definition,
% 3.71/1.93 ( foundationAx
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.71/1.93 => ? [B: $i] :
% 3.71/1.93 ( ( in @ B @ A )
% 3.71/1.93 & ~ ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(wellorderingAx_type,type,
% 3.71/1.93 wellorderingAx: $o ).
% 3.71/1.93
% 3.71/1.93 thf(wellorderingAx,definition,
% 3.71/1.93 ( wellorderingAx
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ? [B: $i] :
% 3.71/1.93 ( ! [C: $i] :
% 3.71/1.93 ( ( in @ C @ B )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ C )
% 3.71/1.93 => ( in @ Xx @ A ) ) )
% 3.71/1.93 & ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( ( in @ Xx @ A )
% 3.71/1.93 & ( in @ Xy @ A ) )
% 3.71/1.93 => ( ! [C: $i] :
% 3.71/1.93 ( ( in @ C @ B )
% 3.71/1.93 => ( ( in @ Xx @ C )
% 3.71/1.93 <=> ( in @ Xy @ C ) ) )
% 3.71/1.93 => ( Xx = Xy ) ) )
% 3.71/1.93 & ! [C: $i,D: $i] :
% 3.71/1.93 ( ( ( in @ C @ B )
% 3.71/1.93 & ( in @ D @ B ) )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ C )
% 3.71/1.93 => ( in @ Xx @ D ) )
% 3.71/1.93 | ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ D )
% 3.71/1.93 => ( in @ Xx @ C ) ) ) )
% 3.71/1.93 & ! [C: $i] :
% 3.71/1.93 ( ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ C )
% 3.71/1.93 => ( in @ Xx @ A ) )
% 3.71/1.93 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.71/1.93 => ? [D: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ D @ B )
% 3.71/1.93 & ( in @ Xx @ C )
% 3.71/1.93 & ~ ? [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ D )
% 3.71/1.93 & ( in @ Xy @ C ) )
% 3.71/1.93 & ! [E: $i] :
% 3.71/1.93 ( ( in @ E @ B )
% 3.71/1.93 => ( ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ E )
% 3.71/1.93 => ( in @ Xy @ D ) )
% 3.71/1.93 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(descr_type,type,
% 3.71/1.93 descr: ( $i > $o ) > $i ).
% 3.71/1.93
% 3.71/1.93 thf(descrp_type,type,
% 3.71/1.93 descrp: $o ).
% 3.71/1.93
% 3.71/1.93 thf(descrp,definition,
% 3.71/1.93 ( descrp
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ( Xphi
% 3.71/1.93 @ ( descr
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstr_type,type,
% 3.71/1.93 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstrI_type,type,
% 3.71/1.93 dsetconstrI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstrI,definition,
% 3.71/1.93 ( dsetconstrI
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 => ( in @ Xx
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstrEL_type,type,
% 3.71/1.93 dsetconstrEL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstrEL,definition,
% 3.71/1.93 ( dsetconstrEL
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstrER_type,type,
% 3.71/1.93 dsetconstrER: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstrER,definition,
% 3.71/1.93 ( dsetconstrER
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.71/1.93 => ( Xphi @ Xx ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exuE1_type,type,
% 3.71/1.93 exuE1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exuE1,definition,
% 3.71/1.93 ( exuE1
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ? [Xx: $i] :
% 3.71/1.93 ( ( Xphi @ Xx )
% 3.71/1.93 & ! [Xy: $i] :
% 3.71/1.93 ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(prop2set_type,type,
% 3.71/1.93 prop2set: $o > $i ).
% 3.71/1.93
% 3.71/1.93 thf(prop2setE_type,type,
% 3.71/1.93 prop2setE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(prop2setE,definition,
% 3.71/1.93 ( prop2setE
% 3.71/1.93 = ( ! [Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.71/1.93 => Xphi ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptysetE_type,type,
% 3.71/1.93 emptysetE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptysetE,definition,
% 3.71/1.93 ( emptysetE
% 3.71/1.93 = ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ emptyset )
% 3.71/1.93 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptysetimpfalse_type,type,
% 3.71/1.93 emptysetimpfalse: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptysetimpfalse,definition,
% 3.71/1.93 ( emptysetimpfalse
% 3.71/1.93 = ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ emptyset )
% 3.71/1.93 => $false ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(notinemptyset_type,type,
% 3.71/1.93 notinemptyset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(notinemptyset,definition,
% 3.71/1.93 ( notinemptyset
% 3.71/1.93 = ( ! [Xx: $i] :
% 3.71/1.93 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exuE3e_type,type,
% 3.71/1.93 exuE3e: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exuE3e,definition,
% 3.71/1.93 ( exuE3e
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setext_type,type,
% 3.71/1.93 setext: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setext,definition,
% 3.71/1.93 ( setext
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ Xx @ B ) )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ A ) )
% 3.71/1.93 => ( A = B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptyI_type,type,
% 3.71/1.93 emptyI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptyI,definition,
% 3.71/1.93 ( emptyI
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ~ ( in @ Xx @ A )
% 3.71/1.93 => ( A = emptyset ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(noeltsimpempty_type,type,
% 3.71/1.93 noeltsimpempty: $o ).
% 3.71/1.93
% 3.71/1.93 thf(noeltsimpempty,definition,
% 3.71/1.93 ( noeltsimpempty
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ~ ( in @ Xx @ A )
% 3.71/1.93 => ( A = emptyset ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setbeta_type,type,
% 3.71/1.93 setbeta: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setbeta,definition,
% 3.71/1.93 ( setbeta
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.71/1.93 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(nonempty_type,type,
% 3.71/1.93 nonempty: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(nonempty,definition,
% 3.71/1.93 ( nonempty
% 3.71/1.93 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(nonemptyE1_type,type,
% 3.71/1.93 nonemptyE1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(nonemptyE1,definition,
% 3.71/1.93 ( nonemptyE1
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ( nonempty @ A )
% 3.71/1.93 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(nonemptyI_type,type,
% 3.71/1.93 nonemptyI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(nonemptyI,definition,
% 3.71/1.93 ( nonemptyI
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 => ( nonempty
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(nonemptyI1_type,type,
% 3.71/1.93 nonemptyI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(nonemptyI1,definition,
% 3.71/1.93 ( nonemptyI1
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.71/1.93 => ( nonempty @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinIL_type,type,
% 3.71/1.93 setadjoinIL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinIL,definition,
% 3.71/1.93 ( setadjoinIL
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptyinunitempty_type,type,
% 3.71/1.93 emptyinunitempty: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptyinunitempty,definition,
% 3.71/1.93 ( emptyinunitempty
% 3.71/1.93 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinIR_type,type,
% 3.71/1.93 setadjoinIR: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinIR,definition,
% 3.71/1.93 ( setadjoinIR
% 3.71/1.93 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinE_type,type,
% 3.71/1.93 setadjoinE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinE,definition,
% 3.71/1.93 ( setadjoinE
% 3.71/1.93 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.71/1.93 => ! [Xphi: $o] :
% 3.71/1.93 ( ( ( Xy = Xx )
% 3.71/1.93 => Xphi )
% 3.71/1.93 => ( ( ( in @ Xy @ A )
% 3.71/1.93 => Xphi )
% 3.71/1.93 => Xphi ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinOr_type,type,
% 3.71/1.93 setadjoinOr: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinOr,definition,
% 3.71/1.93 ( setadjoinOr
% 3.71/1.93 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.71/1.93 => ( ( Xy = Xx )
% 3.71/1.93 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setoftrueEq_type,type,
% 3.71/1.93 setoftrueEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setoftrueEq,definition,
% 3.71/1.93 ( setoftrueEq
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xx: $i] : $true )
% 3.71/1.93 = A ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powersetI_type,type,
% 3.71/1.93 powersetI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetI,definition,
% 3.71/1.93 ( powersetI
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ A ) )
% 3.71/1.93 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptyinPowerset_type,type,
% 3.71/1.93 emptyinPowerset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptyinPowerset,definition,
% 3.71/1.93 ( emptyinPowerset
% 3.71/1.93 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptyInPowerset_type,type,
% 3.71/1.93 emptyInPowerset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptyInPowerset,definition,
% 3.71/1.93 ( emptyInPowerset
% 3.71/1.93 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powersetE_type,type,
% 3.71/1.93 powersetE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetE,definition,
% 3.71/1.93 ( powersetE
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ B @ ( powerset @ A ) )
% 3.71/1.93 => ( ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setunionI_type,type,
% 3.71/1.93 setunionI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setunionI,definition,
% 3.71/1.93 ( setunionI
% 3.71/1.93 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 => ( ( in @ B @ A )
% 3.71/1.93 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setunionE_type,type,
% 3.71/1.93 setunionE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setunionE,definition,
% 3.71/1.93 ( setunionE
% 3.71/1.93 = ( ! [A: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( setunion @ A ) )
% 3.71/1.93 => ! [Xphi: $o] :
% 3.71/1.93 ( ! [B: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 => ( ( in @ B @ A )
% 3.71/1.93 => Xphi ) )
% 3.71/1.93 => Xphi ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subPowSU_type,type,
% 3.71/1.93 subPowSU: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subPowSU,definition,
% 3.71/1.93 ( subPowSU
% 3.71/1.93 = ( ! [A: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exuE2_type,type,
% 3.71/1.93 exuE2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exuE2,definition,
% 3.71/1.93 ( exuE2
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ? [Xx: $i] :
% 3.71/1.93 ! [Xy: $i] :
% 3.71/1.93 ( ( Xphi @ Xy )
% 3.71/1.93 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(nonemptyImpWitness_type,type,
% 3.71/1.93 nonemptyImpWitness: $o ).
% 3.71/1.93
% 3.71/1.93 thf(nonemptyImpWitness,definition,
% 3.71/1.93 ( nonemptyImpWitness
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ( nonempty @ A )
% 3.71/1.93 => ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & $true ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(uniqinunit_type,type,
% 3.71/1.93 uniqinunit: $o ).
% 3.71/1.93
% 3.71/1.93 thf(uniqinunit,definition,
% 3.71/1.93 ( uniqinunit
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(notinsingleton_type,type,
% 3.71/1.93 notinsingleton: $o ).
% 3.71/1.93
% 3.71/1.93 thf(notinsingleton,definition,
% 3.71/1.93 ( notinsingleton
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx != Xy )
% 3.71/1.93 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(eqinunit_type,type,
% 3.71/1.93 eqinunit: $o ).
% 3.71/1.93
% 3.71/1.93 thf(eqinunit,definition,
% 3.71/1.93 ( eqinunit
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx = Xy )
% 3.71/1.93 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(singletonsswitch_type,type,
% 3.71/1.93 singletonsswitch: $o ).
% 3.71/1.93
% 3.71/1.93 thf(singletonsswitch,definition,
% 3.71/1.93 ( singletonsswitch
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.71/1.93 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(upairsetE_type,type,
% 3.71/1.93 upairsetE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairsetE,definition,
% 3.71/1.93 ( upairsetE
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.71/1.93 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.71/1.93 => ( ( Xz = Xx )
% 3.71/1.93 | ( Xz = Xy ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(upairsetIL_type,type,
% 3.71/1.93 upairsetIL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairsetIL,definition,
% 3.71/1.93 ( upairsetIL
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(upairsetIR_type,type,
% 3.71/1.93 upairsetIR: $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairsetIR,definition,
% 3.71/1.93 ( upairsetIR
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptyE1_type,type,
% 3.71/1.93 emptyE1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptyE1,definition,
% 3.71/1.93 ( emptyE1
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ( Xphi @ Xx ) )
% 3.71/1.93 => ( ( ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 = emptyset )
% 3.71/1.93 => $false ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(vacuousDall_type,type,
% 3.71/1.93 vacuousDall: $o ).
% 3.71/1.93
% 3.71/1.93 thf(vacuousDall,definition,
% 3.71/1.93 ( vacuousDall
% 3.71/1.93 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ emptyset )
% 3.71/1.93 => ( Xphi @ Xx ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(quantDeMorgan1_type,type,
% 3.71/1.93 quantDeMorgan1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(quantDeMorgan1,definition,
% 3.71/1.93 ( quantDeMorgan1
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ~ ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( Xphi @ Xx ) )
% 3.71/1.93 => ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(quantDeMorgan2_type,type,
% 3.71/1.93 quantDeMorgan2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(quantDeMorgan2,definition,
% 3.71/1.93 ( quantDeMorgan2
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ~ ( Xphi @ Xx ) )
% 3.71/1.93 => ~ ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(quantDeMorgan3_type,type,
% 3.71/1.93 quantDeMorgan3: $o ).
% 3.71/1.93
% 3.71/1.93 thf(quantDeMorgan3,definition,
% 3.71/1.93 ( quantDeMorgan3
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ~ ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ( Xphi @ Xx ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(quantDeMorgan4_type,type,
% 3.71/1.93 quantDeMorgan4: $o ).
% 3.71/1.93
% 3.71/1.93 thf(quantDeMorgan4,definition,
% 3.71/1.93 ( quantDeMorgan4
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ~ ( Xphi @ Xx ) )
% 3.71/1.93 => ~ ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(prop2setI_type,type,
% 3.71/1.93 prop2setI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(prop2setI,definition,
% 3.71/1.93 ( prop2setI
% 3.71/1.93 = ( ! [Xphi: $o] :
% 3.71/1.93 ( Xphi
% 3.71/1.93 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(set2prop_type,type,
% 3.71/1.93 set2prop: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(prop2set2propI_type,type,
% 3.71/1.93 prop2set2propI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(prop2set2propI,definition,
% 3.71/1.93 ( prop2set2propI
% 3.71/1.93 = ( ! [Xphi: $o] :
% 3.71/1.93 ( Xphi
% 3.71/1.93 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(notdexE_type,type,
% 3.71/1.93 notdexE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(notdexE,definition,
% 3.71/1.93 ( notdexE
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ~ ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ( Xphi @ Xx ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(notdallE_type,type,
% 3.71/1.93 notdallE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(notdallE,definition,
% 3.71/1.93 ( notdallE
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ~ ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( Xphi @ Xx ) )
% 3.71/1.93 => ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exuI1_type,type,
% 3.71/1.93 exuI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exuI1,definition,
% 3.71/1.93 ( exuI1
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ? [Xx: $i] :
% 3.71/1.93 ( ( Xphi @ Xx )
% 3.71/1.93 & ! [Xy: $i] :
% 3.71/1.93 ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) )
% 3.71/1.93 => ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exuI3_type,type,
% 3.71/1.93 exuI3: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exuI3,definition,
% 3.71/1.93 ( exuI3
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.71/1.93 => ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xphi @ Xx )
% 3.71/1.93 => ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) )
% 3.71/1.93 => ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exuI2_type,type,
% 3.71/1.93 exuI2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exuI2,definition,
% 3.71/1.93 ( exuI2
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ? [Xx: $i] :
% 3.71/1.93 ! [Xy: $i] :
% 3.71/1.93 ( ( Xphi @ Xy )
% 3.71/1.93 <=> ( Xy = Xx ) )
% 3.71/1.93 => ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(inCongP_type,type,
% 3.71/1.93 inCongP: $o ).
% 3.71/1.93
% 3.71/1.93 thf(inCongP,definition,
% 3.71/1.93 ( inCongP
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( A = B )
% 3.71/1.93 => ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx = Xy )
% 3.71/1.93 => ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(in__Cong_type,type,
% 3.71/1.93 in__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(in__Cong,definition,
% 3.71/1.93 ( in__Cong
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( A = B )
% 3.71/1.93 => ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx = Xy )
% 3.71/1.93 => ( ( in @ Xx @ A )
% 3.71/1.93 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exuE3u_type,type,
% 3.71/1.93 exuE3u: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exuE3u,definition,
% 3.71/1.93 ( exuE3u
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xphi @ Xx )
% 3.71/1.93 => ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exu__Cong_type,type,
% 3.71/1.93 exu__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exu__Cong,definition,
% 3.71/1.93 ( exu__Cong
% 3.71/1.93 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx = Xy )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 <=> ( Xpsi @ Xy ) ) )
% 3.71/1.93 => ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 <=> ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptyset__Cong_type,type,
% 3.71/1.93 emptyset__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptyset__Cong,definition,
% 3.71/1.93 ( emptyset__Cong
% 3.71/1.93 = ( emptyset = emptyset ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoin__Cong_type,type,
% 3.71/1.93 setadjoin__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoin__Cong,definition,
% 3.71/1.93 ( setadjoin__Cong
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx = Xy )
% 3.71/1.93 => ! [Xz: $i,Xu: $i] :
% 3.71/1.93 ( ( Xz = Xu )
% 3.71/1.93 => ( ( setadjoin @ Xx @ Xz )
% 3.71/1.93 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powerset__Cong_type,type,
% 3.71/1.93 powerset__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powerset__Cong,definition,
% 3.71/1.93 ( powerset__Cong
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( A = B )
% 3.71/1.93 => ( ( powerset @ A )
% 3.71/1.93 = ( powerset @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setunion__Cong_type,type,
% 3.71/1.93 setunion__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setunion__Cong,definition,
% 3.71/1.93 ( setunion__Cong
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( A = B )
% 3.71/1.93 => ( ( setunion @ A )
% 3.71/1.93 = ( setunion @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(omega__Cong_type,type,
% 3.71/1.93 omega__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(omega__Cong,definition,
% 3.71/1.93 ( omega__Cong
% 3.71/1.93 = ( omega = omega ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(exuEu_type,type,
% 3.71/1.93 exuEu: $o ).
% 3.71/1.93
% 3.71/1.93 thf(exuEu,definition,
% 3.71/1.93 ( exuEu
% 3.71/1.93 = ( ! [Xphi: $i > $o] :
% 3.71/1.93 ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xphi @ Xx )
% 3.71/1.93 => ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(descr__Cong_type,type,
% 3.71/1.93 descr__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(descr__Cong,definition,
% 3.71/1.93 ( descr__Cong
% 3.71/1.93 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx = Xy )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 <=> ( Xpsi @ Xy ) ) )
% 3.71/1.93 => ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ( ( exu
% 3.71/1.93 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.71/1.93 => ( ( descr
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 = ( descr
% 3.71/1.93 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstr__Cong_type,type,
% 3.71/1.93 dsetconstr__Cong: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dsetconstr__Cong,definition,
% 3.71/1.93 ( dsetconstr__Cong
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( A = B )
% 3.71/1.93 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( Xx = Xy )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.71/1.93 => ( ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 = ( dsetconstr @ B
% 3.71/1.93 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subset_type,type,
% 3.71/1.93 subset: $i > $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(disjoint_type,type,
% 3.71/1.93 disjoint: $i > $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(setsmeet_type,type,
% 3.71/1.93 setsmeet: $i > $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetI1_type,type,
% 3.71/1.93 subsetI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetI1,definition,
% 3.71/1.93 ( subsetI1
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ Xx @ B ) )
% 3.71/1.93 => ( subset @ A @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(eqimpsubset2_type,type,
% 3.71/1.93 eqimpsubset2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(eqimpsubset2,definition,
% 3.71/1.93 ( eqimpsubset2
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( A = B )
% 3.71/1.93 => ( subset @ B @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(eqimpsubset1_type,type,
% 3.71/1.93 eqimpsubset1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(eqimpsubset1,definition,
% 3.71/1.93 ( eqimpsubset1
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( A = B )
% 3.71/1.93 => ( subset @ A @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subsetI2_type,type,
% 3.71/1.93 subsetI2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetI2,definition,
% 3.71/1.93 ( subsetI2
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ Xx @ B ) )
% 3.71/1.93 => ( subset @ A @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(emptysetsubset_type,type,
% 3.71/1.93 emptysetsubset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(emptysetsubset,definition,
% 3.71/1.93 ( emptysetsubset
% 3.71/1.93 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subsetE_type,type,
% 3.71/1.93 subsetE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetE,definition,
% 3.71/1.93 ( subsetE
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subsetE2_type,type,
% 3.71/1.93 subsetE2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetE2,definition,
% 3.71/1.93 ( subsetE2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( ~ ( in @ Xx @ B )
% 3.71/1.93 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(notsubsetI_type,type,
% 3.71/1.93 notsubsetI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(notsubsetI,definition,
% 3.71/1.93 ( notsubsetI
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ B )
% 3.71/1.93 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(notequalI1_type,type,
% 3.71/1.93 notequalI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(notequalI1,definition,
% 3.71/1.93 ( notequalI1
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ~ ( subset @ A @ B )
% 3.71/1.93 => ( A != B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(notequalI2_type,type,
% 3.71/1.93 notequalI2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(notequalI2,definition,
% 3.71/1.93 ( notequalI2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ B )
% 3.71/1.93 => ( A != B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subsetRefl_type,type,
% 3.71/1.93 subsetRefl: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetRefl,definition,
% 3.71/1.93 ( subsetRefl
% 3.71/1.93 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subsetTrans_type,type,
% 3.71/1.93 subsetTrans: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetTrans,definition,
% 3.71/1.93 ( subsetTrans
% 3.71/1.93 = ( ! [A: $i,B: $i,C: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( ( subset @ B @ C )
% 3.71/1.93 => ( subset @ A @ C ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinSub_type,type,
% 3.71/1.93 setadjoinSub: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinSub,definition,
% 3.71/1.93 ( setadjoinSub
% 3.71/1.93 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinSub2_type,type,
% 3.71/1.93 setadjoinSub2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setadjoinSub2,definition,
% 3.71/1.93 ( setadjoinSub2
% 3.71/1.93 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subset2powerset_type,type,
% 3.71/1.93 subset2powerset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subset2powerset,definition,
% 3.71/1.93 ( subset2powerset
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setextsub_type,type,
% 3.71/1.93 setextsub: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setextsub,definition,
% 3.71/1.93 ( setextsub
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( ( subset @ B @ A )
% 3.71/1.93 => ( A = B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subsetemptysetimpeq_type,type,
% 3.71/1.93 subsetemptysetimpeq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetemptysetimpeq,definition,
% 3.71/1.93 ( subsetemptysetimpeq
% 3.71/1.93 = ( ! [A: $i] :
% 3.71/1.93 ( ( subset @ A @ emptyset )
% 3.71/1.93 => ( A = emptyset ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powersetI1_type,type,
% 3.71/1.93 powersetI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetI1,definition,
% 3.71/1.93 ( powersetI1
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( subset @ B @ A )
% 3.71/1.93 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powersetE1_type,type,
% 3.71/1.93 powersetE1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetE1,definition,
% 3.71/1.93 ( powersetE1
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( in @ B @ ( powerset @ A ) )
% 3.71/1.93 => ( subset @ B @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(inPowerset_type,type,
% 3.71/1.93 inPowerset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(inPowerset,definition,
% 3.71/1.93 ( inPowerset
% 3.71/1.93 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powersetsubset_type,type,
% 3.71/1.93 powersetsubset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetsubset,definition,
% 3.71/1.93 ( powersetsubset
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(sepInPowerset_type,type,
% 3.71/1.93 sepInPowerset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(sepInPowerset,definition,
% 3.71/1.93 ( sepInPowerset
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( in
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 @ ( powerset @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(sepSubset_type,type,
% 3.71/1.93 sepSubset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(sepSubset,definition,
% 3.71/1.93 ( sepSubset
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( subset
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 @ A ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunion_type,type,
% 3.71/1.93 binunion: $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(binunionIL_type,type,
% 3.71/1.93 binunionIL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionIL,definition,
% 3.71/1.93 ( binunionIL
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(upairset2IR_type,type,
% 3.71/1.93 upairset2IR: $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairset2IR,definition,
% 3.71/1.93 ( upairset2IR
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionIR_type,type,
% 3.71/1.93 binunionIR: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionIR,definition,
% 3.71/1.93 ( binunionIR
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionEcases_type,type,
% 3.71/1.93 binunionEcases: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionEcases,definition,
% 3.71/1.93 ( binunionEcases
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.71/1.93 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.71/1.93 => ( ( ( in @ Xx @ A )
% 3.71/1.93 => Xphi )
% 3.71/1.93 => ( ( ( in @ Xx @ B )
% 3.71/1.93 => Xphi )
% 3.71/1.93 => Xphi ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionE_type,type,
% 3.71/1.93 binunionE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionE,definition,
% 3.71/1.93 ( binunionE
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.71/1.93 => ( ( in @ Xx @ A )
% 3.71/1.93 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionLsub_type,type,
% 3.71/1.93 binunionLsub: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionLsub,definition,
% 3.71/1.93 ( binunionLsub
% 3.71/1.93 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionRsub_type,type,
% 3.71/1.93 binunionRsub: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionRsub,definition,
% 3.71/1.93 ( binunionRsub
% 3.71/1.93 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersect_type,type,
% 3.71/1.93 binintersect: $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectI_type,type,
% 3.71/1.93 binintersectI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectI,definition,
% 3.71/1.93 ( binintersectI
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset5_type,type,
% 3.71/1.93 binintersectSubset5: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset5,definition,
% 3.71/1.93 ( binintersectSubset5
% 3.71/1.93 = ( ! [A: $i,B: $i,C: $i] :
% 3.71/1.93 ( ( subset @ C @ A )
% 3.71/1.93 => ( ( subset @ C @ B )
% 3.71/1.93 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectEL_type,type,
% 3.71/1.93 binintersectEL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectEL,definition,
% 3.71/1.93 ( binintersectEL
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectLsub_type,type,
% 3.71/1.93 binintersectLsub: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectLsub,definition,
% 3.71/1.93 ( binintersectLsub
% 3.71/1.93 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset2_type,type,
% 3.71/1.93 binintersectSubset2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset2,definition,
% 3.71/1.93 ( binintersectSubset2
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( ( binintersect @ A @ B )
% 3.71/1.93 = A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset3_type,type,
% 3.71/1.93 binintersectSubset3: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset3,definition,
% 3.71/1.93 ( binintersectSubset3
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( ( binintersect @ A @ B )
% 3.71/1.93 = B )
% 3.71/1.93 => ( subset @ B @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectER_type,type,
% 3.71/1.93 binintersectER: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectER,definition,
% 3.71/1.93 ( binintersectER
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.71/1.93 => ( in @ Xx @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(disjointsetsI1_type,type,
% 3.71/1.93 disjointsetsI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(disjointsetsI1,definition,
% 3.71/1.93 ( disjointsetsI1
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ~ ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ( in @ Xx @ B ) )
% 3.71/1.93 => ( ( binintersect @ A @ B )
% 3.71/1.93 = emptyset ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectRsub_type,type,
% 3.71/1.93 binintersectRsub: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectRsub,definition,
% 3.71/1.93 ( binintersectRsub
% 3.71/1.93 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset4_type,type,
% 3.71/1.93 binintersectSubset4: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset4,definition,
% 3.71/1.93 ( binintersectSubset4
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( subset @ B @ A )
% 3.71/1.93 => ( ( binintersect @ A @ B )
% 3.71/1.93 = B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset1_type,type,
% 3.71/1.93 binintersectSubset1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectSubset1,definition,
% 3.71/1.93 ( binintersectSubset1
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( ( binintersect @ A @ B )
% 3.71/1.93 = A )
% 3.71/1.93 => ( subset @ A @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(bs114d_type,type,
% 3.71/1.93 bs114d: $o ).
% 3.71/1.93
% 3.71/1.93 thf(bs114d,definition,
% 3.71/1.93 ( bs114d
% 3.71/1.93 = ( ! [A: $i,B: $i,C: $i] :
% 3.71/1.93 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.71/1.93 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(regular_type,type,
% 3.71/1.93 regular: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminus_type,type,
% 3.71/1.93 setminus: $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(setminusI_type,type,
% 3.71/1.93 setminusI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusI,definition,
% 3.71/1.93 ( setminusI
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusEL_type,type,
% 3.71/1.93 setminusEL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusEL,definition,
% 3.71/1.93 ( setminusEL
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusER_type,type,
% 3.71/1.93 setminusER: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusER,definition,
% 3.71/1.93 ( setminusER
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.71/1.93 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusSubset2_type,type,
% 3.71/1.93 setminusSubset2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusSubset2,definition,
% 3.71/1.93 ( setminusSubset2
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( subset @ A @ B )
% 3.71/1.93 => ( ( setminus @ A @ B )
% 3.71/1.93 = emptyset ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusERneg_type,type,
% 3.71/1.93 setminusERneg: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusERneg,definition,
% 3.71/1.93 ( setminusERneg
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.71/1.93 => ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusELneg_type,type,
% 3.71/1.93 setminusELneg: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusELneg,definition,
% 3.71/1.93 ( setminusELneg
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.71/1.93 => ( ~ ( in @ Xx @ B )
% 3.71/1.93 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusILneg_type,type,
% 3.71/1.93 setminusILneg: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusILneg,definition,
% 3.71/1.93 ( setminusILneg
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ~ ( in @ Xx @ A )
% 3.71/1.93 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusIRneg_type,type,
% 3.71/1.93 setminusIRneg: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusIRneg,definition,
% 3.71/1.93 ( setminusIRneg
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ B )
% 3.71/1.93 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusLsub_type,type,
% 3.71/1.93 setminusLsub: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusLsub,definition,
% 3.71/1.93 ( setminusLsub
% 3.71/1.93 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusSubset1_type,type,
% 3.71/1.93 setminusSubset1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusSubset1,definition,
% 3.71/1.93 ( setminusSubset1
% 3.71/1.93 = ( ! [A: $i,B: $i] :
% 3.71/1.93 ( ( ( setminus @ A @ B )
% 3.71/1.93 = emptyset )
% 3.71/1.93 => ( subset @ A @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(symdiff_type,type,
% 3.71/1.93 symdiff: $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffE_type,type,
% 3.71/1.93 symdiffE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffE,definition,
% 3.71/1.93 ( symdiffE
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.71/1.93 => ! [Xphi: $o] :
% 3.71/1.93 ( ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ B )
% 3.71/1.93 => Xphi ) )
% 3.71/1.93 => ( ( ~ ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ B )
% 3.71/1.93 => Xphi ) )
% 3.71/1.93 => Xphi ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffI1_type,type,
% 3.71/1.93 symdiffI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffI1,definition,
% 3.71/1.93 ( symdiffI1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffI2_type,type,
% 3.71/1.93 symdiffI2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffI2,definition,
% 3.71/1.93 ( symdiffI2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ~ ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ B )
% 3.71/1.93 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffIneg1_type,type,
% 3.71/1.93 symdiffIneg1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffIneg1,definition,
% 3.71/1.93 ( symdiffIneg1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ B )
% 3.71/1.93 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffIneg2_type,type,
% 3.71/1.93 symdiffIneg2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(symdiffIneg2,definition,
% 3.71/1.93 ( symdiffIneg2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ~ ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ B )
% 3.71/1.93 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(iskpair_type,type,
% 3.71/1.93 iskpair: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(secondinupair_type,type,
% 3.71/1.93 secondinupair: $o ).
% 3.71/1.93
% 3.71/1.93 thf(secondinupair,definition,
% 3.71/1.93 ( secondinupair
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairIL_type,type,
% 3.71/1.93 setukpairIL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairIL,definition,
% 3.71/1.93 ( setukpairIL
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairIR_type,type,
% 3.71/1.93 setukpairIR: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairIR,definition,
% 3.71/1.93 ( setukpairIR
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(kpairiskpair_type,type,
% 3.71/1.93 kpairiskpair: $o ).
% 3.71/1.93
% 3.71/1.93 thf(kpairiskpair,definition,
% 3.71/1.93 ( kpairiskpair
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(kpair_type,type,
% 3.71/1.93 kpair: $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(kpairp_type,type,
% 3.71/1.93 kpairp: $o ).
% 3.71/1.93
% 3.71/1.93 thf(kpairp,definition,
% 3.71/1.93 ( kpairp
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprod_type,type,
% 3.71/1.93 cartprod: $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(singletonsubset_type,type,
% 3.71/1.93 singletonsubset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(singletonsubset,definition,
% 3.71/1.93 ( singletonsubset
% 3.71/1.93 = ( ! [A: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(singletoninpowerset_type,type,
% 3.71/1.93 singletoninpowerset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(singletoninpowerset,definition,
% 3.71/1.93 ( singletoninpowerset
% 3.71/1.93 = ( ! [A: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(singletoninpowunion_type,type,
% 3.71/1.93 singletoninpowunion: $o ).
% 3.71/1.93
% 3.71/1.93 thf(singletoninpowunion,definition,
% 3.71/1.93 ( singletoninpowunion
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(upairset2E_type,type,
% 3.71/1.93 upairset2E: $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairset2E,definition,
% 3.71/1.93 ( upairset2E
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.71/1.93 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.71/1.93 => ( ( Xz = Xx )
% 3.71/1.93 | ( Xz = Xy ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(upairsubunion_type,type,
% 3.71/1.93 upairsubunion: $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairsubunion,definition,
% 3.71/1.93 ( upairsubunion
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(upairinpowunion_type,type,
% 3.71/1.93 upairinpowunion: $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairinpowunion,definition,
% 3.71/1.93 ( upairinpowunion
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ubforcartprodlem1_type,type,
% 3.71/1.93 ubforcartprodlem1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ubforcartprodlem1,definition,
% 3.71/1.93 ( ubforcartprodlem1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ubforcartprodlem2_type,type,
% 3.71/1.93 ubforcartprodlem2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ubforcartprodlem2,definition,
% 3.71/1.93 ( ubforcartprodlem2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ubforcartprodlem3_type,type,
% 3.71/1.93 ubforcartprodlem3: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ubforcartprodlem3,definition,
% 3.71/1.93 ( ubforcartprodlem3
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodpairin_type,type,
% 3.71/1.93 cartprodpairin: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodpairin,definition,
% 3.71/1.93 ( cartprodpairin
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodmempair1_type,type,
% 3.71/1.93 cartprodmempair1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodmempair1,definition,
% 3.71/1.93 ( cartprodmempair1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.71/1.93 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.71/1.93 => ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ? [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 & ( Xu
% 3.71/1.93 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodmempair_type,type,
% 3.71/1.93 cartprodmempair: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodmempair,definition,
% 3.71/1.93 ( cartprodmempair
% 3.71/1.93 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.71/1.93 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.71/1.93 => ( iskpair @ Xu ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setunionE2_type,type,
% 3.71/1.93 setunionE2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setunionE2,definition,
% 3.71/1.93 ( setunionE2
% 3.71/1.93 = ( ! [A: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( setunion @ A ) )
% 3.71/1.93 => ? [X: $i] :
% 3.71/1.93 ( ( in @ X @ A )
% 3.71/1.93 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setunionsingleton1_type,type,
% 3.71/1.93 setunionsingleton1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setunionsingleton1,definition,
% 3.71/1.93 ( setunionsingleton1
% 3.71/1.93 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setunionsingleton2_type,type,
% 3.71/1.93 setunionsingleton2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setunionsingleton2,definition,
% 3.71/1.93 ( setunionsingleton2
% 3.71/1.93 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setunionsingleton_type,type,
% 3.71/1.93 setunionsingleton: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setunionsingleton,definition,
% 3.71/1.93 ( setunionsingleton
% 3.71/1.93 = ( ! [Xx: $i] :
% 3.71/1.93 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.71/1.93 = Xx ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(singleton_type,type,
% 3.71/1.93 singleton: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(singletonprop_type,type,
% 3.71/1.93 singletonprop: $o ).
% 3.71/1.93
% 3.71/1.93 thf(singletonprop,definition,
% 3.71/1.93 ( singletonprop
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 => ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) )
% 3.71/1.93 => ( ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ( Xphi @ Xx ) )
% 3.71/1.93 => ( singleton
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ex1_type,type,
% 3.71/1.93 ex1: $i > ( $i > $o ) > $o ).
% 3.71/1.93
% 3.71/1.93 thf(ex1E1_type,type,
% 3.71/1.93 ex1E1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ex1E1,definition,
% 3.71/1.93 ( ex1E1
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ( ex1 @ A
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ex1I_type,type,
% 3.71/1.93 ex1I: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ex1I,definition,
% 3.71/1.93 ( ex1I
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 => ( ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xy = Xx ) ) )
% 3.71/1.93 => ( ex1 @ A
% 3.71/1.93 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ex1I2_type,type,
% 3.71/1.93 ex1I2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ex1I2,definition,
% 3.71/1.93 ( ex1I2
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 => ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) )
% 3.71/1.93 => ( ? [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 & ( Xphi @ Xx ) )
% 3.71/1.93 => ( ex1 @ A
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(singletonsuniq_type,type,
% 3.71/1.93 singletonsuniq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(singletonsuniq,definition,
% 3.71/1.93 ( singletonsuniq
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( ( setadjoin @ Xx @ emptyset )
% 3.71/1.93 = ( setadjoin @ Xy @ emptyset ) )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(atmost1p_type,type,
% 3.71/1.93 atmost1p: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(atleast2p_type,type,
% 3.71/1.93 atleast2p: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(atmost2p_type,type,
% 3.71/1.93 atmost2p: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairsetp_type,type,
% 3.71/1.93 upairsetp: $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjL1_type,type,
% 3.71/1.93 setukpairinjL1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjL1,definition,
% 3.71/1.93 ( setukpairinjL1
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.71/1.93 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.71/1.93 => ( Xx = Xz ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(kfstsingleton_type,type,
% 3.71/1.93 kfstsingleton: $o ).
% 3.71/1.93
% 3.71/1.93 thf(kfstsingleton,definition,
% 3.71/1.93 ( kfstsingleton
% 3.71/1.93 = ( ! [Xu: $i] :
% 3.71/1.93 ( ( iskpair @ Xu )
% 3.71/1.93 => ( singleton
% 3.71/1.93 @ ( dsetconstr @ ( setunion @ Xu )
% 3.71/1.93 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(theprop_type,type,
% 3.71/1.93 theprop: $o ).
% 3.71/1.93
% 3.71/1.93 thf(theprop,definition,
% 3.71/1.93 ( theprop
% 3.71/1.93 = ( ! [X: $i] :
% 3.71/1.93 ( ( singleton @ X )
% 3.71/1.93 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(kfst_type,type,
% 3.71/1.93 kfst: $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(kfstpairEq_type,type,
% 3.71/1.93 kfstpairEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(kfstpairEq,definition,
% 3.71/1.93 ( kfstpairEq
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.71/1.93 = Xx ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodfstin_type,type,
% 3.71/1.93 cartprodfstin: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodfstin,definition,
% 3.71/1.93 ( cartprodfstin
% 3.71/1.93 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.71/1.93 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.71/1.93 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjL2_type,type,
% 3.71/1.93 setukpairinjL2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjL2,definition,
% 3.71/1.93 ( setukpairinjL2
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.71/1.93 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.71/1.93 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.71/1.93 => ( Xx = Xz ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjL_type,type,
% 3.71/1.93 setukpairinjL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjL,definition,
% 3.71/1.93 ( setukpairinjL
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.71/1.93 ( ( ( kpair @ Xx @ Xy )
% 3.71/1.93 = ( kpair @ Xz @ Xu ) )
% 3.71/1.93 => ( Xx = Xz ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR11_type,type,
% 3.71/1.93 setukpairinjR11: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR11,definition,
% 3.71/1.93 ( setukpairinjR11
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx = Xy )
% 3.71/1.93 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.71/1.93 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR12_type,type,
% 3.71/1.93 setukpairinjR12: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR12,definition,
% 3.71/1.93 ( setukpairinjR12
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xx = Xy )
% 3.71/1.93 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.71/1.93 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR1_type,type,
% 3.71/1.93 setukpairinjR1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR1,definition,
% 3.71/1.93 ( setukpairinjR1
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.71/1.93 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.71/1.93 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.71/1.93 => ( ( Xz = Xu )
% 3.71/1.93 => ( Xy = Xu ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(upairequniteq_type,type,
% 3.71/1.93 upairequniteq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(upairequniteq,definition,
% 3.71/1.93 ( upairequniteq
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.71/1.93 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.71/1.93 = ( setadjoin @ Xz @ emptyset ) )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR2_type,type,
% 3.71/1.93 setukpairinjR2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR2,definition,
% 3.71/1.93 ( setukpairinjR2
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.71/1.93 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.71/1.93 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.71/1.93 => ( Xy = Xu ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR_type,type,
% 3.71/1.93 setukpairinjR: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setukpairinjR,definition,
% 3.71/1.93 ( setukpairinjR
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.71/1.93 ( ( ( kpair @ Xx @ Xy )
% 3.71/1.93 = ( kpair @ Xz @ Xu ) )
% 3.71/1.93 => ( Xy = Xu ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ksndsingleton_type,type,
% 3.71/1.93 ksndsingleton: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ksndsingleton,definition,
% 3.71/1.93 ( ksndsingleton
% 3.71/1.93 = ( ! [Xu: $i] :
% 3.71/1.93 ( ( iskpair @ Xu )
% 3.71/1.93 => ( singleton
% 3.71/1.93 @ ( dsetconstr @ ( setunion @ Xu )
% 3.71/1.93 @ ^ [Xx: $i] :
% 3.71/1.93 ( Xu
% 3.71/1.93 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ksnd_type,type,
% 3.71/1.93 ksnd: $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(ksndpairEq_type,type,
% 3.71/1.93 ksndpairEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ksndpairEq,definition,
% 3.71/1.93 ( ksndpairEq
% 3.71/1.93 = ( ! [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.71/1.93 = Xy ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(kpairsurjEq_type,type,
% 3.71/1.93 kpairsurjEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(kpairsurjEq,definition,
% 3.71/1.93 ( kpairsurjEq
% 3.71/1.93 = ( ! [Xu: $i] :
% 3.71/1.93 ( ( iskpair @ Xu )
% 3.71/1.93 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.71/1.93 = Xu ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodsndin_type,type,
% 3.71/1.93 cartprodsndin: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodsndin,definition,
% 3.71/1.93 ( cartprodsndin
% 3.71/1.93 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.71/1.93 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.71/1.93 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodpairmemEL_type,type,
% 3.71/1.93 cartprodpairmemEL: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodpairmemEL,definition,
% 3.71/1.93 ( cartprodpairmemEL
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodpairmemER_type,type,
% 3.71/1.93 cartprodpairmemER: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodpairmemER,definition,
% 3.71/1.93 ( cartprodpairmemER
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.71/1.93 => ( in @ Xy @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodmempaircEq_type,type,
% 3.71/1.93 cartprodmempaircEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodmempaircEq,definition,
% 3.71/1.93 ( cartprodmempaircEq
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( kpair @ Xx @ Xy )
% 3.71/1.93 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodfstpairEq_type,type,
% 3.71/1.93 cartprodfstpairEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodfstpairEq,definition,
% 3.71/1.93 ( cartprodfstpairEq
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.71/1.93 = Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodsndpairEq_type,type,
% 3.71/1.93 cartprodsndpairEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodsndpairEq,definition,
% 3.71/1.93 ( cartprodsndpairEq
% 3.71/1.93 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.71/1.93 = Xy ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodpairsurjEq_type,type,
% 3.71/1.93 cartprodpairsurjEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(cartprodpairsurjEq,definition,
% 3.71/1.93 ( cartprodpairsurjEq
% 3.71/1.93 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.71/1.93 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.71/1.93 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.71/1.93 = Xu ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(breln_type,type,
% 3.71/1.93 breln: $i > $i > $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstr_type,type,
% 3.71/1.93 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrI_type,type,
% 3.71/1.93 dpsetconstrI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrI,definition,
% 3.71/1.93 ( dpsetconstrI
% 3.71/1.93 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( Xphi @ Xx @ Xy )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ Xy )
% 3.71/1.93 @ ( dpsetconstr @ A @ B
% 3.71/1.93 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrSub_type,type,
% 3.71/1.93 dpsetconstrSub: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrSub,definition,
% 3.71/1.93 ( dpsetconstrSub
% 3.71/1.93 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.71/1.93 ( subset
% 3.71/1.93 @ ( dpsetconstr @ A @ B
% 3.71/1.93 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.71/1.93 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setOfPairsIsBReln_type,type,
% 3.71/1.93 setOfPairsIsBReln: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setOfPairsIsBReln,definition,
% 3.71/1.93 ( setOfPairsIsBReln
% 3.71/1.93 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.71/1.93 ( breln @ A @ B
% 3.71/1.93 @ ( dpsetconstr @ A @ B
% 3.71/1.93 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrERa_type,type,
% 3.71/1.93 dpsetconstrERa: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrERa,definition,
% 3.71/1.93 ( dpsetconstrERa
% 3.71/1.93 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.71/1.93 @ ( dpsetconstr @ A @ B
% 3.71/1.93 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.71/1.93 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrEL1_type,type,
% 3.71/1.93 dpsetconstrEL1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrEL1,definition,
% 3.71/1.93 ( dpsetconstrEL1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ ( kpair @ Xx @ Xy )
% 3.71/1.93 @ ( dpsetconstr @ A @ B
% 3.71/1.93 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrEL2_type,type,
% 3.71/1.93 dpsetconstrEL2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrEL2,definition,
% 3.71/1.93 ( dpsetconstrEL2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ ( kpair @ Xx @ Xy )
% 3.71/1.93 @ ( dpsetconstr @ A @ B
% 3.71/1.93 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.71/1.93 => ( in @ Xy @ B ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrER_type,type,
% 3.71/1.93 dpsetconstrER: $o ).
% 3.71/1.93
% 3.71/1.93 thf(dpsetconstrER,definition,
% 3.71/1.93 ( dpsetconstrER
% 3.71/1.93 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( in @ ( kpair @ Xx @ Xy )
% 3.71/1.93 @ ( dpsetconstr @ A @ B
% 3.71/1.93 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.71/1.93 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(func_type,type,
% 3.71/1.93 func: $i > $i > $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcSet_type,type,
% 3.71/1.93 funcSet: $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(funcImageSingleton_type,type,
% 3.71/1.93 funcImageSingleton: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcImageSingleton,definition,
% 3.71/1.93 ( funcImageSingleton
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( singleton
% 3.71/1.93 @ ( dsetconstr @ B
% 3.71/1.93 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(apProp_type,type,
% 3.71/1.93 apProp: $o ).
% 3.71/1.93
% 3.71/1.93 thf(apProp,definition,
% 3.71/1.93 ( apProp
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in
% 3.71/1.93 @ ( setunion
% 3.71/1.93 @ ( dsetconstr @ B
% 3.71/1.93 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.71/1.93 @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ap_type,type,
% 3.71/1.93 ap: $i > $i > $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(app_type,type,
% 3.71/1.93 app: $o ).
% 3.71/1.93
% 3.71/1.93 thf(app,definition,
% 3.71/1.93 ( app
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(infuncsetfunc_type,type,
% 3.71/1.93 infuncsetfunc: $o ).
% 3.71/1.93
% 3.71/1.93 thf(infuncsetfunc,definition,
% 3.71/1.93 ( infuncsetfunc
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.71/1.93 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ap2p_type,type,
% 3.71/1.93 ap2p: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ap2p,definition,
% 3.71/1.93 ( ap2p
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(funcinfuncset_type,type,
% 3.71/1.93 funcinfuncset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcinfuncset,definition,
% 3.71/1.93 ( funcinfuncset
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(lamProp_type,type,
% 3.71/1.93 lamProp: $o ).
% 3.71/1.93
% 3.71/1.93 thf(lamProp,definition,
% 3.71/1.93 ( lamProp
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.71/1.93 => ( func @ A @ B
% 3.71/1.93 @ ( dpsetconstr @ A @ B
% 3.71/1.93 @ ^ [Xx: $i,Xy: $i] :
% 3.71/1.93 ( ( Xf @ Xx )
% 3.71/1.93 = Xy ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(lam_type,type,
% 3.71/1.93 lam: $i > $i > ( $i > $i ) > $i ).
% 3.71/1.93
% 3.71/1.93 thf(lamp_type,type,
% 3.71/1.93 lamp: $o ).
% 3.71/1.93
% 3.71/1.93 thf(lamp,definition,
% 3.71/1.93 ( lamp
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.71/1.93 => ( func @ A @ B
% 3.71/1.93 @ ( lam @ A @ B
% 3.71/1.93 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(lam2p_type,type,
% 3.71/1.93 lam2p: $o ).
% 3.71/1.93
% 3.71/1.93 thf(lam2p,definition,
% 3.71/1.93 ( lam2p
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.71/1.93 => ( in
% 3.71/1.93 @ ( lam @ A @ B
% 3.71/1.93 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.71/1.93 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(brelnall1_type,type,
% 3.71/1.93 brelnall1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(brelnall1,definition,
% 3.71/1.93 ( brelnall1
% 3.71/1.93 = ( ! [A: $i,B: $i,R: $i] :
% 3.71/1.93 ( ( breln @ A @ B @ R )
% 3.71/1.93 => ! [Xphi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.93 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ R )
% 3.71/1.93 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(brelnall2_type,type,
% 3.71/1.93 brelnall2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(brelnall2,definition,
% 3.71/1.93 ( brelnall2
% 3.71/1.93 = ( ! [A: $i,B: $i,R: $i] :
% 3.71/1.93 ( ( breln @ A @ B @ R )
% 3.71/1.93 => ! [Xphi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.93 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ R )
% 3.71/1.93 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ex1E2_type,type,
% 3.71/1.93 ex1E2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ex1E2,definition,
% 3.71/1.93 ( ex1E2
% 3.71/1.93 = ( ! [A: $i,Xphi: $i > $o] :
% 3.71/1.93 ( ( ex1 @ A
% 3.71/1.93 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( ( Xphi @ Xx )
% 3.71/1.93 => ( ( Xphi @ Xy )
% 3.71/1.93 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(funcGraphProp1_type,type,
% 3.71/1.93 funcGraphProp1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcGraphProp1,definition,
% 3.71/1.93 ( funcGraphProp1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(funcGraphProp3_type,type,
% 3.71/1.93 funcGraphProp3: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcGraphProp3,definition,
% 3.71/1.93 ( funcGraphProp3
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(funcGraphProp2_type,type,
% 3.71/1.93 funcGraphProp2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcGraphProp2,definition,
% 3.71/1.93 ( funcGraphProp2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.71/1.93 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.71/1.93 = Xy ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(funcextLem_type,type,
% 3.71/1.93 funcextLem: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcextLem,definition,
% 3.71/1.93 ( funcextLem
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ! [Xg: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xg )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.71/1.93 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(funcGraphProp4_type,type,
% 3.71/1.93 funcGraphProp4: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcGraphProp4,definition,
% 3.71/1.93 ( funcGraphProp4
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.71/1.93 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.71/1.93 = Xy ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subbreln_type,type,
% 3.71/1.93 subbreln: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subbreln,definition,
% 3.71/1.93 ( subbreln
% 3.71/1.93 = ( ! [A: $i,B: $i,R: $i] :
% 3.71/1.93 ( ( breln @ A @ B @ R )
% 3.71/1.93 => ! [S: $i] :
% 3.71/1.93 ( ( breln @ A @ B @ S )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.71/1.93 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(eqbreln_type,type,
% 3.71/1.93 eqbreln: $o ).
% 3.71/1.93
% 3.71/1.93 thf(eqbreln,definition,
% 3.71/1.93 ( eqbreln
% 3.71/1.93 = ( ! [A: $i,B: $i,R: $i] :
% 3.71/1.93 ( ( breln @ A @ B @ R )
% 3.71/1.93 => ! [S: $i] :
% 3.71/1.93 ( ( breln @ A @ B @ S )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ B )
% 3.71/1.93 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.71/1.93 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.71/1.93 => ( R = S ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(funcext_type,type,
% 3.71/1.93 funcext: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcext,definition,
% 3.71/1.93 ( funcext
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ! [Xg: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xg )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.71/1.93 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.71/1.93 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(funcext2_type,type,
% 3.71/1.93 funcext2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(funcext2,definition,
% 3.71/1.93 ( funcext2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.71/1.93 => ! [Xg: $i] :
% 3.71/1.93 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.71/1.93 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.71/1.93 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ap2apEq1_type,type,
% 3.71/1.93 ap2apEq1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ap2apEq1,definition,
% 3.71/1.93 ( ap2apEq1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.71/1.93 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ap2apEq2_type,type,
% 3.71/1.93 ap2apEq2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ap2apEq2,definition,
% 3.71/1.93 ( ap2apEq2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.71/1.93 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(beta1_type,type,
% 3.71/1.93 beta1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(beta1,definition,
% 3.71/1.93 ( beta1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( ap @ A @ B
% 3.71/1.93 @ ( lam @ A @ B
% 3.71/1.93 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.71/1.93 @ Xx )
% 3.71/1.93 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(eta1_type,type,
% 3.71/1.93 eta1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(eta1,definition,
% 3.71/1.93 ( eta1
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( func @ A @ B @ Xf )
% 3.71/1.93 => ( ( lam @ A @ B
% 3.71/1.93 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.71/1.93 = Xf ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(lam2lamEq_type,type,
% 3.71/1.93 lam2lamEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(lam2lamEq,definition,
% 3.71/1.93 ( lam2lamEq
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.71/1.93 => ( ( lam @ A @ B
% 3.71/1.93 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.71/1.93 = ( lam @ A @ B
% 3.71/1.93 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(beta2_type,type,
% 3.71/1.93 beta2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(beta2,definition,
% 3.71/1.93 ( beta2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( ap @ A @ B
% 3.71/1.93 @ ( lam @ A @ B
% 3.71/1.93 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.71/1.93 @ Xx )
% 3.71/1.93 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(eta2_type,type,
% 3.71/1.93 eta2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(eta2,definition,
% 3.71/1.93 ( eta2
% 3.71/1.93 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.93 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.71/1.93 => ( ( lam @ A @ B
% 3.71/1.93 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.71/1.93 = Xf ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(iffalseProp1_type,type,
% 3.71/1.93 iffalseProp1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(iffalseProp1,definition,
% 3.71/1.93 ( iffalseProp1
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( ~ Xphi
% 3.71/1.93 => ( in @ Xy
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xz: $i] :
% 3.71/1.93 ( ( Xphi
% 3.71/1.93 & ( Xz = Xx ) )
% 3.71/1.93 | ( ~ Xphi
% 3.71/1.93 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(iffalseProp2_type,type,
% 3.71/1.93 iffalseProp2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(iffalseProp2,definition,
% 3.71/1.93 ( iffalseProp2
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( ~ Xphi
% 3.71/1.93 => ( ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xz: $i] :
% 3.71/1.93 ( ( Xphi
% 3.71/1.93 & ( Xz = Xx ) )
% 3.71/1.93 | ( ~ Xphi
% 3.71/1.93 & ( Xz = Xy ) ) ) )
% 3.71/1.93 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(iftrueProp1_type,type,
% 3.71/1.93 iftrueProp1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(iftrueProp1,definition,
% 3.71/1.93 ( iftrueProp1
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( Xphi
% 3.71/1.93 => ( in @ Xx
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xz: $i] :
% 3.71/1.93 ( ( Xphi
% 3.71/1.93 & ( Xz = Xx ) )
% 3.71/1.93 | ( ~ Xphi
% 3.71/1.93 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(iftrueProp2_type,type,
% 3.71/1.93 iftrueProp2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(iftrueProp2,definition,
% 3.71/1.93 ( iftrueProp2
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( Xphi
% 3.71/1.93 => ( ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xz: $i] :
% 3.71/1.93 ( ( Xphi
% 3.71/1.93 & ( Xz = Xx ) )
% 3.71/1.93 | ( ~ Xphi
% 3.71/1.93 & ( Xz = Xy ) ) ) )
% 3.71/1.93 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(ifSingleton_type,type,
% 3.71/1.93 ifSingleton: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ifSingleton,definition,
% 3.71/1.93 ( ifSingleton
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( singleton
% 3.71/1.93 @ ( dsetconstr @ A
% 3.71/1.93 @ ^ [Xz: $i] :
% 3.71/1.93 ( ( Xphi
% 3.71/1.93 & ( Xz = Xx ) )
% 3.71/1.93 | ( ~ Xphi
% 3.71/1.93 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(if_type,type,
% 3.71/1.93 if: $i > $o > $i > $i > $i ).
% 3.71/1.93
% 3.71/1.93 thf(ifp_type,type,
% 3.71/1.93 ifp: $o ).
% 3.71/1.93
% 3.71/1.93 thf(ifp,definition,
% 3.71/1.93 ( ifp
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(theeq_type,type,
% 3.71/1.93 theeq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(theeq,definition,
% 3.71/1.93 ( theeq
% 3.71/1.93 = ( ! [X: $i] :
% 3.71/1.93 ( ( singleton @ X )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ X )
% 3.71/1.93 => ( ( setunion @ X )
% 3.71/1.93 = Xx ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(iftrue_type,type,
% 3.71/1.93 iftrue: $o ).
% 3.71/1.93
% 3.71/1.93 thf(iftrue,definition,
% 3.71/1.93 ( iftrue
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( Xphi
% 3.71/1.93 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.71/1.93 = Xx ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(iffalse_type,type,
% 3.71/1.93 iffalse: $o ).
% 3.71/1.93
% 3.71/1.93 thf(iffalse,definition,
% 3.71/1.93 ( iffalse
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( ~ Xphi
% 3.71/1.93 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.71/1.93 = Xy ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(iftrueorfalse_type,type,
% 3.71/1.93 iftrueorfalse: $o ).
% 3.71/1.93
% 3.71/1.93 thf(iftrueorfalse,definition,
% 3.71/1.93 ( iftrueorfalse
% 3.71/1.93 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ! [Xy: $i] :
% 3.71/1.93 ( ( in @ Xy @ A )
% 3.71/1.93 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectT_lem_type,type,
% 3.71/1.93 binintersectT_lem: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectT_lem,definition,
% 3.71/1.93 ( binintersectT_lem
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionT_lem_type,type,
% 3.71/1.93 binunionT_lem: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionT_lem,definition,
% 3.71/1.93 ( binunionT_lem
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powersetT_lem_type,type,
% 3.71/1.93 powersetT_lem: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetT_lem,definition,
% 3.71/1.93 ( powersetT_lem
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setminusT_lem_type,type,
% 3.71/1.93 setminusT_lem: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setminusT_lem,definition,
% 3.71/1.93 ( setminusT_lem
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementT_lem_type,type,
% 3.71/1.93 complementT_lem: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementT_lem,definition,
% 3.71/1.93 ( complementT_lem
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(setextT_type,type,
% 3.71/1.93 setextT: $o ).
% 3.71/1.93
% 3.71/1.93 thf(setextT,definition,
% 3.71/1.93 ( setextT
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ X )
% 3.71/1.93 => ( in @ Xx @ Y ) ) )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ Y )
% 3.71/1.93 => ( in @ Xx @ X ) ) )
% 3.71/1.93 => ( X = Y ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(subsetTI_type,type,
% 3.71/1.93 subsetTI: $o ).
% 3.71/1.93
% 3.71/1.93 thf(subsetTI,definition,
% 3.71/1.93 ( subsetTI
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ X )
% 3.71/1.93 => ( in @ Xx @ Y ) ) )
% 3.71/1.93 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powersetTI1_type,type,
% 3.71/1.93 powersetTI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetTI1,definition,
% 3.71/1.93 ( powersetTI1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ X )
% 3.71/1.93 => ( in @ Xx @ Y ) ) )
% 3.71/1.93 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(powersetTE1_type,type,
% 3.71/1.93 powersetTE1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(powersetTE1,definition,
% 3.71/1.93 ( powersetTE1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ X @ ( powerset @ Y ) )
% 3.71/1.93 => ( ( in @ Xx @ X )
% 3.71/1.93 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementTI1_type,type,
% 3.71/1.93 complementTI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementTI1,definition,
% 3.71/1.93 ( complementTI1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ X )
% 3.71/1.93 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementTE1_type,type,
% 3.71/1.93 complementTE1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementTE1,definition,
% 3.71/1.93 ( complementTE1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.71/1.93 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectTELcontra_type,type,
% 3.71/1.93 binintersectTELcontra: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectTELcontra,definition,
% 3.71/1.93 ( binintersectTELcontra
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ X )
% 3.71/1.93 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectTERcontra_type,type,
% 3.71/1.93 binintersectTERcontra: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binintersectTERcontra,definition,
% 3.71/1.93 ( binintersectTERcontra
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ Y )
% 3.71/1.93 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(contrasubsetT_type,type,
% 3.71/1.93 contrasubsetT: $o ).
% 3.71/1.93
% 3.71/1.93 thf(contrasubsetT,definition,
% 3.71/1.93 ( contrasubsetT
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.71/1.93 => ( ( in @ Xx @ Y )
% 3.71/1.93 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(contrasubsetT1_type,type,
% 3.71/1.93 contrasubsetT1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(contrasubsetT1,definition,
% 3.71/1.93 ( contrasubsetT1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( subset @ X @ Y )
% 3.71/1.93 => ( ~ ( in @ Xx @ Y )
% 3.71/1.93 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(contrasubsetT2_type,type,
% 3.71/1.93 contrasubsetT2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(contrasubsetT2,definition,
% 3.71/1.93 ( contrasubsetT2
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ X @ Y )
% 3.71/1.93 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(contrasubsetT3_type,type,
% 3.71/1.93 contrasubsetT3: $o ).
% 3.71/1.93
% 3.71/1.93 thf(contrasubsetT3,definition,
% 3.71/1.93 ( contrasubsetT3
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.71/1.93 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementI1_type,type,
% 3.71/1.93 doubleComplementI1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementI1,definition,
% 3.71/1.93 ( doubleComplementI1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ X )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementE1_type,type,
% 3.71/1.93 doubleComplementE1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementE1,definition,
% 3.71/1.93 ( doubleComplementE1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.71/1.93 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementSub1_type,type,
% 3.71/1.93 doubleComplementSub1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementSub1,definition,
% 3.71/1.93 ( doubleComplementSub1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementSub2_type,type,
% 3.71/1.93 doubleComplementSub2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementSub2,definition,
% 3.71/1.93 ( doubleComplementSub2
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementEq_type,type,
% 3.71/1.93 doubleComplementEq: $o ).
% 3.71/1.93
% 3.71/1.93 thf(doubleComplementEq,definition,
% 3.71/1.93 ( doubleComplementEq
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ( X
% 3.71/1.93 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementTnotintersectT_type,type,
% 3.71/1.93 complementTnotintersectT: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementTnotintersectT,definition,
% 3.71/1.93 ( complementTnotintersectT
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.71/1.93 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementImpComplementIntersect_type,type,
% 3.71/1.93 complementImpComplementIntersect: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementImpComplementIntersect,definition,
% 3.71/1.93 ( complementImpComplementIntersect
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementSubsetComplementIntersect_type,type,
% 3.71/1.93 complementSubsetComplementIntersect: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementSubsetComplementIntersect,definition,
% 3.71/1.93 ( complementSubsetComplementIntersect
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementInPowersetComplementIntersect_type,type,
% 3.71/1.93 complementInPowersetComplementIntersect: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementInPowersetComplementIntersect,definition,
% 3.71/1.93 ( complementInPowersetComplementIntersect
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(contraSubsetComplement_type,type,
% 3.71/1.93 contraSubsetComplement: $o ).
% 3.71/1.93
% 3.71/1.93 thf(contraSubsetComplement,definition,
% 3.71/1.93 ( contraSubsetComplement
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ Y )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementTcontraSubset_type,type,
% 3.71/1.93 complementTcontraSubset: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementTcontraSubset,definition,
% 3.71/1.93 ( complementTcontraSubset
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.71/1.93 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionTILcontra_type,type,
% 3.71/1.93 binunionTILcontra: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionTILcontra,definition,
% 3.71/1.93 ( binunionTILcontra
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.71/1.93 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionTIRcontra_type,type,
% 3.71/1.93 binunionTIRcontra: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionTIRcontra,definition,
% 3.71/1.93 ( binunionTIRcontra
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.71/1.93 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(inIntersectImpInUnion_type,type,
% 3.71/1.93 inIntersectImpInUnion: $o ).
% 3.71/1.93
% 3.71/1.93 thf(inIntersectImpInUnion,definition,
% 3.71/1.93 ( inIntersectImpInUnion
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.71/1.93 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(inIntersectImpInUnion2_type,type,
% 3.71/1.93 inIntersectImpInUnion2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(inIntersectImpInUnion2,definition,
% 3.71/1.93 ( inIntersectImpInUnion2
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.71/1.93 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(inIntersectImpInIntersectUnions_type,type,
% 3.71/1.93 inIntersectImpInIntersectUnions: $o ).
% 3.71/1.93
% 3.71/1.93 thf(inIntersectImpInIntersectUnions,definition,
% 3.71/1.93 ( inIntersectImpInIntersectUnions
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.71/1.93 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(intersectInPowersetIntersectUnions_type,type,
% 3.71/1.93 intersectInPowersetIntersectUnions: $o ).
% 3.71/1.93
% 3.71/1.93 thf(intersectInPowersetIntersectUnions,definition,
% 3.71/1.93 ( intersectInPowersetIntersectUnions
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(inComplementUnionImpNotIn1_type,type,
% 3.71/1.93 inComplementUnionImpNotIn1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(inComplementUnionImpNotIn1,definition,
% 3.71/1.93 ( inComplementUnionImpNotIn1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.71/1.93 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(inComplementUnionImpInComplement1_type,type,
% 3.71/1.93 inComplementUnionImpInComplement1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(inComplementUnionImpInComplement1,definition,
% 3.71/1.93 ( inComplementUnionImpInComplement1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionTE_type,type,
% 3.71/1.93 binunionTE: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionTE,definition,
% 3.71/1.93 ( binunionTE
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xphi: $o,Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.71/1.93 => ( ( ( in @ Xx @ X )
% 3.71/1.93 => Xphi )
% 3.71/1.93 => ( ( ( in @ Xx @ Y )
% 3.71/1.93 => Xphi )
% 3.71/1.93 => Xphi ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(binunionTEcontra_type,type,
% 3.71/1.93 binunionTEcontra: $o ).
% 3.71/1.93
% 3.71/1.93 thf(binunionTEcontra,definition,
% 3.71/1.93 ( binunionTEcontra
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ~ ( in @ Xx @ X )
% 3.71/1.93 => ( ~ ( in @ Xx @ Y )
% 3.71/1.93 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2a1_type,type,
% 3.71/1.93 demorgan2a1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2a1,definition,
% 3.71/1.93 ( demorgan2a1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(complementUnionInPowersetComplement_type,type,
% 3.71/1.93 complementUnionInPowersetComplement: $o ).
% 3.71/1.93
% 3.71/1.93 thf(complementUnionInPowersetComplement,definition,
% 3.71/1.93 ( complementUnionInPowersetComplement
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2a2_type,type,
% 3.71/1.93 demorgan2a2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2a2,definition,
% 3.71/1.93 ( demorgan2a2
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan1a_type,type,
% 3.71/1.93 demorgan1a: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan1a,definition,
% 3.71/1.93 ( demorgan1a
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.71/1.93 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan1b_type,type,
% 3.71/1.93 demorgan1b: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan1b,definition,
% 3.71/1.93 ( demorgan1b
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan1_type,type,
% 3.71/1.93 demorgan1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan1,definition,
% 3.71/1.93 ( demorgan1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.71/1.93 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2a_type,type,
% 3.71/1.93 demorgan2a: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2a,definition,
% 3.71/1.93 ( demorgan2a
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.71/1.93 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2b2_type,type,
% 3.71/1.93 demorgan2b2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2b2,definition,
% 3.71/1.93 ( demorgan2b2
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.71/1.93 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2b_type,type,
% 3.71/1.93 demorgan2b: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2b,definition,
% 3.71/1.93 ( demorgan2b
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.93 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.71/1.93 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2_type,type,
% 3.71/1.93 demorgan2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(demorgan2,definition,
% 3.71/1.93 ( demorgan2
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.71/1.93 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule0_type,type,
% 3.71/1.93 woz13rule0: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule0,definition,
% 3.71/1.93 ( woz13rule0
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.71/1.93 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule1_type,type,
% 3.71/1.93 woz13rule1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule1,definition,
% 3.71/1.93 ( woz13rule1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ X @ Z )
% 3.71/1.93 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule2_type,type,
% 3.71/1.93 woz13rule2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule2,definition,
% 3.71/1.93 ( woz13rule2
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ Y @ Z )
% 3.71/1.93 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule3_type,type,
% 3.71/1.93 woz13rule3: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule3,definition,
% 3.71/1.93 ( woz13rule3
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ X @ Y )
% 3.71/1.93 => ( ( subset @ X @ Z )
% 3.71/1.93 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule4_type,type,
% 3.71/1.93 woz13rule4: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz13rule4,definition,
% 3.71/1.93 ( woz13rule4
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ! [W: $i] :
% 3.71/1.93 ( ( in @ W @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ X @ Z )
% 3.71/1.93 => ( ( subset @ Y @ W )
% 3.71/1.93 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_1_type,type,
% 3.71/1.93 woz1_1: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_1,definition,
% 3.71/1.93 ( woz1_1
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_2_type,type,
% 3.71/1.93 woz1_2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_2,definition,
% 3.71/1.93 ( woz1_2
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ! [W: $i] :
% 3.71/1.93 ( ( in @ W @ ( powerset @ A ) )
% 3.71/1.93 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.71/1.93 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_3_type,type,
% 3.71/1.93 woz1_3: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_3,definition,
% 3.71/1.93 ( woz1_3
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ! [Z: $i] :
% 3.71/1.93 ( ( in @ Z @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_4_type,type,
% 3.71/1.93 woz1_4: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_4,definition,
% 3.71/1.93 ( woz1_4
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.71/1.93 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_5_type,type,
% 3.71/1.93 woz1_5: $o ).
% 3.71/1.93
% 3.71/1.93 thf(woz1_5,definition,
% 3.71/1.93 ( woz1_5
% 3.71/1.93 = ( ! [A: $i,X: $i] :
% 3.71/1.93 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.93 => ! [Y: $i] :
% 3.71/1.93 ( ( in @ Y @ ( powerset @ A ) )
% 3.71/1.93 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.71/1.93
% 3.71/1.93 thf(breln1_type,type,
% 3.71/1.93 breln1: $i > $i > $o ).
% 3.71/1.93
% 3.71/1.93 thf(breln1all2_type,type,
% 3.71/1.93 breln1all2: $o ).
% 3.71/1.93
% 3.71/1.93 thf(breln1all2,definition,
% 3.71/1.93 ( breln1all2
% 3.71/1.93 = ( ! [A: $i,R: $i] :
% 3.71/1.93 ( ( breln1 @ A @ R )
% 3.71/1.93 => ! [Xphi: $i > $o] :
% 3.71/1.93 ( ! [Xx: $i] :
% 3.71/1.93 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ R )
% 3.71/1.94 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1Set_type,type,
% 3.71/1.94 breln1Set: $i > $i ).
% 3.71/1.94
% 3.71/1.94 thf(breln1SetBreln1_type,type,
% 3.71/1.94 breln1SetBreln1: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1SetBreln1,definition,
% 3.71/1.94 ( breln1SetBreln1
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( in @ R @ ( breln1Set @ A ) )
% 3.71/1.94 => ( breln1 @ A @ R ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(transitive_type,type,
% 3.71/1.94 transitive: $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(antisymmetric_type,type,
% 3.71/1.94 antisymmetric: $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(reflexive_type,type,
% 3.71/1.94 reflexive: $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(refltransitive_type,type,
% 3.71/1.94 refltransitive: $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(refllinearorder_type,type,
% 3.71/1.94 refllinearorder: $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(reflwellordering_type,type,
% 3.71/1.94 reflwellordering: $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(choice2fnsingleton_type,type,
% 3.71/1.94 choice2fnsingleton: $o ).
% 3.71/1.94
% 3.71/1.94 thf(choice2fnsingleton,definition,
% 3.71/1.94 ( choice2fnsingleton
% 3.71/1.94 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.71/1.94 ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ? [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ B )
% 3.71/1.94 & ( Xphi @ Xx @ Xy ) ) )
% 3.71/1.94 => ! [R: $i] :
% 3.71/1.94 ( ( in @ R @ ( breln1Set @ B ) )
% 3.71/1.94 => ( ( reflwellordering @ B @ R )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ( singleton
% 3.71/1.94 @ ( dsetconstr @ B
% 3.71/1.94 @ ^ [Xy: $i] :
% 3.71/1.94 ( ( Xphi @ Xx @ Xy )
% 3.71/1.94 & ! [Xz: $i] :
% 3.71/1.94 ( ( in @ Xz @ B )
% 3.71/1.94 => ( ( Xphi @ Xx @ Xz )
% 3.71/1.94 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(setOfPairsIsBReln1_type,type,
% 3.71/1.94 setOfPairsIsBReln1: $o ).
% 3.71/1.94
% 3.71/1.94 thf(setOfPairsIsBReln1,definition,
% 3.71/1.94 ( setOfPairsIsBReln1
% 3.71/1.94 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.71/1.94 ( breln1 @ A
% 3.71/1.94 @ ( dpsetconstr @ A @ A
% 3.71/1.94 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1all1_type,type,
% 3.71/1.94 breln1all1: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1all1,definition,
% 3.71/1.94 ( breln1all1
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [Xphi: $i > $o] :
% 3.71/1.94 ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ R )
% 3.71/1.94 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(subbreln1_type,type,
% 3.71/1.94 subbreln1: $o ).
% 3.71/1.94
% 3.71/1.94 thf(subbreln1,definition,
% 3.71/1.94 ( subbreln1
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.71/1.94 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(eqbreln1_type,type,
% 3.71/1.94 eqbreln1: $o ).
% 3.71/1.94
% 3.71/1.94 thf(eqbreln1,definition,
% 3.71/1.94 ( eqbreln1
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.71/1.94 => ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.71/1.94 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.71/1.94 => ( R = S ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1invset_type,type,
% 3.71/1.94 breln1invset: $i > $i > $i ).
% 3.71/1.94
% 3.71/1.94 thf(breln1invprop_type,type,
% 3.71/1.94 breln1invprop: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1invprop,definition,
% 3.71/1.94 ( breln1invprop
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1invI_type,type,
% 3.71/1.94 breln1invI: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1invI,definition,
% 3.71/1.94 ( breln1invI
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1invE_type,type,
% 3.71/1.94 breln1invE: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1invE,definition,
% 3.71/1.94 ( breln1invE
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.71/1.94 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compset_type,type,
% 3.71/1.94 breln1compset: $i > $i > $i > $i ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compprop_type,type,
% 3.71/1.94 breln1compprop: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compprop,definition,
% 3.71/1.94 ( breln1compprop
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compI_type,type,
% 3.71/1.94 breln1compI: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compI,definition,
% 3.71/1.94 ( breln1compI
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ! [Xz: $i] :
% 3.71/1.94 ( ( in @ Xz @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.71/1.94 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.71/1.94 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compE_type,type,
% 3.71/1.94 breln1compE: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compE,definition,
% 3.71/1.94 ( breln1compE
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.71/1.94 => ? [Xz: $i] :
% 3.71/1.94 ( ( in @ Xz @ A )
% 3.71/1.94 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.71/1.94 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compEex_type,type,
% 3.71/1.94 breln1compEex: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1compEex,definition,
% 3.71/1.94 ( breln1compEex
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.71/1.94 => ! [Xphi: $o] :
% 3.71/1.94 ( ! [Xz: $i] :
% 3.71/1.94 ( ( in @ Xz @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.71/1.94 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.71/1.94 => Xphi ) ) )
% 3.71/1.94 => Xphi ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionprop_type,type,
% 3.71/1.94 breln1unionprop: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionprop,definition,
% 3.71/1.94 ( breln1unionprop
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionIL_type,type,
% 3.71/1.94 breln1unionIL: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionIL,definition,
% 3.71/1.94 ( breln1unionIL
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionIR_type,type,
% 3.71/1.94 breln1unionIR: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionIR,definition,
% 3.71/1.94 ( breln1unionIR
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.71/1.94 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionI_type,type,
% 3.71/1.94 breln1unionI: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionI,definition,
% 3.71/1.94 ( breln1unionI
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.71/1.94 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionE_type,type,
% 3.71/1.94 breln1unionE: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionE,definition,
% 3.71/1.94 ( breln1unionE
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionEcases_type,type,
% 3.71/1.94 breln1unionEcases: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionEcases,definition,
% 3.71/1.94 ( breln1unionEcases
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.71/1.94 => ! [Xphi: $o] :
% 3.71/1.94 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.71/1.94 => Xphi )
% 3.71/1.94 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.71/1.94 => Xphi )
% 3.71/1.94 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionCommutes_type,type,
% 3.71/1.94 breln1unionCommutes: $o ).
% 3.71/1.94
% 3.71/1.94 thf(breln1unionCommutes,definition,
% 3.71/1.94 ( breln1unionCommutes
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ( ( binunion @ R @ S )
% 3.71/1.94 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(woz2Ex_type,type,
% 3.71/1.94 woz2Ex: $o ).
% 3.71/1.94
% 3.71/1.94 thf(woz2Ex,definition,
% 3.71/1.94 ( woz2Ex
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ( R
% 3.71/1.94 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(woz2W_type,type,
% 3.71/1.94 woz2W: $o ).
% 3.71/1.94
% 3.71/1.94 thf(woz2W,definition,
% 3.71/1.94 ( woz2W
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.71/1.94 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(woz2A_type,type,
% 3.71/1.94 woz2A: $o ).
% 3.71/1.94
% 3.71/1.94 thf(woz2A,definition,
% 3.71/1.94 ( woz2A
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [T: $i] :
% 3.71/1.94 ( ( breln1 @ A @ T )
% 3.71/1.94 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.71/1.94 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(woz2B_type,type,
% 3.71/1.94 woz2B: $o ).
% 3.71/1.94
% 3.71/1.94 thf(woz2B,definition,
% 3.71/1.94 ( woz2B
% 3.71/1.94 = ( ! [A: $i,R: $i] :
% 3.71/1.94 ( ( breln1 @ A @ R )
% 3.71/1.94 => ! [S: $i] :
% 3.71/1.94 ( ( breln1 @ A @ S )
% 3.71/1.94 => ! [T: $i] :
% 3.71/1.94 ( ( breln1 @ A @ T )
% 3.71/1.94 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.71/1.94 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(image1Ex_type,type,
% 3.71/1.94 image1Ex: $o ).
% 3.71/1.94
% 3.71/1.94 thf(image1Ex,definition,
% 3.71/1.94 ( image1Ex
% 3.71/1.94 = ( ! [A: $i,Xf: $i > $i] :
% 3.71/1.94 ? [B: $i] :
% 3.71/1.94 ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ B )
% 3.71/1.94 <=> ? [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 & ( Xx
% 3.71/1.94 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(image1Ex1_type,type,
% 3.71/1.94 image1Ex1: $o ).
% 3.71/1.94
% 3.71/1.94 thf(image1Ex1,definition,
% 3.71/1.94 ( image1Ex1
% 3.71/1.94 = ( ! [A: $i,Xf: $i > $i] :
% 3.71/1.94 ( exu
% 3.71/1.94 @ ^ [B: $i] :
% 3.71/1.94 ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ B )
% 3.71/1.94 <=> ? [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 & ( Xx
% 3.71/1.94 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(image1_type,type,
% 3.71/1.94 image1: $i > ( $i > $i ) > $i ).
% 3.71/1.94
% 3.71/1.94 thf(image1Equiv_type,type,
% 3.71/1.94 image1Equiv: $o ).
% 3.71/1.94
% 3.71/1.94 thf(image1Equiv,definition,
% 3.71/1.94 ( image1Equiv
% 3.71/1.94 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.71/1.94 ( ( in @ Xx
% 3.71/1.94 @ ( image1 @ A
% 3.71/1.94 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.71/1.94 <=> ? [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 & ( Xx
% 3.71/1.94 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(image1E_type,type,
% 3.71/1.94 image1E: $o ).
% 3.71/1.94
% 3.71/1.94 thf(image1E,definition,
% 3.71/1.94 ( image1E
% 3.71/1.94 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.71/1.94 ( ( in @ Xx
% 3.71/1.94 @ ( image1 @ A
% 3.71/1.94 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.71/1.94 => ? [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 & ( Xx
% 3.71/1.94 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(image1I_type,type,
% 3.71/1.94 image1I: $o ).
% 3.71/1.94
% 3.71/1.94 thf(image1I,definition,
% 3.71/1.94 ( image1I
% 3.71/1.94 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.71/1.94 ( ? [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ A )
% 3.71/1.94 & ( Xx
% 3.71/1.94 = ( Xf @ Xy ) ) )
% 3.71/1.94 => ( in @ Xx
% 3.71/1.94 @ ( image1 @ A
% 3.71/1.94 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(injective_type,type,
% 3.71/1.94 injective: $i > $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(injFuncSet_type,type,
% 3.71/1.94 injFuncSet: $i > $i > $i ).
% 3.71/1.94
% 3.71/1.94 thf(injFuncInInjFuncSet_type,type,
% 3.71/1.94 injFuncInInjFuncSet: $o ).
% 3.71/1.94
% 3.71/1.94 thf(injFuncInInjFuncSet,definition,
% 3.71/1.94 ( injFuncInInjFuncSet
% 3.71/1.94 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.94 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.71/1.94 => ( ( injective @ A @ B @ Xf )
% 3.71/1.94 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(injFuncSetFuncIn_type,type,
% 3.71/1.94 injFuncSetFuncIn: $o ).
% 3.71/1.94
% 3.71/1.94 thf(injFuncSetFuncIn,definition,
% 3.71/1.94 ( injFuncSetFuncIn
% 3.71/1.94 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.94 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.71/1.94 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(injFuncSetFuncInj_type,type,
% 3.71/1.94 injFuncSetFuncInj: $o ).
% 3.71/1.94
% 3.71/1.94 thf(injFuncSetFuncInj,definition,
% 3.71/1.94 ( injFuncSetFuncInj
% 3.71/1.94 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.71/1.94 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.71/1.94 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(surjective_type,type,
% 3.71/1.94 surjective: $i > $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(surjFuncSet_type,type,
% 3.71/1.94 surjFuncSet: $i > $i > $i ).
% 3.71/1.94
% 3.71/1.94 thf(surjFuncSetFuncIn_type,type,
% 3.71/1.94 surjFuncSetFuncIn: $o ).
% 3.71/1.94
% 3.71/1.94 thf(surjFuncSetFuncIn,definition,
% 3.71/1.94 ( surjFuncSetFuncIn
% 3.71/1.94 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.71/1.94 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 3.71/1.94 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(surjFuncSetFuncSurj_type,type,
% 3.71/1.94 surjFuncSetFuncSurj: $o ).
% 3.71/1.94
% 3.71/1.94 thf(surjFuncSetFuncSurj,definition,
% 3.71/1.94 ( surjFuncSetFuncSurj
% 3.71/1.94 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.71/1.94 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 3.71/1.94 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(leftInvIsSurj_type,type,
% 3.71/1.94 leftInvIsSurj: $o ).
% 3.71/1.94
% 3.71/1.94 thf(leftInvIsSurj,definition,
% 3.71/1.94 ( leftInvIsSurj
% 3.71/1.94 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.71/1.94 ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.71/1.94 => ! [Xg: $i] :
% 3.71/1.94 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 3.71/1.94 => ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 3.71/1.94 = Xx ) )
% 3.71/1.94 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(surjCantorThm_type,type,
% 3.71/1.94 surjCantorThm: $o ).
% 3.71/1.94
% 3.71/1.94 thf(surjCantorThm,definition,
% 3.71/1.94 ( surjCantorThm
% 3.71/1.94 = ( ! [A: $i,Xf: $i] :
% 3.71/1.94 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 3.71/1.94 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(foundation2_type,type,
% 3.71/1.94 foundation2: $o ).
% 3.71/1.94
% 3.71/1.94 thf(foundation2,definition,
% 3.71/1.94 ( foundation2
% 3.71/1.94 = ( ! [A: $i] :
% 3.71/1.94 ( ( nonempty @ A )
% 3.71/1.94 => ? [X: $i] :
% 3.71/1.94 ( ( in @ X @ A )
% 3.71/1.94 & ( ( binintersect @ X @ A )
% 3.71/1.94 = emptyset ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(notinself_type,type,
% 3.71/1.94 notinself: $o ).
% 3.71/1.94
% 3.71/1.94 thf(notinself,definition,
% 3.71/1.94 ( notinself
% 3.71/1.94 = ( ! [A: $i] :
% 3.71/1.94 ~ ( in @ A @ A ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(notinself2_type,type,
% 3.71/1.94 notinself2: $o ).
% 3.71/1.94
% 3.71/1.94 thf(notinself2,definition,
% 3.71/1.94 ( notinself2
% 3.71/1.94 = ( ! [A: $i,B: $i] :
% 3.71/1.94 ( ( in @ A @ B )
% 3.71/1.94 => ~ ( in @ B @ A ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(omegaS_type,type,
% 3.71/1.94 omegaS: $i > $i ).
% 3.71/1.94
% 3.71/1.94 thf(omegaSp_type,type,
% 3.71/1.94 omegaSp: $o ).
% 3.71/1.94
% 3.71/1.94 thf(omegaSp,definition,
% 3.71/1.94 ( omegaSp
% 3.71/1.94 = ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ omega )
% 3.71/1.94 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(omegaSclos_type,type,
% 3.71/1.94 omegaSclos: $o ).
% 3.71/1.94
% 3.71/1.94 thf(omegaSclos,definition,
% 3.71/1.94 ( omegaSclos
% 3.71/1.94 = ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ omega )
% 3.71/1.94 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(peano0notS_type,type,
% 3.71/1.94 peano0notS: $o ).
% 3.71/1.94
% 3.71/1.94 thf(peano0notS,definition,
% 3.71/1.94 ( peano0notS
% 3.71/1.94 = ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ omega )
% 3.71/1.94 => ( ( omegaS @ Xx )
% 3.71/1.94 != emptyset ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(peano3_type,type,
% 3.71/1.94 peano3: $i > $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(peano4_type,type,
% 3.71/1.94 peano4: $i > $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(peano5_type,type,
% 3.71/1.94 peano5: $i > $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(peanoSinj_type,type,
% 3.71/1.94 peanoSinj: $o ).
% 3.71/1.94
% 3.71/1.94 thf(peanoSinj,definition,
% 3.71/1.94 ( peanoSinj
% 3.71/1.94 = ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ omega )
% 3.71/1.94 => ! [Xy: $i] :
% 3.71/1.94 ( ( in @ Xy @ omega )
% 3.71/1.94 => ( ( ( omegaS @ Xx )
% 3.71/1.94 = ( omegaS @ Xy ) )
% 3.71/1.94 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(peano_type,type,
% 3.71/1.94 peano: $i > $i > $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(transitiveset_type,type,
% 3.71/1.94 transitiveset: $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(transitiveset,definition,
% 3.71/1.94 ( transitiveset
% 3.71/1.94 = ( ^ [A: $i] :
% 3.71/1.94 ! [X: $i] :
% 3.71/1.94 ( ( in @ X @ A )
% 3.71/1.94 => ( subset @ X @ A ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(transitivesetOp1_type,type,
% 3.71/1.94 transitivesetOp1: $o ).
% 3.71/1.94
% 3.71/1.94 thf(transitivesetOp1,definition,
% 3.71/1.94 ( transitivesetOp1
% 3.71/1.94 = ( ! [X: $i] :
% 3.71/1.94 ( ( transitiveset @ X )
% 3.71/1.94 => ! [A: $i] :
% 3.71/1.94 ( ( in @ A @ X )
% 3.71/1.94 => ( subset @ A @ X ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(binintTransitive_type,type,
% 3.71/1.94 binintTransitive: $o ).
% 3.71/1.94
% 3.71/1.94 thf(binintTransitive,definition,
% 3.71/1.94 ( binintTransitive
% 3.71/1.94 = ( ! [X: $i] :
% 3.71/1.94 ( ( transitiveset @ X )
% 3.71/1.94 => ! [Y: $i] :
% 3.71/1.94 ( ( transitiveset @ Y )
% 3.71/1.94 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(transitivesetOp2_type,type,
% 3.71/1.94 transitivesetOp2: $o ).
% 3.71/1.94
% 3.71/1.94 thf(transitivesetOp2,definition,
% 3.71/1.94 ( transitivesetOp2
% 3.71/1.94 = ( ! [X: $i] :
% 3.71/1.94 ( ( transitiveset @ X )
% 3.71/1.94 => ! [A: $i,Xx: $i] :
% 3.71/1.94 ( ( in @ A @ X )
% 3.71/1.94 => ( ( in @ Xx @ A )
% 3.71/1.94 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(setunionTransitive_type,type,
% 3.71/1.94 setunionTransitive: $o ).
% 3.71/1.94
% 3.71/1.94 thf(setunionTransitive,definition,
% 3.71/1.94 ( setunionTransitive
% 3.71/1.94 = ( ! [X: $i] :
% 3.71/1.94 ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ X )
% 3.71/1.94 => ( transitiveset @ Xx ) )
% 3.71/1.94 => ( transitiveset @ ( setunion @ X ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(stricttotalorderedByIn_type,type,
% 3.71/1.94 stricttotalorderedByIn: $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(stricttotalorderedByIn,definition,
% 3.71/1.94 ( stricttotalorderedByIn
% 3.71/1.94 = ( ^ [A: $i] :
% 3.71/1.94 ( ! [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ A )
% 3.71/1.94 => ! [X: $i] :
% 3.71/1.94 ( ( in @ X @ A )
% 3.71/1.94 => ! [Y: $i] :
% 3.71/1.94 ( ( in @ Y @ A )
% 3.71/1.94 => ( ( ( in @ Xx @ X )
% 3.71/1.94 & ( in @ X @ Y ) )
% 3.71/1.94 => ( in @ Xx @ Y ) ) ) ) )
% 3.71/1.94 & ! [X: $i] :
% 3.71/1.94 ( ( in @ X @ A )
% 3.71/1.94 => ! [Y: $i] :
% 3.71/1.94 ( ( in @ Y @ A )
% 3.71/1.94 => ( ( X = Y )
% 3.71/1.94 | ( in @ X @ Y )
% 3.71/1.94 | ( in @ Y @ X ) ) ) )
% 3.71/1.94 & ! [X: $i] :
% 3.71/1.94 ( ( in @ X @ A )
% 3.71/1.94 => ~ ( in @ X @ X ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(wellorderedByIn_type,type,
% 3.71/1.94 wellorderedByIn: $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(wellorderedByIn,definition,
% 3.71/1.94 ( wellorderedByIn
% 3.71/1.94 = ( ^ [A: $i] :
% 3.71/1.94 ( ( stricttotalorderedByIn @ A )
% 3.71/1.94 & ! [X: $i] :
% 3.71/1.94 ( ( in @ X @ ( powerset @ A ) )
% 3.71/1.94 => ( ( nonempty @ X )
% 3.71/1.94 => ? [Xx: $i] :
% 3.71/1.94 ( ( in @ Xx @ X )
% 3.71/1.94 & ! [Y: $i] :
% 3.71/1.94 ( ( in @ Y @ X )
% 3.71/1.94 => ( ( Xx = Y )
% 3.71/1.94 | ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(ordinal_type,type,
% 3.71/1.94 ordinal: $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(ordinal,definition,
% 3.71/1.94 ( ordinal
% 3.71/1.94 = ( ^ [Xx: $i] :
% 3.71/1.94 ( ( transitiveset @ Xx )
% 3.71/1.94 & ( wellorderedByIn @ Xx ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(limitOrdinal_type,type,
% 3.71/1.94 limitOrdinal: $i > $o ).
% 3.71/1.94
% 3.71/1.94 thf(ordinalMinLem1_type,type,
% 3.71/1.94 ordinalMinLem1: $o ).
% 3.71/1.94
% 3.71/1.94 thf(ordinalMinLem1,definition,
% 3.71/1.94 ( ordinalMinLem1
% 3.71/1.94 = ( ! [X: $i] :
% 3.71/1.94 ( ( ordinal @ X )
% 3.71/1.94 => ! [Y: $i] :
% 3.71/1.94 ( ( ordinal @ Y )
% 3.71/1.94 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(ordinalTransSet_type,type,
% 3.71/1.94 ordinalTransSet: $o ).
% 3.71/1.94
% 3.71/1.94 thf(ordinalTransSet,definition,
% 3.71/1.94 ( ordinalTransSet
% 3.71/1.94 = ( ! [X: $i] :
% 3.71/1.94 ( ( ordinal @ X )
% 3.71/1.94 => ! [Xx: $i,A: $i] :
% 3.71/1.94 ( ( in @ A @ X )
% 3.71/1.94 => ( ( in @ Xx @ A )
% 3.71/1.94 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.71/1.94
% 3.71/1.94 thf(ordinalTransSet1,conjecture,
% 3.71/1.94 ( setextAx
% 3.71/1.94 => ( emptysetAx
% 3.71/1.94 => ( setadjoinAx
% 3.71/1.94 => ( powersetAx
% 3.71/1.94 => ( setunionAx
% 3.71/1.94 => ( omega0Ax
% 3.71/1.94 => ( omegaSAx
% 3.71/1.94 => ( omegaIndAx
% 3.71/1.94 => ( replAx
% 3.71/1.94 => ( foundationAx
% 3.71/1.94 => ( wellorderingAx
% 3.71/1.94 => ( descrp
% 3.71/1.94 => ( dsetconstrI
% 3.71/1.94 => ( dsetconstrEL
% 3.71/1.94 => ( dsetconstrER
% 3.71/1.94 => ( exuE1
% 3.71/1.94 => ( prop2setE
% 3.71/1.94 => ( emptysetE
% 3.71/1.94 => ( emptysetimpfalse
% 3.71/1.94 => ( notinemptyset
% 3.71/1.94 => ( exuE3e
% 3.71/1.94 => ( setext
% 3.71/1.94 => ( emptyI
% 3.71/1.94 => ( noeltsimpempty
% 3.71/1.94 => ( setbeta
% 3.71/1.94 => ( nonemptyE1
% 3.71/1.94 => ( nonemptyI
% 3.71/1.94 => ( nonemptyI1
% 3.71/1.94 => ( setadjoinIL
% 3.71/1.94 => ( emptyinunitempty
% 3.71/1.94 => ( setadjoinIR
% 3.71/1.94 => ( setadjoinE
% 3.71/1.94 => ( setadjoinOr
% 3.71/1.94 => ( setoftrueEq
% 3.71/1.94 => ( powersetI
% 3.71/1.94 => ( emptyinPowerset
% 3.71/1.94 => ( emptyInPowerset
% 3.71/1.94 => ( powersetE
% 3.71/1.94 => ( setunionI
% 3.71/1.94 => ( setunionE
% 3.71/1.94 => ( subPowSU
% 3.71/1.94 => ( exuE2
% 3.71/1.94 => ( nonemptyImpWitness
% 3.71/1.94 => ( uniqinunit
% 3.71/1.94 => ( notinsingleton
% 3.71/1.94 => ( eqinunit
% 3.71/1.94 => ( singletonsswitch
% 3.71/1.94 => ( upairsetE
% 3.71/1.94 => ( upairsetIL
% 3.71/1.94 => ( upairsetIR
% 3.71/1.94 => ( emptyE1
% 3.71/1.94 => ( vacuousDall
% 3.71/1.94 => ( quantDeMorgan1
% 3.71/1.94 => ( quantDeMorgan2
% 3.71/1.94 => ( quantDeMorgan3
% 3.71/1.94 => ( quantDeMorgan4
% 3.71/1.94 => ( prop2setI
% 3.71/1.94 => ( prop2set2propI
% 3.71/1.94 => ( notdexE
% 3.71/1.94 => ( notdallE
% 3.71/1.94 => ( exuI1
% 3.71/1.94 => ( exuI3
% 3.71/1.94 => ( exuI2
% 3.71/1.94 => ( inCongP
% 3.71/1.94 => ( in__Cong
% 3.71/1.94 => ( exuE3u
% 3.71/1.94 => ( exu__Cong
% 3.71/1.94 => ( emptyset__Cong
% 3.71/1.94 => ( setadjoin__Cong
% 3.71/1.94 => ( powerset__Cong
% 3.71/1.94 => ( setunion__Cong
% 3.71/1.94 => ( omega__Cong
% 3.71/1.94 => ( exuEu
% 3.71/1.94 => ( descr__Cong
% 3.71/1.94 => ( dsetconstr__Cong
% 3.71/1.94 => ( subsetI1
% 3.71/1.94 => ( eqimpsubset2
% 3.71/1.94 => ( eqimpsubset1
% 3.71/1.94 => ( subsetI2
% 3.71/1.94 => ( emptysetsubset
% 3.71/1.94 => ( subsetE
% 3.71/1.94 => ( subsetE2
% 3.71/1.94 => ( notsubsetI
% 3.71/1.94 => ( notequalI1
% 3.71/1.94 => ( notequalI2
% 3.71/1.94 => ( subsetRefl
% 3.71/1.94 => ( subsetTrans
% 3.71/1.94 => ( setadjoinSub
% 3.71/1.94 => ( setadjoinSub2
% 3.71/1.94 => ( subset2powerset
% 3.71/1.94 => ( setextsub
% 3.71/1.94 => ( subsetemptysetimpeq
% 3.71/1.94 => ( powersetI1
% 3.71/1.94 => ( powersetE1
% 3.71/1.94 => ( inPowerset
% 3.71/1.94 => ( powersetsubset
% 3.71/1.94 => ( sepInPowerset
% 3.71/1.94 => ( sepSubset
% 3.71/1.94 => ( binunionIL
% 3.71/1.94 => ( upairset2IR
% 3.71/1.94 => ( binunionIR
% 3.71/1.94 => ( binunionEcases
% 3.71/1.94 => ( binunionE
% 3.71/1.94 => ( binunionLsub
% 3.71/1.94 => ( binunionRsub
% 3.71/1.94 => ( binintersectI
% 3.71/1.94 => ( binintersectSubset5
% 3.71/1.94 => ( binintersectEL
% 3.71/1.94 => ( binintersectLsub
% 3.71/1.94 => ( binintersectSubset2
% 3.71/1.94 => ( binintersectSubset3
% 3.71/1.94 => ( binintersectER
% 3.71/1.94 => ( disjointsetsI1
% 3.71/1.94 => ( binintersectRsub
% 3.71/1.94 => ( binintersectSubset4
% 3.71/1.94 => ( binintersectSubset1
% 3.71/1.94 => ( bs114d
% 3.71/1.94 => ( setminusI
% 3.71/1.94 => ( setminusEL
% 3.71/1.94 => ( setminusER
% 3.71/1.94 => ( setminusSubset2
% 3.71/1.94 => ( setminusERneg
% 3.71/1.94 => ( setminusELneg
% 3.71/1.94 => ( setminusILneg
% 3.71/1.94 => ( setminusIRneg
% 3.71/1.94 => ( setminusLsub
% 3.71/1.94 => ( setminusSubset1
% 3.71/1.94 => ( symdiffE
% 3.71/1.94 => ( symdiffI1
% 3.71/1.94 => ( symdiffI2
% 3.71/1.94 => ( symdiffIneg1
% 3.71/1.94 => ( symdiffIneg2
% 3.71/1.94 => ( secondinupair
% 3.71/1.94 => ( setukpairIL
% 3.71/1.94 => ( setukpairIR
% 3.71/1.94 => ( kpairiskpair
% 3.71/1.94 => ( kpairp
% 3.71/1.94 => ( singletonsubset
% 3.71/1.94 => ( singletoninpowerset
% 3.71/1.94 => ( singletoninpowunion
% 3.71/1.94 => ( upairset2E
% 3.71/1.94 => ( upairsubunion
% 3.71/1.94 => ( upairinpowunion
% 3.71/1.94 => ( ubforcartprodlem1
% 3.71/1.94 => ( ubforcartprodlem2
% 3.71/1.94 => ( ubforcartprodlem3
% 3.71/1.94 => ( cartprodpairin
% 3.71/1.94 => ( cartprodmempair1
% 3.71/1.94 => ( cartprodmempair
% 3.71/1.94 => ( setunionE2
% 3.71/1.94 => ( setunionsingleton1
% 3.71/1.94 => ( setunionsingleton2
% 3.71/1.94 => ( setunionsingleton
% 3.71/1.94 => ( singletonprop
% 3.71/1.94 => ( ex1E1
% 3.71/1.94 => ( ex1I
% 3.71/1.94 => ( ex1I2
% 3.71/1.94 => ( singletonsuniq
% 3.71/1.94 => ( setukpairinjL1
% 3.71/1.94 => ( kfstsingleton
% 3.71/1.94 => ( theprop
% 3.71/1.94 => ( kfstpairEq
% 3.71/1.94 => ( cartprodfstin
% 3.71/1.94 => ( setukpairinjL2
% 3.71/1.94 => ( setukpairinjL
% 3.71/1.94 => ( setukpairinjR11
% 3.71/1.94 => ( setukpairinjR12
% 3.71/1.94 => ( setukpairinjR1
% 3.71/1.94 => ( upairequniteq
% 3.71/1.94 => ( setukpairinjR2
% 3.71/1.94 => ( setukpairinjR
% 3.71/1.94 => ( ksndsingleton
% 3.71/1.94 => ( ksndpairEq
% 3.71/1.94 => ( kpairsurjEq
% 3.71/1.94 => ( cartprodsndin
% 3.71/1.94 => ( cartprodpairmemEL
% 3.71/1.94 => ( cartprodpairmemER
% 3.71/1.94 => ( cartprodmempaircEq
% 3.71/1.94 => ( cartprodfstpairEq
% 3.71/1.94 => ( cartprodsndpairEq
% 3.71/1.94 => ( cartprodpairsurjEq
% 3.71/1.94 => ( dpsetconstrI
% 3.71/1.94 => ( dpsetconstrSub
% 3.71/1.94 => ( setOfPairsIsBReln
% 3.71/1.94 => ( dpsetconstrERa
% 3.71/1.94 => ( dpsetconstrEL1
% 3.71/1.94 => ( dpsetconstrEL2
% 3.71/1.94 => ( dpsetconstrER
% 3.71/1.94 => ( funcImageSingleton
% 3.71/1.94 => ( apProp
% 3.71/1.94 => ( app
% 3.71/1.94 => ( infuncsetfunc
% 3.71/1.94 => ( ap2p
% 3.71/1.94 => ( funcinfuncset
% 3.71/1.94 => ( lamProp
% 3.71/1.94 => ( lamp
% 3.71/1.94 => ( lam2p
% 3.71/1.94 => ( brelnall1
% 3.71/1.94 => ( brelnall2
% 3.71/1.94 => ( ex1E2
% 3.71/1.94 => ( funcGraphProp1
% 3.71/1.94 => ( funcGraphProp3
% 3.71/1.94 => ( funcGraphProp2
% 3.71/1.94 => ( funcextLem
% 3.71/1.94 => ( funcGraphProp4
% 3.71/1.94 => ( subbreln
% 3.71/1.94 => ( eqbreln
% 3.71/1.94 => ( funcext
% 3.71/1.94 => ( funcext2
% 3.71/1.94 => ( ap2apEq1
% 3.71/1.94 => ( ap2apEq2
% 3.71/1.94 => ( beta1
% 3.71/1.94 => ( eta1
% 3.71/1.94 => ( lam2lamEq
% 3.71/1.94 => ( beta2
% 3.71/1.94 => ( eta2
% 3.71/1.94 => ( iffalseProp1
% 3.71/1.94 => ( iffalseProp2
% 3.71/1.94 => ( iftrueProp1
% 3.71/1.94 => ( iftrueProp2
% 3.71/1.94 => ( ifSingleton
% 3.71/1.94 => ( ifp
% 3.71/1.94 => ( theeq
% 3.71/1.94 => ( iftrue
% 3.71/1.94 => ( iffalse
% 3.71/1.94 => ( iftrueorfalse
% 3.71/1.94 => ( binintersectT_lem
% 3.71/1.94 => ( binunionT_lem
% 3.71/1.94 => ( powersetT_lem
% 3.71/1.94 => ( setminusT_lem
% 3.71/1.94 => ( complementT_lem
% 3.71/1.94 => ( setextT
% 3.71/1.94 => ( subsetTI
% 3.71/1.94 => ( powersetTI1
% 3.71/1.94 => ( powersetTE1
% 3.71/1.94 => ( complementTI1
% 3.71/1.94 => ( complementTE1
% 3.71/1.94 => ( binintersectTELcontra
% 3.71/1.94 => ( binintersectTERcontra
% 3.71/1.94 => ( contrasubsetT
% 3.71/1.94 => ( contrasubsetT1
% 3.71/1.94 => ( contrasubsetT2
% 3.71/1.94 => ( contrasubsetT3
% 3.71/1.94 => ( doubleComplementI1
% 3.71/1.94 => ( doubleComplementE1
% 3.71/1.94 => ( doubleComplementSub1
% 3.71/1.94 => ( doubleComplementSub2
% 3.71/1.94 => ( doubleComplementEq
% 3.71/1.94 => ( complementTnotintersectT
% 3.71/1.94 => ( complementImpComplementIntersect
% 3.71/1.94 => ( complementSubsetComplementIntersect
% 3.71/1.94 => ( complementInPowersetComplementIntersect
% 3.71/1.94 => ( contraSubsetComplement
% 3.71/1.94 => ( complementTcontraSubset
% 3.71/1.94 => ( binunionTILcontra
% 3.71/1.94 => ( binunionTIRcontra
% 3.71/1.94 => ( inIntersectImpInUnion
% 3.71/1.94 => ( inIntersectImpInUnion2
% 3.71/1.94 => ( inIntersectImpInIntersectUnions
% 3.71/1.94 => ( intersectInPowersetIntersectUnions
% 3.71/1.94 => ( inComplementUnionImpNotIn1
% 3.71/1.94 => ( inComplementUnionImpInComplement1
% 3.71/1.94 => ( binunionTE
% 3.71/1.94 => ( binunionTEcontra
% 3.71/1.94 => ( demorgan2a1
% 3.71/1.94 => ( complementUnionInPowersetComplement
% 3.71/1.94 => ( demorgan2a2
% 3.71/1.94 => ( demorgan1a
% 3.71/1.94 => ( demorgan1b
% 3.71/1.94 => ( demorgan1
% 3.71/1.94 => ( demorgan2a
% 3.71/1.94 => ( demorgan2b2
% 3.71/1.94 => ( demorgan2b
% 3.71/1.94 => ( demorgan2
% 3.71/1.94 => ( woz13rule0
% 3.71/1.94 => ( woz13rule1
% 3.71/1.94 => ( woz13rule2
% 3.71/1.94 => ( woz13rule3
% 3.71/1.94 => ( woz13rule4
% 3.71/1.94 => ( woz1_1
% 3.71/1.94 => ( woz1_2
% 3.71/1.94 => ( woz1_3
% 3.71/1.94 => ( woz1_4
% 3.71/1.94 => ( woz1_5
% 3.71/1.94 => ( breln1all2
% 3.71/1.94 => ( breln1SetBreln1
% 3.71/1.94 => ( choice2fnsingleton
% 3.71/1.94 => ( setOfPairsIsBReln1
% 3.71/1.94 => ( breln1all1
% 3.71/1.94 => ( subbreln1
% 3.71/1.94 => ( eqbreln1
% 3.71/1.94 => ( breln1invprop
% 3.71/1.94 => ( breln1invI
% 3.71/1.94 => ( breln1invE
% 3.71/1.94 => ( breln1compprop
% 3.71/1.94 => ( breln1compI
% 3.71/1.94 => ( breln1compE
% 3.71/1.94 => ( breln1compEex
% 3.71/1.94 => ( breln1unionprop
% 3.71/1.94 => ( breln1unionIL
% 3.71/1.94 => ( breln1unionIR
% 3.71/1.94 => ( breln1unionI
% 3.71/1.94 => ( breln1unionE
% 3.71/1.94 => ( breln1unionEcases
% 3.71/1.94 => ( breln1unionCommutes
% 3.71/1.94 => ( woz2Ex
% 3.71/1.94 => ( woz2W
% 3.71/1.94 => ( woz2A
% 3.71/1.94 => ( woz2B
% 3.71/1.94 => ( image1Ex
% 3.71/1.94 => ( image1Ex1
% 3.71/1.94 => ( image1Equiv
% 3.71/1.94 => ( image1E
% 3.71/1.94 => ( image1I
% 3.71/1.94 => ( injFuncInInjFuncSet
% 3.71/1.94 => ( injFuncSetFuncIn
% 3.71/1.94 => ( injFuncSetFuncInj
% 3.71/1.94 => ( surjFuncSetFuncIn
% 3.71/1.94 => ( surjFuncSetFuncSurj
% 3.71/1.94 => ( leftInvIsSurj
% 3.71/1.94 => ( surjCantorThm
% 3.71/1.94 => ( foundation2
% 3.71/1.94 => ( notinself
% 3.71/1.94 => ( notinself2
% 3.71/1.94 => ( omegaSp
% 3.71/1.94 => ( omegaSclos
% 3.71/1.94 => ( peano0notS
% 3.71/1.94 => ( peanoSinj
% 3.71/1.94 => ( transitivesetOp1
% 3.71/1.94 => ( binintTransitive
% 3.71/1.94 => ( transitivesetOp2
% 3.71/1.94 => ( setunionTransitive
% 3.71/1.94 => ( ordinalMinLem1
% 3.71/1.94 => ( ordinalTransSet
% 3.71/1.94 => ! [X: $i] :
% 3.71/1.94 ( ( ordinal @ X )
% 3.71/1.94 => ! [A: $i] :
% 3.71/1.94 ( ( in @ A @ X )
% 3.71/1.94 => ( subset @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.71/1.94 ------------------------
% 3.71/1.99 % (10005)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/4Mi)
% 3.71/1.99 % (10007)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/2Mi)
% 3.71/1.99 % (10004)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/183Mi)
% 3.71/1.99 % (10006)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/27Mi)
% 3.71/1.99 % (10008)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/2Mi)
% 3.71/1.99 % (10009)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/275Mi)
% 3.71/1.99 % (10010)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/18Mi)
% 3.71/1.99 % (10007)Instruction limit reached!
% 3.71/1.99 % (10007)------------------------------
% 3.71/1.99 % (10007)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/1.99 % (10007)Termination reason: Unknown
% 3.71/1.99 % (10007)Termination phase: shuffling
% 3.71/1.99
% 3.71/1.99 % (10008)Instruction limit reached!
% 3.71/1.99 % (10008)------------------------------
% 3.71/1.99 % (10008)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/1.99 % (10007)Memory used [KB]: 1663
% 3.71/1.99 % (10007)Time elapsed: 0.003 s
% 3.71/1.99 % (10007)Instructions burned: 2 (million)
% 3.71/1.99 % (10007)------------------------------
% 3.71/1.99 % (10007)------------------------------
% 3.71/1.99 % (10008)Termination reason: Unknown
% 3.71/1.99 % (10008)Termination phase: shuffling
% 3.71/1.99
% 3.71/1.99 % (10008)Memory used [KB]: 1663
% 3.71/1.99 % (10008)Time elapsed: 0.003 s
% 3.71/1.99 % (10008)Instructions burned: 2 (million)
% 3.71/1.99 % (10008)------------------------------
% 3.71/1.99 % (10008)------------------------------
% 3.71/1.99 % (10005)Instruction limit reached!
% 3.71/1.99 % (10005)------------------------------
% 3.71/1.99 % (10005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/1.99 % (10005)Termination reason: Unknown
% 3.71/1.99 % (10005)Termination phase: shuffling
% 3.71/1.99
% 3.71/1.99 % (10005)Memory used [KB]: 1791
% 3.71/1.99 % (10005)Time elapsed: 0.004 s
% 3.71/1.99 % (10005)Instructions burned: 4 (million)
% 3.71/1.99 % (10005)------------------------------
% 3.71/1.99 % (10005)------------------------------
% 3.71/2.00 % (10010)Instruction limit reached!
% 3.71/2.00 % (10010)------------------------------
% 3.71/2.00 % (10010)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/2.00 % (10010)Termination reason: Unknown
% 3.71/2.00 % (10010)Termination phase: shuffling
% 3.71/2.00
% 3.71/2.00 % (10010)Memory used [KB]: 1918
% 3.71/2.00 % (10010)Time elapsed: 0.011 s
% 3.71/2.00 % (10010)Instructions burned: 18 (million)
% 3.71/2.00 % (10010)------------------------------
% 3.71/2.00 % (10010)------------------------------
% 3.71/2.00 % (10006)Instruction limit reached!
% 3.71/2.00 % (10006)------------------------------
% 3.71/2.00 % (10006)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/2.00 % (10006)Termination reason: Unknown
% 3.71/2.00 % (10006)Termination phase: shuffling
% 3.71/2.00
% 3.71/2.00 % (10006)Memory used [KB]: 2174
% 3.71/2.00 % (10006)Time elapsed: 0.015 s
% 3.71/2.00 % (10006)Instructions burned: 28 (million)
% 3.71/2.00 % (10006)------------------------------
% 3.71/2.00 % (10006)------------------------------
% 3.71/2.01 % (10011)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/3Mi)
% 3.71/2.01 % (10012)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/37Mi)
% 3.71/2.01 % (10013)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/15Mi)
% 3.71/2.01 % (10011)Instruction limit reached!
% 3.71/2.01 % (10011)------------------------------
% 3.71/2.01 % (10011)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/2.01 % (10011)Termination reason: Unknown
% 3.71/2.01 % (10011)Termination phase: shuffling
% 3.71/2.01
% 3.71/2.01 % (10011)Memory used [KB]: 1663
% 3.71/2.01 % (10011)Time elapsed: 0.004 s
% 3.71/2.01 % (10011)Instructions burned: 4 (million)
% 3.71/2.01 % (10011)------------------------------
% 3.71/2.01 % (10011)------------------------------
% 3.71/2.01 % (10014)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/3Mi)
% 3.71/2.01 % (10013)Instruction limit reached!
% 3.71/2.01 % (10013)------------------------------
% 3.71/2.01 % (10013)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/2.01 % (10013)Termination reason: Unknown
% 3.71/2.01 % (10013)Termination phase: shuffling
% 3.71/2.01
% 3.71/2.01 % (10013)Memory used [KB]: 1918
% 3.71/2.01 % (10013)Time elapsed: 0.010 s
% 3.71/2.01 % (10013)Instructions burned: 17 (million)
% 3.71/2.01 % (10013)------------------------------
% 3.71/2.01 % (10013)------------------------------
% 3.71/2.02 % (10014)Instruction limit reached!
% 3.71/2.02 % (10014)------------------------------
% 3.71/2.02 % (10014)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/2.02 % (10014)Termination reason: Unknown
% 3.71/2.02 % (10014)Termination phase: shuffling
% 3.71/2.02
% 3.71/2.02 % (10014)Memory used [KB]: 1663
% 3.71/2.02 % (10014)Time elapsed: 0.004 s
% 3.71/2.02 % (10014)Instructions burned: 4 (million)
% 3.71/2.02 % (10014)------------------------------
% 3.71/2.02 % (10014)------------------------------
% 3.71/2.02 % (10015)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/1041Mi)
% 3.71/2.02 % (10012)Instruction limit reached!
% 3.71/2.02 % (10012)------------------------------
% 3.71/2.02 % (10012)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.71/2.02 % (10012)Termination reason: Unknown
% 3.71/2.02 % (10012)Termination phase: shuffling
% 3.71/2.02
% 3.71/2.02 % (10012)Memory used [KB]: 2430
% 3.71/2.02 % (10012)Time elapsed: 0.019 s
% 3.71/2.02 % (10012)Instructions burned: 38 (million)
% 3.71/2.02 % (10012)------------------------------
% 3.71/2.02 % (10012)------------------------------
% 3.71/2.02 % (10016)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/7Mi)
% 4.07/2.03 % (10016)Instruction limit reached!
% 4.07/2.03 % (10016)------------------------------
% 4.07/2.03 % (10016)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.03 % (10016)Termination reason: Unknown
% 4.07/2.03 % (10016)Termination phase: shuffling
% 4.07/2.03
% 4.07/2.03 % (10016)Memory used [KB]: 1791
% 4.07/2.03 % (10016)Time elapsed: 0.005 s
% 4.07/2.03 % (10016)Instructions burned: 7 (million)
% 4.07/2.03 % (10016)------------------------------
% 4.07/2.03 % (10016)------------------------------
% 4.07/2.03 % (10017)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/16Mi)
% 4.07/2.03 % (10018)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/3Mi)
% 4.07/2.03 % (10018)Instruction limit reached!
% 4.07/2.03 % (10018)------------------------------
% 4.07/2.03 % (10018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.03 % (10018)Termination reason: Unknown
% 4.07/2.03 % (10018)Termination phase: shuffling
% 4.07/2.03
% 4.07/2.03 % (10018)Memory used [KB]: 1663
% 4.07/2.03 % (10018)Time elapsed: 0.004 s
% 4.07/2.03 % (10018)Instructions burned: 4 (million)
% 4.07/2.03 % (10018)------------------------------
% 4.07/2.03 % (10018)------------------------------
% 4.07/2.04 % (10017)Instruction limit reached!
% 4.07/2.04 % (10017)------------------------------
% 4.07/2.04 % (10017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.04 % (10017)Termination reason: Unknown
% 4.07/2.04 % (10017)Termination phase: shuffling
% 4.07/2.04
% 4.07/2.04 % (10017)Memory used [KB]: 1918
% 4.07/2.04 % (10017)Time elapsed: 0.010 s
% 4.07/2.04 % (10017)Instructions burned: 17 (million)
% 4.07/2.04 % (10017)------------------------------
% 4.07/2.04 % (10017)------------------------------
% 4.07/2.04 % (10019)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/3Mi)
% 4.07/2.04 % (10019)Instruction limit reached!
% 4.07/2.04 % (10019)------------------------------
% 4.07/2.04 % (10019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.04 % (10019)Termination reason: Unknown
% 4.07/2.04 % (10019)Termination phase: shuffling
% 4.07/2.04
% 4.07/2.04 % (10019)Memory used [KB]: 1663
% 4.07/2.04 % (10019)Time elapsed: 0.004 s
% 4.07/2.04 % (10019)Instructions burned: 4 (million)
% 4.07/2.04 % (10019)------------------------------
% 4.07/2.04 % (10019)------------------------------
% 4.07/2.04 % (10020)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/7Mi)
% 4.07/2.05 % (10020)Instruction limit reached!
% 4.07/2.05 % (10020)------------------------------
% 4.07/2.05 % (10020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.05 % (10020)Termination reason: Unknown
% 4.07/2.05 % (10020)Termination phase: shuffling
% 4.07/2.05
% 4.07/2.05 % (10020)Memory used [KB]: 1791
% 4.07/2.05 % (10020)Time elapsed: 0.005 s
% 4.07/2.05 % (10020)Instructions burned: 7 (million)
% 4.07/2.05 % (10020)------------------------------
% 4.07/2.05 % (10020)------------------------------
% 4.07/2.05 % (10021)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/3Mi)
% 4.07/2.05 % (10021)Instruction limit reached!
% 4.07/2.05 % (10021)------------------------------
% 4.07/2.05 % (10021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.05 % (10021)Termination reason: Unknown
% 4.07/2.05 % (10021)Termination phase: shuffling
% 4.07/2.05
% 4.07/2.05 % (10021)Memory used [KB]: 1663
% 4.07/2.05 % (10021)Time elapsed: 0.003 s
% 4.07/2.05 % (10021)Instructions burned: 3 (million)
% 4.07/2.05 % (10021)------------------------------
% 4.07/2.05 % (10021)------------------------------
% 4.07/2.06 % (10022)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/4Mi)
% 4.07/2.06 % (10022)Instruction limit reached!
% 4.07/2.06 % (10022)------------------------------
% 4.07/2.06 % (10022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.06 % (10022)Termination reason: Unknown
% 4.07/2.06 % (10022)Termination phase: shuffling
% 4.07/2.06
% 4.07/2.06 % (10022)Memory used [KB]: 1663
% 4.07/2.06 % (10022)Time elapsed: 0.004 s
% 4.07/2.06 % (10022)Instructions burned: 5 (million)
% 4.07/2.06 % (10022)------------------------------
% 4.07/2.06 % (10022)------------------------------
% 4.07/2.06 % (10023)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/18Mi)
% 4.07/2.06 % (10024)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_9905 for (2999ds/710Mi)
% 4.07/2.07 % (10023)Instruction limit reached!
% 4.07/2.07 % (10023)------------------------------
% 4.07/2.07 % (10023)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.07 % (10023)Termination reason: Unknown
% 4.07/2.07 % (10023)Termination phase: shuffling
% 4.07/2.07
% 4.07/2.07 % (10023)Memory used [KB]: 2046
% 4.07/2.07 % (10023)Time elapsed: 0.010 s
% 4.07/2.07 % (10023)Instructions burned: 18 (million)
% 4.07/2.07 % (10023)------------------------------
% 4.07/2.07 % (10023)------------------------------
% 4.07/2.07 % (10025)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/6Mi)
% 4.07/2.07 % (10004)Instruction limit reached!
% 4.07/2.07 % (10004)------------------------------
% 4.07/2.07 % (10004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.07 % (10004)Termination reason: Unknown
% 4.07/2.07 % (10004)Termination phase: Property scanning
% 4.07/2.07
% 4.07/2.07 % (10004)Memory used [KB]: 3198
% 4.07/2.07 % (10004)Time elapsed: 0.084 s
% 4.07/2.07 % (10004)Instructions burned: 184 (million)
% 4.07/2.07 % (10004)------------------------------
% 4.07/2.07 % (10004)------------------------------
% 4.07/2.07 % (10025)Instruction limit reached!
% 4.07/2.07 % (10025)------------------------------
% 4.07/2.07 % (10025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.07 % (10025)Termination reason: Unknown
% 4.07/2.07 % (10025)Termination phase: shuffling
% 4.07/2.07
% 4.07/2.07 % (10025)Memory used [KB]: 1791
% 4.07/2.07 % (10025)Time elapsed: 0.005 s
% 4.07/2.07 % (10025)Instructions burned: 7 (million)
% 4.07/2.07 % (10025)------------------------------
% 4.07/2.07 % (10025)------------------------------
% 4.07/2.07 % (10026)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/902Mi)
% 4.07/2.08 % (10027)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/21Mi)
% 4.07/2.09 % (10028)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/5Mi)
% 4.07/2.09 % (10029)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/6Mi)
% 4.07/2.09 % (10028)Instruction limit reached!
% 4.07/2.09 % (10028)------------------------------
% 4.07/2.09 % (10028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.09 % (10028)Termination reason: Unknown
% 4.07/2.09 % (10028)Termination phase: shuffling
% 4.07/2.09
% 4.07/2.09 % (10028)Memory used [KB]: 1791
% 4.07/2.09 % (10028)Time elapsed: 0.005 s
% 4.07/2.09 % (10028)Instructions burned: 7 (million)
% 4.07/2.09 % (10028)------------------------------
% 4.07/2.09 % (10028)------------------------------
% 4.07/2.09 % (10029)Instruction limit reached!
% 4.07/2.09 % (10029)------------------------------
% 4.07/2.09 % (10029)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.09 % (10029)Termination reason: Unknown
% 4.07/2.09 % (10029)Termination phase: shuffling
% 4.07/2.09
% 4.07/2.09 % (10029)Memory used [KB]: 1791
% 4.07/2.09 % (10029)Time elapsed: 0.005 s
% 4.07/2.09 % (10029)Instructions burned: 7 (million)
% 4.07/2.09 % (10029)------------------------------
% 4.07/2.09 % (10029)------------------------------
% 4.07/2.09 % (10027)Instruction limit reached!
% 4.07/2.09 % (10027)------------------------------
% 4.07/2.09 % (10027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.09 % (10027)Termination reason: Unknown
% 4.07/2.09 % (10027)Termination phase: shuffling
% 4.07/2.09
% 4.07/2.09 % (10027)Memory used [KB]: 2046
% 4.07/2.09 % (10027)Time elapsed: 0.012 s
% 4.07/2.09 % (10027)Instructions burned: 22 (million)
% 4.07/2.09 % (10027)------------------------------
% 4.07/2.09 % (10027)------------------------------
% 4.07/2.11 % (10030)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/377Mi)
% 4.07/2.11 % (10031)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/779Mi)
% 4.07/2.11 % (10032)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/19Mi)
% 4.07/2.12 % (10009)Instruction limit reached!
% 4.07/2.12 % (10009)------------------------------
% 4.07/2.12 % (10009)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.12 % (10009)Termination reason: Unknown
% 4.07/2.12 % (10009)Termination phase: Property scanning
% 4.07/2.12
% 4.07/2.12 % (10009)Memory used [KB]: 6012
% 4.07/2.12 % (10009)Time elapsed: 0.132 s
% 4.07/2.12 % (10009)Instructions burned: 275 (million)
% 4.07/2.12 % (10009)------------------------------
% 4.07/2.12 % (10009)------------------------------
% 4.07/2.12 % (10032)Instruction limit reached!
% 4.07/2.12 % (10032)------------------------------
% 4.07/2.12 % (10032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.12 % (10032)Termination reason: Unknown
% 4.07/2.12 % (10032)Termination phase: shuffling
% 4.07/2.12
% 4.07/2.12 % (10032)Memory used [KB]: 2046
% 4.07/2.12 % (10032)Time elapsed: 0.011 s
% 4.07/2.12 % (10032)Instructions burned: 20 (million)
% 4.07/2.12 % (10032)------------------------------
% 4.07/2.12 % (10032)------------------------------
% 4.52/2.14 % (10033)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/879Mi)
% 4.52/2.14 % (10034)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/17Mi)
% 4.52/2.14 % (10034)Instruction limit reached!
% 4.52/2.14 % (10034)------------------------------
% 4.52/2.14 % (10034)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.52/2.14 % (10034)Termination reason: Unknown
% 4.52/2.14 % (10034)Termination phase: shuffling
% 4.52/2.14
% 4.52/2.14 % (10034)Memory used [KB]: 1918
% 4.52/2.14 % (10034)Time elapsed: 0.009 s
% 4.52/2.14 % (10034)Instructions burned: 17 (million)
% 4.52/2.14 % (10034)------------------------------
% 4.52/2.14 % (10034)------------------------------
% 4.52/2.16 % (10035)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_9905 for (2998ds/3Mi)
% 4.52/2.16 % (10035)Instruction limit reached!
% 4.52/2.16 % (10035)------------------------------
% 4.52/2.16 % (10035)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.52/2.16 % (10035)Termination reason: Unknown
% 4.52/2.16 % (10035)Termination phase: shuffling
% 4.52/2.16
% 4.52/2.16 % (10035)Memory used [KB]: 1663
% 4.52/2.16 % (10035)Time elapsed: 0.004 s
% 4.52/2.16 % (10035)Instructions burned: 4 (million)
% 4.52/2.16 % (10035)------------------------------
% 4.52/2.16 % (10035)------------------------------
% 4.52/2.18 % (10036)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_9905 for (2997ds/30Mi)
% 4.52/2.19 % (10036)Instruction limit reached!
% 4.52/2.19 % (10036)------------------------------
% 4.52/2.19 % (10036)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.52/2.19 % (10036)Termination reason: Unknown
% 4.52/2.19 % (10036)Termination phase: shuffling
% 4.52/2.19
% 4.52/2.19 % (10036)Memory used [KB]: 2302
% 4.52/2.19 % (10036)Time elapsed: 0.016 s
% 4.52/2.19 % (10036)Instructions burned: 31 (million)
% 4.52/2.19 % (10036)------------------------------
% 4.52/2.19 % (10036)------------------------------
% 4.59/2.21 % (10037)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_9905 for (2997ds/127Mi)
% 4.59/2.27 % (10037)Instruction limit reached!
% 4.59/2.27 % (10037)------------------------------
% 4.59/2.27 % (10037)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.59/2.27 % (10037)Termination reason: Unknown
% 4.59/2.27 % (10037)Termination phase: Preprocessing 3
% 4.59/2.27
% 4.59/2.27 % (10037)Memory used [KB]: 2942
% 4.59/2.27 % (10037)Time elapsed: 0.059 s
% 4.59/2.27 % (10037)Instructions burned: 128 (million)
% 4.59/2.27 % (10037)------------------------------
% 4.59/2.27 % (10037)------------------------------
% 4.59/2.28 % (10030)Instruction limit reached!
% 4.59/2.28 % (10030)------------------------------
% 4.59/2.28 % (10030)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.59/2.28 % (10030)Termination reason: Unknown
% 4.59/2.28 % (10030)Termination phase: Saturation
% 4.59/2.28
% 4.59/2.28 % (10030)Memory used [KB]: 9466
% 4.59/2.28 % (10030)Time elapsed: 0.171 s
% 4.59/2.28 % (10030)Instructions burned: 378 (million)
% 4.59/2.28 % (10030)------------------------------
% 4.59/2.28 % (10030)------------------------------
% 4.59/2.28 % (10038)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_9905 for (2996ds/100Mi)
% 4.59/2.29 % (10039)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_9905 for (2996ds/3Mi)
% 4.59/2.29 % (10039)Instruction limit reached!
% 4.59/2.29 % (10039)------------------------------
% 4.59/2.29 % (10039)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.59/2.29 % (10039)Termination reason: Unknown
% 4.59/2.29 % (10039)Termination phase: shuffling
% 4.59/2.29
% 4.59/2.29 % (10039)Memory used [KB]: 1663
% 4.59/2.29 % (10039)Time elapsed: 0.004 s
% 4.59/2.29 % (10039)Instructions burned: 4 (million)
% 4.59/2.29 % (10039)------------------------------
% 4.59/2.29 % (10039)------------------------------
% 4.86/2.31 % (10040)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_9905 for (2996ds/20Mi)
% 4.86/2.32 % (10040)Instruction limit reached!
% 4.86/2.32 % (10040)------------------------------
% 4.86/2.32 % (10040)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.86/2.32 % (10040)Termination reason: Unknown
% 4.86/2.32 % (10040)Termination phase: shuffling
% 4.86/2.32
% 4.86/2.32 % (10040)Memory used [KB]: 2046
% 4.86/2.32 % (10040)Time elapsed: 0.012 s
% 4.86/2.32 % (10040)Instructions burned: 20 (million)
% 4.86/2.32 % (10040)------------------------------
% 4.86/2.32 % (10040)------------------------------
% 4.86/2.33 % (10038)Instruction limit reached!
% 4.86/2.33 % (10038)------------------------------
% 4.86/2.33 % (10038)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.86/2.33 % (10038)Termination reason: Unknown
% 4.86/2.33 % (10038)Termination phase: Property scanning
% 4.86/2.33
% 4.86/2.33 % (10038)Memory used [KB]: 2814
% 4.86/2.33 % (10038)Time elapsed: 0.047 s
% 4.86/2.33 % (10038)Instructions burned: 100 (million)
% 4.86/2.33 % (10038)------------------------------
% 4.86/2.33 % (10038)------------------------------
% 4.86/2.34 % (10041)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_9905 for (2996ds/86Mi)
% 4.86/2.35 % (10042)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_9905 for (2996ds/107Mi)
% 4.86/2.38 % (10041)Instruction limit reached!
% 4.86/2.38 % (10041)------------------------------
% 4.86/2.38 % (10041)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.86/2.38 % (10041)Termination reason: Unknown
% 4.86/2.38 % (10041)Termination phase: Property scanning
% 4.86/2.38
% 4.86/2.38 % (10041)Memory used [KB]: 2942
% 4.86/2.38 % (10041)Time elapsed: 0.041 s
% 4.86/2.38 % (10041)Instructions burned: 87 (million)
% 4.86/2.38 % (10041)------------------------------
% 4.86/2.38 % (10041)------------------------------
% 4.86/2.39 % (10042)Instruction limit reached!
% 4.86/2.39 % (10042)------------------------------
% 4.86/2.39 % (10042)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.86/2.39 % (10042)Termination reason: Unknown
% 4.86/2.39 % (10042)Termination phase: SInE selection
% 4.86/2.39
% 4.86/2.39 % (10042)Memory used [KB]: 2942
% 4.86/2.39 % (10042)Time elapsed: 0.050 s
% 4.86/2.39 % (10042)Instructions burned: 107 (million)
% 4.86/2.39 % (10042)------------------------------
% 4.86/2.39 % (10042)------------------------------
% 4.86/2.39 % (10043)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_9905 for (2995ds/39Mi)
% 5.07/2.41 % (10044)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_9905 for (2995ds/448Mi)
% 5.07/2.41 % (10043)Instruction limit reached!
% 5.07/2.41 % (10043)------------------------------
% 5.07/2.41 % (10043)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.07/2.41 % (10043)Termination reason: Unknown
% 5.07/2.41 % (10043)Termination phase: shuffling
% 5.07/2.41
% 5.07/2.41 % (10043)Memory used [KB]: 2430
% 5.07/2.41 % (10043)Time elapsed: 0.020 s
% 5.07/2.41 % (10043)Instructions burned: 39 (million)
% 5.07/2.41 % (10043)------------------------------
% 5.07/2.41 % (10043)------------------------------
% 5.07/2.41 % (10024)Instruction limit reached!
% 5.07/2.41 % (10024)------------------------------
% 5.07/2.41 % (10024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.07/2.41 % (10024)Termination reason: Unknown
% 5.07/2.41 % (10024)Termination phase: Saturation
% 5.07/2.41
% 5.07/2.41 % (10024)Memory used [KB]: 11897
% 5.07/2.41 % (10024)Time elapsed: 0.353 s
% 5.07/2.41 % (10024)Instructions burned: 710 (million)
% 5.07/2.41 % (10024)------------------------------
% 5.07/2.41 % (10024)------------------------------
% 5.07/2.42 % (10031)Instruction limit reached!
% 5.07/2.42 % (10031)------------------------------
% 5.07/2.42 % (10031)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.07/2.42 % (10031)Termination reason: Unknown
% 5.07/2.42 % (10031)Termination phase: Saturation
% 5.07/2.42
% 5.07/2.42 % (10031)Memory used [KB]: 8059
% 5.07/2.42 % (10031)Time elapsed: 0.314 s
% 5.07/2.42 % (10031)Instructions burned: 781 (million)
% 5.07/2.42 % (10031)------------------------------
% 5.07/2.42 % (10031)------------------------------
% 5.07/2.43 % (10015)Instruction limit reached!
% 5.07/2.43 % (10015)------------------------------
% 5.07/2.43 % (10015)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.07/2.43 % (10015)Termination reason: Unknown
% 5.07/2.43 % (10015)Termination phase: Saturation
% 5.07/2.43
% 5.07/2.43 % (10015)Memory used [KB]: 17142
% 5.07/2.43 % (10015)Time elapsed: 0.410 s
% 5.07/2.43 % (10015)Instructions burned: 1042 (million)
% 5.07/2.43 % (10015)------------------------------
% 5.07/2.43 % (10015)------------------------------
% 5.07/2.43 % (10045)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_9905 for (2995ds/46Mi)
% 5.07/2.43 % (10046)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_9905 for (2995ds/98Mi)
% 5.07/2.44 % (10047)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_9905 for (2995ds/507Mi)
% 5.07/2.44 % (10048)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_9905 for (2995ds/149Mi)
% 5.07/2.45 % (10045)Instruction limit reached!
% 5.07/2.45 % (10045)------------------------------
% 5.07/2.45 % (10045)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.07/2.45 % (10045)Termination reason: Unknown
% 5.07/2.45 % (10045)Termination phase: shuffling
% 5.07/2.45
% 5.07/2.45 % (10045)Memory used [KB]: 2558
% 5.07/2.45 % (10045)Time elapsed: 0.024 s
% 5.07/2.45 % (10045)Instructions burned: 47 (million)
% 5.07/2.45 % (10045)------------------------------
% 5.07/2.45 % (10045)------------------------------
% 5.07/2.47 % (10049)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_9905 for (2994ds/39Mi)
% 5.07/2.48 % (10046)Instruction limit reached!
% 5.07/2.48 % (10046)------------------------------
% 5.07/2.48 % (10046)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.07/2.48 % (10046)Termination reason: Unknown
% 5.07/2.48 % (10046)Termination phase: Property scanning
% 5.07/2.48
% 5.07/2.48 % (10046)Memory used [KB]: 3198
% 5.07/2.48 % (10046)Time elapsed: 0.047 s
% 5.07/2.48 % (10046)Instructions burned: 100 (million)
% 5.07/2.48 % (10046)------------------------------
% 5.07/2.48 % (10046)------------------------------
% 5.07/2.49 % (10049)Instruction limit reached!
% 5.07/2.49 % (10049)------------------------------
% 5.07/2.49 % (10049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.07/2.49 % (10049)Termination reason: Unknown
% 5.07/2.49 % (10049)Termination phase: shuffling
% 5.07/2.49
% 5.07/2.49 % (10049)Memory used [KB]: 2430
% 5.07/2.49 % (10049)Time elapsed: 0.020 s
% 5.07/2.49 % (10049)Instructions burned: 39 (million)
% 5.07/2.49 % (10049)------------------------------
% 5.07/2.49 % (10049)------------------------------
% 5.07/2.49 % (10050)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_9905 for (2994ds/6Mi)
% 6.37/2.50 % (10050)Instruction limit reached!
% 6.37/2.50 % (10050)------------------------------
% 6.37/2.50 % (10050)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.50 % (10050)Termination reason: Unknown
% 6.37/2.50 % (10050)Termination phase: shuffling
% 6.37/2.50
% 6.37/2.50 % (10050)Memory used [KB]: 1791
% 6.37/2.50 % (10050)Time elapsed: 0.005 s
% 6.37/2.50 % (10050)Instructions burned: 6 (million)
% 6.37/2.50 % (10050)------------------------------
% 6.37/2.50 % (10050)------------------------------
% 6.37/2.50 % (10051)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_9905 for (2994ds/1087Mi)
% 6.37/2.51 % (10048)Instruction limit reached!
% 6.37/2.51 % (10048)------------------------------
% 6.37/2.51 % (10048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.51 % (10048)Termination reason: Unknown
% 6.37/2.51 % (10048)Termination phase: Function definition elimination
% 6.37/2.51
% 6.37/2.51 % (10048)Memory used [KB]: 3070
% 6.37/2.51 % (10048)Time elapsed: 0.069 s
% 6.37/2.51 % (10048)Instructions burned: 151 (million)
% 6.37/2.51 % (10048)------------------------------
% 6.37/2.51 % (10048)------------------------------
% 6.37/2.51 % (10052)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_9905 for (2994ds/47Mi)
% 6.37/2.53 % (10026)Instruction limit reached!
% 6.37/2.53 % (10026)------------------------------
% 6.37/2.53 % (10026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.53 % (10026)Termination reason: Unknown
% 6.37/2.53 % (10026)Termination phase: Saturation
% 6.37/2.53
% 6.37/2.53 % (10026)Memory used [KB]: 13560
% 6.37/2.53 % (10026)Time elapsed: 0.455 s
% 6.37/2.53 % (10026)Instructions burned: 903 (million)
% 6.37/2.53 % (10026)------------------------------
% 6.37/2.53 % (10026)------------------------------
% 6.37/2.53 % (10053)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_9905 for (2994ds/44Mi)
% 6.37/2.54 % (10052)Instruction limit reached!
% 6.37/2.54 % (10052)------------------------------
% 6.37/2.54 % (10052)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.54 % (10052)Termination reason: Unknown
% 6.37/2.54 % (10052)Termination phase: shuffling
% 6.37/2.54
% 6.37/2.54 % (10052)Memory used [KB]: 2558
% 6.37/2.54 % (10052)Time elapsed: 0.024 s
% 6.37/2.54 % (10052)Instructions burned: 48 (million)
% 6.37/2.54 % (10052)------------------------------
% 6.37/2.54 % (10052)------------------------------
% 6.37/2.54 % (10054)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_9905 for (2994ds/432Mi)
% 6.37/2.55 % (10053)Instruction limit reached!
% 6.37/2.55 % (10053)------------------------------
% 6.37/2.55 % (10053)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.55 % (10053)Termination reason: Unknown
% 6.37/2.55 % (10053)Termination phase: shuffling
% 6.37/2.55
% 6.37/2.55 % (10053)Memory used [KB]: 2558
% 6.37/2.55 % (10053)Time elapsed: 0.023 s
% 6.37/2.55 % (10053)Instructions burned: 45 (million)
% 6.37/2.55 % (10053)------------------------------
% 6.37/2.55 % (10053)------------------------------
% 6.37/2.55 % (10033)Instruction limit reached!
% 6.37/2.55 % (10033)------------------------------
% 6.37/2.55 % (10033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.55 % (10033)Termination reason: Unknown
% 6.37/2.55 % (10033)Termination phase: Saturation
% 6.37/2.55
% 6.37/2.55 % (10033)Memory used [KB]: 10874
% 6.37/2.55 % (10033)Time elapsed: 0.418 s
% 6.37/2.55 % (10033)Instructions burned: 879 (million)
% 6.37/2.55 % (10033)------------------------------
% 6.37/2.55 % (10033)------------------------------
% 6.37/2.55 % (10055)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_9905 for (2994ds/62Mi)
% 6.37/2.57 % (10056)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/31Mi)
% 6.37/2.57 % (10057)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/96Mi)
% 6.37/2.58 % (10056)Instruction limit reached!
% 6.37/2.58 % (10056)------------------------------
% 6.37/2.58 % (10056)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.58 % (10056)Termination reason: Unknown
% 6.37/2.58 % (10056)Termination phase: shuffling
% 6.37/2.58
% 6.37/2.58 % (10056)Memory used [KB]: 2430
% 6.37/2.58 % (10056)Time elapsed: 0.017 s
% 6.37/2.58 % (10056)Instructions burned: 32 (million)
% 6.37/2.58 % (10056)------------------------------
% 6.37/2.58 % (10056)------------------------------
% 6.37/2.58 % (10055)Instruction limit reached!
% 6.37/2.58 % (10055)------------------------------
% 6.37/2.58 % (10055)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.58 % (10055)Termination reason: Unknown
% 6.37/2.58 % (10055)Termination phase: shuffling
% 6.37/2.58
% 6.37/2.58 % (10055)Memory used [KB]: 2814
% 6.37/2.58 % (10055)Time elapsed: 0.030 s
% 6.37/2.58 % (10055)Instructions burned: 62 (million)
% 6.37/2.58 % (10055)------------------------------
% 6.37/2.58 % (10055)------------------------------
% 6.37/2.59 % (10044)Instruction limit reached!
% 6.37/2.59 % (10044)------------------------------
% 6.37/2.59 % (10044)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.37/2.59 % (10044)Termination reason: Unknown
% 6.37/2.59 % (10044)Termination phase: Saturation
% 6.37/2.59
% 6.37/2.59 % (10044)Memory used [KB]: 7931
% 6.37/2.59 % (10044)Time elapsed: 0.186 s
% 6.37/2.59 % (10044)Instructions burned: 448 (million)
% 6.37/2.59 % (10044)------------------------------
% 6.37/2.59 % (10044)------------------------------
% 7.73/2.60 % (10058)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/453Mi)
% 7.73/2.60 % (10059)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/71Mi)
% 7.73/2.61 % (10060)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/83Mi)
% 7.73/2.61 % (10057)Instruction limit reached!
% 7.73/2.61 % (10057)------------------------------
% 7.73/2.61 % (10057)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.61 % (10057)Termination reason: Unknown
% 7.73/2.61 % (10057)Termination phase: Property scanning
% 7.73/2.61
% 7.73/2.61 % (10057)Memory used [KB]: 2942
% 7.73/2.61 % (10057)Time elapsed: 0.046 s
% 7.73/2.61 % (10057)Instructions burned: 98 (million)
% 7.73/2.61 % (10057)------------------------------
% 7.73/2.61 % (10057)------------------------------
% 7.73/2.63 % (10061)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/11Mi)
% 7.73/2.63 % (10059)Instruction limit reached!
% 7.73/2.63 % (10059)------------------------------
% 7.73/2.63 % (10059)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.63 % (10059)Termination reason: Unknown
% 7.73/2.63 % (10059)Termination phase: Property scanning
% 7.73/2.63
% 7.73/2.63 % (10059)Memory used [KB]: 2942
% 7.73/2.63 % (10059)Time elapsed: 0.034 s
% 7.73/2.63 % (10059)Instructions burned: 72 (million)
% 7.73/2.63 % (10059)------------------------------
% 7.73/2.63 % (10059)------------------------------
% 7.73/2.63 % (10061)Instruction limit reached!
% 7.73/2.63 % (10061)------------------------------
% 7.73/2.63 % (10061)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.63 % (10061)Termination reason: Unknown
% 7.73/2.63 % (10061)Termination phase: shuffling
% 7.73/2.63
% 7.73/2.63 % (10061)Memory used [KB]: 1918
% 7.73/2.63 % (10061)Time elapsed: 0.007 s
% 7.73/2.63 % (10061)Instructions burned: 11 (million)
% 7.73/2.63 % (10061)------------------------------
% 7.73/2.63 % (10061)------------------------------
% 7.73/2.65 % (10062)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/41Mi)
% 7.73/2.65 % (10060)Instruction limit reached!
% 7.73/2.65 % (10060)------------------------------
% 7.73/2.65 % (10060)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.65 % (10060)Termination reason: Unknown
% 7.73/2.65 % (10060)Termination phase: Property scanning
% 7.73/2.65
% 7.73/2.65 % (10060)Memory used [KB]: 2814
% 7.73/2.65 % (10060)Time elapsed: 0.040 s
% 7.73/2.65 % (10060)Instructions burned: 83 (million)
% 7.73/2.65 % (10060)------------------------------
% 7.73/2.65 % (10060)------------------------------
% 7.73/2.65 % (10063)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/6Mi)
% 7.73/2.65 % (10063)Instruction limit reached!
% 7.73/2.65 % (10063)------------------------------
% 7.73/2.65 % (10063)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.65 % (10063)Termination reason: Unknown
% 7.73/2.65 % (10063)Termination phase: shuffling
% 7.73/2.65
% 7.73/2.65 % (10063)Memory used [KB]: 1791
% 7.73/2.65 % (10063)Time elapsed: 0.005 s
% 7.73/2.65 % (10063)Instructions burned: 7 (million)
% 7.73/2.65 % (10063)------------------------------
% 7.73/2.65 % (10063)------------------------------
% 7.73/2.67 % (10064)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_9905 for (2993ds/150Mi)
% 7.73/2.67 % (10062)Instruction limit reached!
% 7.73/2.67 % (10062)------------------------------
% 7.73/2.67 % (10062)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.67 % (10062)Termination reason: Unknown
% 7.73/2.67 % (10062)Termination phase: shuffling
% 7.73/2.67
% 7.73/2.67 % (10062)Memory used [KB]: 2558
% 7.73/2.67 % (10062)Time elapsed: 0.021 s
% 7.73/2.67 % (10062)Instructions burned: 42 (million)
% 7.73/2.67 % (10062)------------------------------
% 7.73/2.67 % (10062)------------------------------
% 7.73/2.67 % (10065)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_9905 for (2992ds/343Mi)
% 7.73/2.67 % (10047)Instruction limit reached!
% 7.73/2.67 % (10047)------------------------------
% 7.73/2.67 % (10047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.73/2.67 % (10047)Termination reason: Unknown
% 7.73/2.67 % (10047)Termination phase: Saturation
% 7.73/2.67
% 7.73/2.67 % (10047)Memory used [KB]: 8443
% 7.73/2.67 % (10047)Time elapsed: 0.239 s
% 7.73/2.67 % (10047)Instructions burned: 508 (million)
% 7.73/2.67 % (10047)------------------------------
% 7.73/2.67 % (10047)------------------------------
% 7.73/2.68 % (10066)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_9905 for (2992ds/109Mi)
% 7.73/2.69 % (10067)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_9905 for (2992ds/253Mi)
% 8.23/2.72 % (10054)Instruction limit reached!
% 8.23/2.72 % (10054)------------------------------
% 8.23/2.72 % (10054)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.23/2.72 % (10054)Termination reason: Unknown
% 8.23/2.72 % (10054)Termination phase: Saturation
% 8.23/2.72
% 8.23/2.72 % (10054)Memory used [KB]: 3454
% 8.23/2.72 % (10054)Time elapsed: 0.181 s
% 8.23/2.72 % (10054)Instructions burned: 433 (million)
% 8.23/2.72 % (10054)------------------------------
% 8.23/2.72 % (10054)------------------------------
% 8.23/2.73 % (10066)Instruction limit reached!
% 8.23/2.73 % (10066)------------------------------
% 8.23/2.73 % (10066)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.23/2.73 % (10066)Termination reason: Unknown
% 8.23/2.73 % (10066)Termination phase: SInE selection
% 8.23/2.73
% 8.23/2.73 % (10066)Memory used [KB]: 2942
% 8.23/2.73 % (10066)Time elapsed: 0.052 s
% 8.23/2.73 % (10066)Instructions burned: 109 (million)
% 8.23/2.73 % (10066)------------------------------
% 8.23/2.73 % (10066)------------------------------
% 8.23/2.74 % (10064)Instruction limit reached!
% 8.23/2.74 % (10064)------------------------------
% 8.23/2.74 % (10064)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.23/2.74 % (10064)Termination reason: Unknown
% 8.23/2.74 % (10064)Termination phase: Preprocessing 3
% 8.23/2.74
% 8.23/2.74 % (10064)Memory used [KB]: 5373
% 8.23/2.74 % (10064)Time elapsed: 0.075 s
% 8.23/2.74 % (10064)Instructions burned: 151 (million)
% 8.23/2.74 % (10064)------------------------------
% 8.23/2.74 % (10064)------------------------------
% 8.23/2.74 % (10068)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_9905 for (2992ds/140Mi)
% 8.23/2.74 % (10051)First to succeed.
% 8.23/2.75 % (10069)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_9905 for (2992ds/614Mi)
% 8.23/2.75 % (10051)Refutation found. Thanks to Tanya!
% 8.23/2.75 % SZS status Theorem for DTF2THF_9905
% 8.23/2.75 % SZS output start Proof for DTF2THF_9905
% 8.23/2.75 thf(func_def_0, type, in: $i > $i > $o).
% 8.23/2.75 thf(func_def_1, type, exu: ($i > $o) > $o).
% 8.23/2.75 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 8.23/2.75 thf(func_def_8, type, powerset: $i > $i).
% 8.23/2.75 thf(func_def_10, type, setunion: $i > $i).
% 8.23/2.75 thf(func_def_19, type, descr: ($i > $o) > $i).
% 8.23/2.75 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_26, type, prop2set: $o > $i).
% 8.23/2.75 thf(func_def_36, type, nonempty: $i > $o).
% 8.23/2.75 thf(func_def_69, type, set2prop: $i > $o).
% 8.23/2.75 thf(func_def_88, type, subset: $i > $i > $o).
% 8.23/2.75 thf(func_def_89, type, disjoint: $i > $i > $o).
% 8.23/2.75 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 8.23/2.75 thf(func_def_114, type, binunion: $i > $i > $i).
% 8.23/2.75 thf(func_def_122, type, binintersect: $i > $i > $i).
% 8.23/2.75 thf(func_def_135, type, regular: $i > $o).
% 8.23/2.75 thf(func_def_136, type, setminus: $i > $i > $i).
% 8.23/2.75 thf(func_def_147, type, symdiff: $i > $i > $i).
% 8.23/2.75 thf(func_def_153, type, iskpair: $i > $o).
% 8.23/2.75 thf(func_def_158, type, kpair: $i > $i > $i).
% 8.23/2.75 thf(func_def_160, type, cartprod: $i > $i > $i).
% 8.23/2.75 thf(func_def_177, type, singleton: $i > $o).
% 8.23/2.75 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 8.23/2.75 thf(func_def_184, type, atmost1p: $i > $o).
% 8.23/2.75 thf(func_def_185, type, atleast2p: $i > $o).
% 8.23/2.75 thf(func_def_186, type, atmost2p: $i > $o).
% 8.23/2.75 thf(func_def_187, type, upairsetp: $i > $o).
% 8.23/2.75 thf(func_def_191, type, kfst: $i > $i).
% 8.23/2.75 thf(func_def_203, type, ksnd: $i > $i).
% 8.23/2.75 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 8.23/2.75 thf(func_def_222, type, func: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_223, type, funcSet: $i > $i > $i).
% 8.23/2.75 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 8.23/2.75 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 8.23/2.75 thf(func_def_323, type, breln1: $i > $i > $o).
% 8.23/2.75 thf(func_def_325, type, breln1Set: $i > $i).
% 8.23/2.75 thf(func_def_327, type, transitive: $i > $i > $o).
% 8.23/2.75 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 8.23/2.75 thf(func_def_329, type, reflexive: $i > $i > $o).
% 8.23/2.75 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 8.23/2.75 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 8.23/2.75 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 8.23/2.75 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 8.23/2.75 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 8.23/2.75 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 8.23/2.75 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 8.23/2.75 thf(func_def_378, type, omegaS: $i > $i).
% 8.23/2.75 thf(func_def_382, type, peano3: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_383, type, peano4: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_384, type, peano5: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_386, type, peano: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_387, type, transitiveset: $i > $o).
% 8.23/2.75 thf(func_def_392, type, stricttotalorderedByIn: $i > $o).
% 8.23/2.75 thf(func_def_393, type, wellorderedByIn: $i > $o).
% 8.23/2.75 thf(func_def_394, type, ordinal: $i > $o).
% 8.23/2.75 thf(func_def_395, type, limitOrdinal: $i > $o).
% 8.23/2.75 thf(func_def_413, type, sP0: $i > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_414, type, sP1: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_415, type, sP2: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_417, type, sP4: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_418, type, sP5: $i > $o > $i > $o).
% 8.23/2.75 thf(func_def_420, type, sP7: $i > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_421, type, sP8: $i > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_423, type, sP10: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_424, type, sP11: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_427, type, sP14: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_428, type, sP15: $i > ($i > $o) > $o).
% 8.23/2.75 thf(func_def_430, type, sP17: ($i > $o) > $o).
% 8.23/2.75 thf(func_def_431, type, sP18: $i > ($i > $i) > $i > $o).
% 8.23/2.75 thf(func_def_433, type, sP20: ($i > $o) > $o).
% 8.23/2.75 thf(func_def_434, type, sP21: $i > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_435, type, sP22: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_436, type, sP23: $i > $i > ($i > $i) > $o).
% 8.23/2.75 thf(func_def_438, type, sP25: $i > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_439, type, sP26: ($i > $i) > $i > $i > $o).
% 8.23/2.75 thf(func_def_442, type, sP29: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_445, type, sP32: $i > $i > $o).
% 8.23/2.75 thf(func_def_447, type, sP34: ($i > $o) > ($i > $o) > $i > $i > $o).
% 8.23/2.75 thf(func_def_448, type, sP35: $o > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_449, type, sP36: $i > $o > $i > $i > $o).
% 8.23/2.75 thf(func_def_451, type, sP38: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_454, type, sP41: $i > $o).
% 8.23/2.75 thf(func_def_455, type, sP42: $i > $i > $o).
% 8.23/2.75 thf(func_def_457, type, sP44: $i > $i > $o).
% 8.23/2.75 thf(func_def_459, type, sP46: $i > ($i > $o) > $o).
% 8.23/2.75 thf(func_def_460, type, sP47: $i > ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_461, type, sP48: $i > $i > $o > $o).
% 8.23/2.75 thf(func_def_462, type, sP49: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_463, type, sP50: $i > $o).
% 8.23/2.75 thf(func_def_464, type, sP51: $i > $o).
% 8.23/2.75 thf(func_def_465, type, sP52: ($i > $o) > $o).
% 8.23/2.75 thf(func_def_466, type, sP53: ($i > $i > $o) > $i > $i > $o).
% 8.23/2.75 thf(func_def_467, type, sP54: ($i > $i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_470, type, sP57: $o > $i > $i > $o).
% 8.23/2.75 thf(func_def_472, type, sP59: ($i > $o) > $o).
% 8.23/2.75 thf(func_def_473, type, sP60: ($i > $i) > $i > $i > $o).
% 8.23/2.75 thf(func_def_475, type, sP62: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_476, type, sP63: $i > $i > ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_477, type, sP64: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_478, type, sP65: $i > ($i > $i) > $i > $o).
% 8.23/2.75 thf(func_def_480, type, sP67: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_481, type, sP68: $i > ($i > $o) > $o).
% 8.23/2.75 thf(func_def_484, type, sP71: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_485, type, sP72: $i > $i > ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_486, type, sP73: $i > ($i > $i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_487, type, sP74: $i > ($i > $i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_488, type, sP75: $i > $i > $i > $i > $i > $o > $o).
% 8.23/2.75 thf(func_def_489, type, sP76: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_491, type, sP78: ($i > $o) > $o).
% 8.23/2.75 thf(func_def_492, type, sP79: $o > $i > $i > $o).
% 8.23/2.75 thf(func_def_493, type, sP80: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_494, type, sP81: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_495, type, sP82: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_496, type, sP83: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_497, type, sP84: $i > $i > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_500, type, sP87: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_501, type, sP88: $i > $i > $o).
% 8.23/2.75 thf(func_def_502, type, sP89: $i > $i > $o).
% 8.23/2.75 thf(func_def_503, type, sP90: $i > $o).
% 8.23/2.75 thf(func_def_504, type, sP91: $i > $i > $o).
% 8.23/2.75 thf(func_def_505, type, sP92: $i > $i > $o).
% 8.23/2.75 thf(func_def_506, type, sP93: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_507, type, sP94: $i > $i > ($i > $o) > $o).
% 8.23/2.75 thf(func_def_509, type, sP96: $o > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_510, type, sP97: $i > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_511, type, sP98: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_512, type, sP99: $i > ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_513, type, sP100: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_515, type, sP102: $i > ($i > $o) > $o).
% 8.23/2.75 thf(func_def_517, type, sP104: ($i > $o) > $i > $o).
% 8.23/2.75 thf(func_def_519, type, sP106: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_521, type, sK108: $i > $o).
% 8.23/2.75 thf(func_def_535, type, sK122: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_556, type, sK143: $i > $i).
% 8.23/2.75 thf(func_def_558, type, sK145: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_559, type, sK146: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_560, type, sK147: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_561, type, sK148: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_569, type, sK156: $i > $i).
% 8.23/2.75 thf(func_def_570, type, sK157: ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_580, type, sK167: $i > $i > $o).
% 8.23/2.75 thf(func_def_581, type, sK168: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_589, type, sK176: $i > $o > $i > $i).
% 8.23/2.75 thf(func_def_612, type, sK199: $i > $o).
% 8.23/2.75 thf(func_def_614, type, sK201: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_623, type, sK210: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_624, type, sK211: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_625, type, sK212: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_626, type, sK213: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_637, type, sK224: $i > $i).
% 8.23/2.75 thf(func_def_638, type, sK225: ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_644, type, sK231: $i > $i > $o).
% 8.23/2.75 thf(func_def_656, type, sK243: $i > $o).
% 8.23/2.75 thf(func_def_662, type, sK249: $i > $o).
% 8.23/2.75 thf(func_def_666, type, sK253: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_667, type, sK254: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_668, type, sK255: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_670, type, sK257: $i > $o).
% 8.23/2.75 thf(func_def_679, type, sK266: $i > $o).
% 8.23/2.75 thf(func_def_680, type, sK267: $i > $o).
% 8.23/2.75 thf(func_def_683, type, sK270: $i > $i > $i).
% 8.23/2.75 thf(func_def_693, type, sK280: $i > $i > $i).
% 8.23/2.75 thf(func_def_696, type, sK283: $i > $i).
% 8.23/2.75 thf(func_def_703, type, sK290: $i > $i > $o).
% 8.23/2.75 thf(func_def_707, type, sK294: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_708, type, sK295: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_716, type, sK303: ($i > $o) > $i).
% 8.23/2.75 thf(func_def_717, type, sK304: $i > $o).
% 8.23/2.75 thf(func_def_719, type, sK306: $i > $o).
% 8.23/2.75 thf(func_def_720, type, sK307: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_721, type, sK308: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_746, type, sK333: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_747, type, sK334: ($i > $o) > $i).
% 8.23/2.75 thf(func_def_748, type, sK335: $i > $o).
% 8.23/2.75 thf(func_def_751, type, sK338: $i > $i).
% 8.23/2.75 thf(func_def_754, type, sK341: $i > ($i > $i) > $i > $i).
% 8.23/2.75 thf(func_def_760, type, sK347: ($i > $o) > $i).
% 8.23/2.75 thf(func_def_761, type, sK348: ($i > $o) > $i).
% 8.23/2.75 thf(func_def_762, type, sK349: $i > $o).
% 8.23/2.75 thf(func_def_764, type, sK351: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_765, type, sK352: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_766, type, sK353: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_767, type, sK354: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_783, type, sK370: $i > $i).
% 8.23/2.75 thf(func_def_784, type, sK371: ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_785, type, sK372: $i > $i > ($i > $i) > $i).
% 8.23/2.75 thf(func_def_794, type, sK381: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_795, type, sK382: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_809, type, sK396: $i > $o).
% 8.23/2.75 thf(func_def_814, type, sK401: $i > $i > $o).
% 8.23/2.75 thf(func_def_831, type, sK418: $i > $i).
% 8.23/2.75 thf(func_def_832, type, sK419: ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_833, type, sK420: ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_834, type, sK421: $i > ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_841, type, sK428: $i > $i > $i).
% 8.23/2.75 thf(func_def_850, type, sK437: $i > $o).
% 8.23/2.75 thf(func_def_852, type, sK439: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_853, type, sK440: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_858, type, sK445: $i > $i).
% 8.23/2.75 thf(func_def_859, type, sK446: ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_862, type, sK449: $i > $i > $i).
% 8.23/2.75 thf(func_def_863, type, sK450: $i > $i > $i).
% 8.23/2.75 thf(func_def_869, type, sK456: ($i > $o) > ($i > $o) > $i > $i > $i).
% 8.23/2.75 thf(func_def_870, type, sK457: ($i > $o) > ($i > $o) > $i > $i > $i).
% 8.23/2.75 thf(func_def_873, type, sK460: $i > $o).
% 8.23/2.75 thf(func_def_874, type, sK461: $i > $o).
% 8.23/2.75 thf(func_def_888, type, sK475: $i > $i).
% 8.23/2.75 thf(func_def_890, type, sK477: $i > ($i > $i) > $i > $i).
% 8.23/2.75 thf(func_def_892, type, sK479: $i > $o).
% 8.23/2.75 thf(func_def_893, type, sK480: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_894, type, sK481: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_909, type, sK496: $i > $i).
% 8.23/2.75 thf(func_def_927, type, sK514: $i > $i > $i).
% 8.23/2.75 thf(func_def_928, type, sK515: $i > $i).
% 8.23/2.75 thf(func_def_945, type, sK532: $i > $i > $i).
% 8.23/2.75 thf(func_def_948, type, sK535: $i > $i > $i).
% 8.23/2.75 thf(func_def_956, type, sK543: $i > ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_957, type, sK544: $i > ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_958, type, sK545: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_961, type, sK548: $i > $o).
% 8.23/2.75 thf(func_def_965, type, sK552: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_991, type, sK578: $i > $o).
% 8.23/2.75 thf(func_def_1007, type, sK594: $i > $i).
% 8.23/2.75 thf(func_def_1008, type, sK595: $i > $i).
% 8.23/2.75 thf(func_def_1014, type, sK601: ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1015, type, sK602: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1016, type, sK603: $i > $o).
% 8.23/2.75 thf(func_def_1032, type, sK619: $i > $i > $o).
% 8.23/2.75 thf(func_def_1053, type, sK640: $i > $o).
% 8.23/2.75 thf(func_def_1057, type, sK644: $i > $i).
% 8.23/2.75 thf(func_def_1059, type, sK646: $i > $i > $o).
% 8.23/2.75 thf(func_def_1060, type, sK647: ($i > $i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1061, type, sK648: $i > ($i > $i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1062, type, sK649: ($i > $i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1063, type, sK650: ($i > $i > $o) > $i > $i > $i).
% 8.23/2.75 thf(func_def_1084, type, sK671: $i > $i > $i).
% 8.23/2.75 thf(func_def_1118, type, sK705: ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1119, type, sK706: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1120, type, sK707: $i > $o).
% 8.23/2.75 thf(func_def_1146, type, sK733: $i > $o).
% 8.23/2.75 thf(func_def_1148, type, sK735: $i > $i).
% 8.23/2.75 thf(func_def_1149, type, sK736: $i > $i).
% 8.23/2.75 thf(func_def_1150, type, sK737: ($i > $i) > $i > $i).
% 8.23/2.75 thf(func_def_1151, type, sK738: ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_1159, type, sK746: $i > $i > ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1160, type, sK747: $i > $i > ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1161, type, sK748: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1165, type, sK752: $i > $o).
% 8.23/2.75 thf(func_def_1172, type, sK759: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1173, type, sK760: $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1182, type, sK769: $i > $i).
% 8.23/2.75 thf(func_def_1183, type, sK770: ($i > $i) > $i > $i > $i).
% 8.23/2.75 thf(func_def_1184, type, sK771: $i > ($i > $i) > $i > $i).
% 8.23/2.75 thf(func_def_1192, type, sK779: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1193, type, sK780: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1194, type, sK781: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1196, type, sK783: $i > $o).
% 8.23/2.75 thf(func_def_1220, type, sK807: $i > $i > $i).
% 8.23/2.75 thf(func_def_1242, type, sK829: $i > $i).
% 8.23/2.75 thf(func_def_1244, type, sK831: $i > ($i > $i) > $i > $i).
% 8.23/2.75 thf(func_def_1245, type, sK832: $i > $i > $o).
% 8.23/2.75 thf(func_def_1273, type, sK860: $i > $i).
% 8.23/2.75 thf(func_def_1303, type, sK890: $i > $i > ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1304, type, sK891: $i > $i > ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1305, type, sK892: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1309, type, sK896: $i > $o).
% 8.23/2.75 thf(func_def_1317, type, sK904: $i > $i > ($i > $i > $o) > $i).
% 8.23/2.75 thf(func_def_1318, type, sK905: $i > ($i > $i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1319, type, sK906: $i > ($i > $i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1320, type, sK907: $i > ($i > $i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1321, type, sK908: $i > $i > $o).
% 8.23/2.75 thf(func_def_1331, type, sK918: $i > $i > $o).
% 8.23/2.75 thf(func_def_1336, type, sK923: $i > $i > $i > $i > $i > $o > $i).
% 8.23/2.75 thf(func_def_1351, type, sK938: $i > $o).
% 8.23/2.75 thf(func_def_1353, type, sK940: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1354, type, sK941: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1359, type, sK946: $i > $o).
% 8.23/2.75 thf(func_def_1368, type, sK955: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1369, type, sK956: ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1370, type, sK957: $i > $o).
% 8.23/2.75 thf(func_def_1371, type, sK958: $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_1372, type, sK959: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1376, type, sK963: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1377, type, sK964: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1392, type, sK979: $i > $i).
% 8.23/2.75 thf(func_def_1395, type, sK982: $i > $i).
% 8.23/2.75 thf(func_def_1409, type, sK996: $i > $o).
% 8.23/2.75 thf(func_def_1414, type, sK1001: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1415, type, sK1002: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1435, type, sK1022: $i > $i > $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1442, type, sK1029: $i > $o).
% 8.23/2.75 thf(func_def_1443, type, sK1030: $i > $o).
% 8.23/2.75 thf(func_def_1444, type, sK1031: ($i > $o) > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1445, type, sK1032: ($i > $o) > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1449, type, sK1036: $i > $o).
% 8.23/2.75 thf(func_def_1450, type, sK1037: $i > $o).
% 8.23/2.75 thf(func_def_1451, type, sK1038: ($i > $o) > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1452, type, sK1039: ($i > $o) > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1453, type, sK1040: $i > $i > $i).
% 8.23/2.75 thf(func_def_1454, type, sK1041: $i > $i > $i).
% 8.23/2.75 thf(func_def_1455, type, sK1042: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1456, type, sK1043: $i > $i > $i).
% 8.23/2.75 thf(func_def_1457, type, sK1044: $i > $i > $i).
% 8.23/2.75 thf(func_def_1458, type, sK1045: $i > $i).
% 8.23/2.75 thf(func_def_1459, type, sK1046: $i > $i).
% 8.23/2.75 thf(func_def_1460, type, sK1047: $i > $i).
% 8.23/2.75 thf(func_def_1461, type, sK1048: $i > $i).
% 8.23/2.75 thf(func_def_1462, type, sK1049: $i > $i > $i).
% 8.23/2.75 thf(func_def_1463, type, sK1050: $i > $i > $i).
% 8.23/2.75 thf(func_def_1464, type, sK1051: $i > $i > $i).
% 8.23/2.75 thf(func_def_1465, type, sK1052: $i > $i > $i).
% 8.23/2.75 thf(func_def_1466, type, sK1053: $i > $i > $i).
% 8.23/2.75 thf(func_def_1467, type, sK1054: $i > $i > $i).
% 8.23/2.75 thf(func_def_1468, type, sK1055: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1469, type, sK1056: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1471, type, sK1058: $i > $i).
% 8.23/2.75 thf(func_def_1474, type, sK1061: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1475, type, sK1062: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1486, type, sK1073: $i > $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1488, type, sK1075: $i > $o).
% 8.23/2.75 thf(func_def_1494, type, sK1081: $i > $i > $i).
% 8.23/2.75 thf(func_def_1500, type, sK1087: $i > $i > $i > $i > $o).
% 8.23/2.75 thf(func_def_1511, type, sK1098: $i > $i).
% 8.23/2.75 thf(func_def_1512, type, sK1099: $i > ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1513, type, sK1100: $i > ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1514, type, sK1101: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1517, type, sK1104: $i > $o).
% 8.23/2.75 thf(func_def_1522, type, sK1109: $i > $o).
% 8.23/2.75 thf(func_def_1525, type, sK1112: $i > $o).
% 8.23/2.75 thf(func_def_1526, type, sK1113: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1527, type, sK1114: $i > ($i > $o) > $i).
% 8.23/2.75 thf(func_def_1532, type, sK1119: $i > $o).
% 8.23/2.75 thf(func_def_1533, type, sK1120: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1534, type, sK1121: ($i > $o) > $i > $i).
% 8.23/2.75 thf(func_def_1542, type, sK1129: $i > $o).
% 8.23/2.75 thf(func_def_1559, type, sK1146: $i > $i > $i > $i).
% 8.23/2.75 thf(func_def_1584, type, sK1171: $i > $i > $o).
% 8.23/2.75 thf(func_def_1608, type, ph1195: !>[X0: $tType]:(X0)).
% 8.23/2.75 thf(f8599,plain,(
% 8.23/2.75 $false),
% 8.23/2.75 inference(subsumption_resolution,[status(thm)],[f8598,f3827])).
% 8.23/2.75 thf(f3827,plain,(
% 8.23/2.75 ((in @ sK142 @ sK141) = $true)),
% 8.23/2.75 inference(cnf_transformation,[status(thm)],[f1716])).
% 8.23/2.75 thf(f1716,plain,(
% 8.23/2.75 (notinself = $true) & (breln1unionIL = $true) & (powersetAx = $true) & (omegaSAx = $true) & (setukpairinjL = $true) & (setadjoinSub = $true) & (binintersectSubset1 = $true) & (ifSingleton = $true) & (setadjoinIL = $true) & (binintersectER = $true) & (image1Ex = $true) & (descr__Cong = $true) & (setunionsingleton2 = $true) & (powersetTE1 = $true) & (cartprodmempair1 = $true) & (omega0Ax = $true) & (setukpairinjR2 = $true) & (ordinalTransSet = $true) & (setunionE = $true) & (upairsetE = $true) & (theeq = $true) & (subsetTI = $true) & (funcext = $true) & (setminusERneg = $true) & (wellorderingAx = $true) & (setukpairIL = $true) & (setunionI = $true) & (ap2apEq2 = $true) & (choice2fnsingleton = $true) & (powersetTI1 = $true) & (setadjoinSub2 = $true) & (notinemptyset = $true) & (binunionRsub = $true) & (cartprodpairsurjEq = $true) & (emptysetAx = $true) & (cartprodmempair = $true) & (cartprodfstpairEq = $true) & (cartprodmempaircEq = $true) & (setukpairinjL1 = $true) & (breln1unionI = $true) & (surjFuncSetFuncSurj = $true) & (singletonsuniq = $true) & (cartprodsndin = $true) & (complementUnionInPowersetComplement = $true) & (ubforcartprodlem1 = $true) & (ordinalMinLem1 = $true) & (breln1compprop = $true) & (subPowSU = $true) & (complementTnotintersectT = $true) & (singletonsubset = $true) & (woz13rule2 = $true) & (surjFuncSetFuncIn = $true) & (binintersectSubset5 = $true) & (iffalseProp1 = $true) & (binunionE = $true) & (exuI3 = $true) & (breln1invI = $true) & (doubleComplementSub1 = $true) & (foundation2 = $true) & (ex1I = $true) & (demorgan2a = $true) & (eqimpsubset2 = $true) & (setoftrueEq = $true) & (eqbreln = $true) & (vacuousDall = $true) & (cartprodpairmemER = $true) & (setminusLsub = $true) & (funcGraphProp2 = $true) & (inCongP = $true) & (contrasubsetT3 = $true) & (bs114d = $true) & (binintTransitive = $true) & (notdexE = $true) & (demorgan2 = $true) & (setukpairinjL2 = $true) & (dsetconstrI = $true) & (binunionIL = $true) & (setextAx = $true) & (upairequniteq = $true) & (inComplementUnionImpNotIn1 = $true) & (setunionTransitive = $true) & (subbreln1 = $true) & (demorgan1a = $true) & (exuE3e = $true) & (ubforcartprodlem2 = $true) & (upairsetIL = $true) & (dpsetconstrER = $true) & (eqimpsubset1 = $true) & (omegaSp = $true) & (binunionTILcontra = $true) & (binunionTIRcontra = $true) & (quantDeMorgan1 = $true) & (woz1_3 = $true) & (emptyinunitempty = $true) & (powersetI = $true) & (ksndsingleton = $true) & (setminusER = $true) & (quantDeMorgan3 = $true) & (brelnall1 = $true) & (setminusI = $true) & (sepInPowerset = $true) & (nonemptyE1 = $true) & (iffalseProp2 = $true) & (dsetconstrEL = $true) & (emptyI = $true) & (symdiffI2 = $true) & (binunionEcases = $true) & (kfstpairEq = $true) & (notequalI1 = $true) & (powersetE = $true) & (woz1_1 = $true) & (symdiffI1 = $true) & (iffalse = $true) & (nonemptyI1 = $true) & (lamProp = $true) & (setminusIRneg = $true) & (breln1invE = $true) & (singletonsswitch = $true) & (ex1I2 = $true) & (setextsub = $true) & (omega__Cong = $true) & (transitivesetOp2 = $true) & (singletoninpowunion = $true) & (binintersectEL = $true) & (($true = (ordinal @ sK141)) & (((in @ sK142 @ sK141) = $true) & ((subset @ sK142 @ sK141) != $true))) & (exu__Cong = $true) & (funcGraphProp3 = $true) & (funcextLem = $true) & (kpairp = $true) & (breln1compEex = $true) & (eqbreln1 = $true) & (setadjoinE = $true) & (binintersectSubset2 = $true) & (ex1E2 = $true) & (binunionT_lem = $true) & (beta1 = $true) & (breln1unionCommutes = $true) & (replAx = $true) & (sepSubset = $true) & (breln1invprop = $true) & (woz13rule4 = $true) & (demorgan1b = $true) & (setunionsingleton1 = $true) & (in__Cong = $true) & (binunionIR = $true) & (funcext2 = $true) & (prop2setI = $true) & (funcGraphProp4 = $true) & (powersetT_lem = $true) & (surjCantorThm = $true) & (apProp = $true) & (inIntersectImpInUnion2 = $true) & (binunionTE = $true) & (complementTcontraSubset = $true) & (binintersectSubset4 = $true) & (intersectInPowersetIntersectUnions = $true) & (setunionE2 = $true) & (powerset__Cong = $true) & (breln1unionIR = $true) & (injFuncSetFuncIn = $true) & (setbeta = $true) & (binunionLsub = $true) & (ap2p = $true) & (dpsetconstrI = $true) & (complementT_lem = $true) & (subbreln = $true) & (subsetemptysetimpeq = $true) & (breln1compE = $true) & (setukpairinjR12 = $true) & (subsetI1 = $true) & (lam2lamEq = $true) & (inComplementUnionImpInComplement1 = $true) & (omegaIndAx = $true) & (iftrueorfalse = $true) & (complementImpComplementIntersect = $true) & (breln1all2 = $true) & (complementSubsetComplementIntersect = $true) & (ap2apEq1 = $true) & (cartprodpairmemEL = $true) & (iftrue = $true) & (exuE1 = $true) & (breln1SetBreln1 = $true) & (setadjoinOr = $true) & (binintersectI = $true) & (dpsetconstrERa = $true) & (setadjoinAx = $true) & (exuE2 = $true) & (setukpairinjR11 = $true) & (demorgan2a1 = $true) & (contraSubsetComplement = $true) & (dsetconstrER = $true) & (inIntersectImpInIntersectUnions = $true) & (woz2A = $true) & (binintersectTERcontra = $true) & (setOfPairsIsBReln1 = $true) & (woz1_4 = $true) & (exuI1 = $true) & (iftrueProp2 = $true) & (subsetE2 = $true) & (woz2B = $true) & (funcGraphProp1 = $true) & (contrasubsetT = $true) & (setminusILneg = $true) & (upairinpowunion = $true) & (leftInvIsSurj = $true) & (doubleComplementE1 = $true) & (lamp = $true) & (emptysetE = $true) & (demorgan2a2 = $true) & (exuE3u = $true) & (upairset2E = $true) & (woz13rule0 = $true) & (binintersectT_lem = $true) & (beta2 = $true) & (breln1unionE = $true) & (symdiffIneg1 = $true) & (inPowerset = $true) & (image1I = $true) & (binintersectRsub = $true) & (breln1all1 = $true) & (notdallE = $true) & (demorgan2b = $true) & (theprop = $true) & (contrasubsetT2 = $true) & (setminusSubset2 = $true) & (setminusELneg = $true) & (descrp = $true) & (demorgan2b2 = $true) & (demorgan1 = $true) & (subset2powerset = $true) & (exuEu = $true) & (breln1unionEcases = $true) & (omegaSclos = $true) & (setminusEL = $true) & (woz13rule3 = $true) & (dsetconstr__Cong = $true) & (notequalI2 = $true) & (symdiffE = $true) & (setminusT_lem = $true) & (image1Ex1 = $true) & (dpsetconstrEL1 = $true) & (setOfPairsIsBReln = $true) & (funcinfuncset = $true) & (iftrueProp1 = $true) & (binintersectTELcontra = $true) & (exuI2 = $true) & (eta1 = $true) & (setext = $true) & (setunion__Cong = $true) & (transitivesetOp1 = $true) & (emptyInPowerset = $true) & (setadjoinIR = $true) & (inIntersectImpInUnion = $true) & (upairsetIR = $true) & (disjointsetsI1 = $true) & (nonemptyI = $true) & (kpairiskpair = $true) & (kfstsingleton = $true) & (injFuncInInjFuncSet = $true) & (woz1_5 = $true) & (setunionsingleton = $true) & (kpairsurjEq = $true) & (ubforcartprodlem3 = $true) & (ex1E1 = $true) & (image1E = $true) & (binunionTEcontra = $true) & (injFuncSetFuncInj = $true) & (peano0notS = $true) & (subsetTrans = $true) & (funcImageSingleton = $true) & (contrasubsetT1 = $true) & (dpsetconstrSub = $true) & (notinsingleton = $true) & (quantDeMorgan4 = $true) & (woz1_2 = $true) & (notsubsetI = $true) & (singletoninpowerset = $true) & (setukpairIR = $true) & (lam2p = $true) & (complementTI1 = $true) & (complementInPowersetComplementIntersect = $true) & (noeltsimpempty = $true) & (foundationAx = $true) & (eqinunit = $true) & (emptyinPowerset = $true) & (powersetE1 = $true) & (powersetI1 = $true) & (setukpairinjR1 = $true) & (setunionAx = $true) & (setukpairinjR = $true) & (cartprodfstin = $true) & (emptysetsubset = $true) & (peanoSinj = $true) & (woz13rule1 = $true) & (breln1unionprop = $true) & (cartprodsndpairEq = $true) & (singletonprop = $true) & (setminusSubset1 = $true) & (doubleComplementSub2 = $true) & (cartprodpairin = $true) & (woz2W = $true) & (doubleComplementEq = $true) & (upairsubunion = $true) & (prop2set2propI = $true) & (ifp = $true) & (upairset2IR = $true) & (eta2 = $true) & (subsetRefl = $true) & (setadjoin__Cong = $true) & (emptysetimpfalse = $true) & (binintersectSubset3 = $true) & (breln1compI = $true) & (powersetsubset = $true) & (uniqinunit = $true) & (complementTE1 = $true) & (emptyE1 = $true) & (nonemptyImpWitness = $true) & (subsetE = $true) & (app = $true) & (ksndpairEq = $true) & (infuncsetfunc = $true) & (woz2Ex = $true) & (setextT = $true) & (symdiffIneg2 = $true) & (quantDeMorgan2 = $true) & (brelnall2 = $true) & (subsetI2 = $true) & (dpsetconstrEL2 = $true) & (doubleComplementI1 = $true) & (image1Equiv = $true) & (notinself2 = $true) & (prop2setE = $true) & (secondinupair = $true) & (binintersectLsub = $true) & (emptyset__Cong = $true)),
% 8.23/2.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK141,sK142])],[f1362,f1715,f1714])).
% 8.23/2.75 thf(f1714,plain,(
% 8.23/2.75 ? [X0 : $i] : (((ordinal @ X0) = $true) & ? [X1 : $i] : (((in @ X1 @ X0) = $true) & ((subset @ X1 @ X0) != $true))) => (($true = (ordinal @ sK141)) & ? [X1 : $i] : (((in @ X1 @ sK141) = $true) & ((subset @ X1 @ sK141) != $true)))),
% 8.23/2.75 introduced(definition,[],[choice_axiom])).
% 8.23/2.75 thf(f1715,plain,(
% 8.23/2.75 ? [X1 : $i] : (((in @ X1 @ sK141) = $true) & ((subset @ X1 @ sK141) != $true)) => (((in @ sK142 @ sK141) = $true) & ((subset @ sK142 @ sK141) != $true))),
% 8.23/2.75 introduced(definition,[],[choice_axiom])).
% 8.23/2.75 thf(f1362,plain,(
% 8.23/2.75 (notinself = $true) & (breln1unionIL = $true) & (powersetAx = $true) & (omegaSAx = $true) & (setukpairinjL = $true) & (setadjoinSub = $true) & (binintersectSubset1 = $true) & (ifSingleton = $true) & (setadjoinIL = $true) & (binintersectER = $true) & (image1Ex = $true) & (descr__Cong = $true) & (setunionsingleton2 = $true) & (powersetTE1 = $true) & (cartprodmempair1 = $true) & (omega0Ax = $true) & (setukpairinjR2 = $true) & (ordinalTransSet = $true) & (setunionE = $true) & (upairsetE = $true) & (theeq = $true) & (subsetTI = $true) & (funcext = $true) & (setminusERneg = $true) & (wellorderingAx = $true) & (setukpairIL = $true) & (setunionI = $true) & (ap2apEq2 = $true) & (choice2fnsingleton = $true) & (powersetTI1 = $true) & (setadjoinSub2 = $true) & (notinemptyset = $true) & (binunionRsub = $true) & (cartprodpairsurjEq = $true) & (emptysetAx = $true) & (cartprodmempair = $true) & (cartprodfstpairEq = $true) & (cartprodmempaircEq = $true) & (setukpairinjL1 = $true) & (breln1unionI = $true) & (surjFuncSetFuncSurj = $true) & (singletonsuniq = $true) & (cartprodsndin = $true) & (complementUnionInPowersetComplement = $true) & (ubforcartprodlem1 = $true) & (ordinalMinLem1 = $true) & (breln1compprop = $true) & (subPowSU = $true) & (complementTnotintersectT = $true) & (singletonsubset = $true) & (woz13rule2 = $true) & (surjFuncSetFuncIn = $true) & (binintersectSubset5 = $true) & (iffalseProp1 = $true) & (binunionE = $true) & (exuI3 = $true) & (breln1invI = $true) & (doubleComplementSub1 = $true) & (foundation2 = $true) & (ex1I = $true) & (demorgan2a = $true) & (eqimpsubset2 = $true) & (setoftrueEq = $true) & (eqbreln = $true) & (vacuousDall = $true) & (cartprodpairmemER = $true) & (setminusLsub = $true) & (funcGraphProp2 = $true) & (inCongP = $true) & (contrasubsetT3 = $true) & (bs114d = $true) & (binintTransitive = $true) & (notdexE = $true) & (demorgan2 = $true) & (setukpairinjL2 = $true) & (dsetconstrI = $true) & (binunionIL = $true) & (setextAx = $true) & (upairequniteq = $true) & (inComplementUnionImpNotIn1 = $true) & (setunionTransitive = $true) & (subbreln1 = $true) & (demorgan1a = $true) & (exuE3e = $true) & (ubforcartprodlem2 = $true) & (upairsetIL = $true) & (dpsetconstrER = $true) & (eqimpsubset1 = $true) & (omegaSp = $true) & (binunionTILcontra = $true) & (binunionTIRcontra = $true) & (quantDeMorgan1 = $true) & (woz1_3 = $true) & (emptyinunitempty = $true) & (powersetI = $true) & (ksndsingleton = $true) & (setminusER = $true) & (quantDeMorgan3 = $true) & (brelnall1 = $true) & (setminusI = $true) & (sepInPowerset = $true) & (nonemptyE1 = $true) & (iffalseProp2 = $true) & (dsetconstrEL = $true) & (emptyI = $true) & (symdiffI2 = $true) & (binunionEcases = $true) & (kfstpairEq = $true) & (notequalI1 = $true) & (powersetE = $true) & (woz1_1 = $true) & (symdiffI1 = $true) & (iffalse = $true) & (nonemptyI1 = $true) & (lamProp = $true) & (setminusIRneg = $true) & (breln1invE = $true) & (singletonsswitch = $true) & (ex1I2 = $true) & (setextsub = $true) & (omega__Cong = $true) & (transitivesetOp2 = $true) & (singletoninpowunion = $true) & (binintersectEL = $true) & ? [X0 : $i] : (((ordinal @ X0) = $true) & ? [X1 : $i] : (((in @ X1 @ X0) = $true) & ((subset @ X1 @ X0) != $true))) & (exu__Cong = $true) & (funcGraphProp3 = $true) & (funcextLem = $true) & (kpairp = $true) & (breln1compEex = $true) & (eqbreln1 = $true) & (setadjoinE = $true) & (binintersectSubset2 = $true) & (ex1E2 = $true) & (binunionT_lem = $true) & (beta1 = $true) & (breln1unionCommutes = $true) & (replAx = $true) & (sepSubset = $true) & (breln1invprop = $true) & (woz13rule4 = $true) & (demorgan1b = $true) & (setunionsingleton1 = $true) & (in__Cong = $true) & (binunionIR = $true) & (funcext2 = $true) & (prop2setI = $true) & (funcGraphProp4 = $true) & (powersetT_lem = $true) & (surjCantorThm = $true) & (apProp = $true) & (inIntersectImpInUnion2 = $true) & (binunionTE = $true) & (complementTcontraSubset = $true) & (binintersectSubset4 = $true) & (intersectInPowersetIntersectUnions = $true) & (setunionE2 = $true) & (powerset__Cong = $true) & (breln1unionIR = $true) & (injFuncSetFuncIn = $true) & (setbeta = $true) & (binunionLsub = $true) & (ap2p = $true) & (dpsetconstrI = $true) & (complementT_lem = $true) & (subbreln = $true) & (subsetemptysetimpeq = $true) & (breln1compE = $true) & (setukpairinjR12 = $true) & (subsetI1 = $true) & (lam2lamEq = $true) & (inComplementUnionImpInComplement1 = $true) & (omegaIndAx = $true) & (iftrueorfalse = $true) & (complementImpComplementIntersect = $true) & (breln1all2 = $true) & (complementSubsetComplementIntersect = $true) & (ap2apEq1 = $true) & (cartprodpairmemEL = $true) & (iftrue = $true) & (exuE1 = $true) & (breln1SetBreln1 = $true) & (setadjoinOr = $true) & (binintersectI = $true) & (dpsetconstrERa = $true) & (setadjoinAx = $true) & (exuE2 = $true) & (setukpairinjR11 = $true) & (demorgan2a1 = $true) & (contraSubsetComplement = $true) & (dsetconstrER = $true) & (inIntersectImpInIntersectUnions = $true) & (woz2A = $true) & (binintersectTERcontra = $true) & (setOfPairsIsBReln1 = $true) & (woz1_4 = $true) & (exuI1 = $true) & (iftrueProp2 = $true) & (subsetE2 = $true) & (woz2B = $true) & (funcGraphProp1 = $true) & (contrasubsetT = $true) & (setminusILneg = $true) & (upairinpowunion = $true) & (leftInvIsSurj = $true) & (doubleComplementE1 = $true) & (lamp = $true) & (emptysetE = $true) & (demorgan2a2 = $true) & (exuE3u = $true) & (upairset2E = $true) & (woz13rule0 = $true) & (binintersectT_lem = $true) & (beta2 = $true) & (breln1unionE = $true) & (symdiffIneg1 = $true) & (inPowerset = $true) & (image1I = $true) & (binintersectRsub = $true) & (breln1all1 = $true) & (notdallE = $true) & (demorgan2b = $true) & (theprop = $true) & (contrasubsetT2 = $true) & (setminusSubset2 = $true) & (setminusELneg = $true) & (descrp = $true) & (demorgan2b2 = $true) & (demorgan1 = $true) & (subset2powerset = $true) & (exuEu = $true) & (breln1unionEcases = $true) & (omegaSclos = $true) & (setminusEL = $true) & (woz13rule3 = $true) & (dsetconstr__Cong = $true) & (notequalI2 = $true) & (symdiffE = $true) & (setminusT_lem = $true) & (image1Ex1 = $true) & (dpsetconstrEL1 = $true) & (setOfPairsIsBReln = $true) & (funcinfuncset = $true) & (iftrueProp1 = $true) & (binintersectTELcontra = $true) & (exuI2 = $true) & (eta1 = $true) & (setext = $true) & (setunion__Cong = $true) & (transitivesetOp1 = $true) & (emptyInPowerset = $true) & (setadjoinIR = $true) & (inIntersectImpInUnion = $true) & (upairsetIR = $true) & (disjointsetsI1 = $true) & (nonemptyI = $true) & (kpairiskpair = $true) & (kfstsingleton = $true) & (injFuncInInjFuncSet = $true) & (woz1_5 = $true) & (setunionsingleton = $true) & (kpairsurjEq = $true) & (ubforcartprodlem3 = $true) & (ex1E1 = $true) & (image1E = $true) & (binunionTEcontra = $true) & (injFuncSetFuncInj = $true) & (peano0notS = $true) & (subsetTrans = $true) & (funcImageSingleton = $true) & (contrasubsetT1 = $true) & (dpsetconstrSub = $true) & (notinsingleton = $true) & (quantDeMorgan4 = $true) & (woz1_2 = $true) & (notsubsetI = $true) & (singletoninpowerset = $true) & (setukpairIR = $true) & (lam2p = $true) & (complementTI1 = $true) & (complementInPowersetComplementIntersect = $true) & (noeltsimpempty = $true) & (foundationAx = $true) & (eqinunit = $true) & (emptyinPowerset = $true) & (powersetE1 = $true) & (powersetI1 = $true) & (setukpairinjR1 = $true) & (setunionAx = $true) & (setukpairinjR = $true) & (cartprodfstin = $true) & (emptysetsubset = $true) & (peanoSinj = $true) & (woz13rule1 = $true) & (breln1unionprop = $true) & (cartprodsndpairEq = $true) & (singletonprop = $true) & (setminusSubset1 = $true) & (doubleComplementSub2 = $true) & (cartprodpairin = $true) & (woz2W = $true) & (doubleComplementEq = $true) & (upairsubunion = $true) & (prop2set2propI = $true) & (ifp = $true) & (upairset2IR = $true) & (eta2 = $true) & (subsetRefl = $true) & (setadjoin__Cong = $true) & (emptysetimpfalse = $true) & (binintersectSubset3 = $true) & (breln1compI = $true) & (powersetsubset = $true) & (uniqinunit = $true) & (complementTE1 = $true) & (emptyE1 = $true) & (nonemptyImpWitness = $true) & (subsetE = $true) & (app = $true) & (ksndpairEq = $true) & (infuncsetfunc = $true) & (woz2Ex = $true) & (setextT = $true) & (symdiffIneg2 = $true) & (quantDeMorgan2 = $true) & (brelnall2 = $true) & (subsetI2 = $true) & (dpsetconstrEL2 = $true) & (doubleComplementI1 = $true) & (image1Equiv = $true) & (notinself2 = $true) & (prop2setE = $true) & (secondinupair = $true) & (binintersectLsub = $true) & (emptyset__Cong = $true)),
% 8.23/2.75 inference(flattening,[status(thm)],[f1361])).
% 8.23/2.75 thf(f1361,plain,(
% 8.23/2.75 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (((ordinal @ X0) = $true) & ? [X1 : $i] : (((in @ X1 @ X0) = $true) & ((subset @ X1 @ X0) != $true))) & (ordinalTransSet = $true)) & (ordinalMinLem1 = $true)) & (setunionTransitive = $true)) & (transitivesetOp2 = $true)) & (binintTransitive = $true)) & (transitivesetOp1 = $true)) & (peanoSinj = $true)) & (peano0notS = $true)) & (omegaSclos = $true)) & (omegaSp = $true)) & (notinself2 = $true)) & (notinself = $true)) & (foundation2 = $true)) & (surjCantorThm = $true)) & (leftInvIsSurj = $true)) & (surjFuncSetFuncSurj = $true)) & (surjFuncSetFuncIn = $true)) & (injFuncSetFuncInj = $true)) & (injFuncSetFuncIn = $true)) & (injFuncInInjFuncSet = $true)) & (image1I = $true)) & (image1E = $true)) & (image1Equiv = $true)) & (image1Ex1 = $true)) & (image1Ex = $true)) & (woz2B = $true)) & (woz2A = $true)) & (woz2W = $true)) & (woz2Ex = $true)) & (breln1unionCommutes = $true)) & (breln1unionEcases = $true)) & (breln1unionE = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 8.23/2.75 inference(ennf_transformation,[status(thm)],[f943])).
% 8.23/2.75 thf(f943,plain,(
% 8.23/2.75 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ((breln1unionE = $true) => ((breln1unionEcases = $true) => ((breln1unionCommutes = $true) => ((woz2Ex = $true) => ((woz2W = $true) => ((woz2A = $true) => ((woz2B = $true) => ((image1Ex = $true) => ((image1Ex1 = $true) => ((image1Equiv = $true) => ((image1E = $true) => ((image1I = $true) => ((injFuncInInjFuncSet = $true) => ((injFuncSetFuncIn = $true) => ((injFuncSetFuncInj = $true) => ((surjFuncSetFuncIn = $true) => ((surjFuncSetFuncSurj = $true) => ((leftInvIsSurj = $true) => ((surjCantorThm = $true) => ((foundation2 = $true) => ((notinself = $true) => ((notinself2 = $true) => ((omegaSp = $true) => ((omegaSclos = $true) => ((peano0notS = $true) => ((peanoSinj = $true) => ((transitivesetOp1 = $true) => ((binintTransitive = $true) => ((transitivesetOp2 = $true) => ((setunionTransitive = $true) => ((ordinalMinLem1 = $true) => ((ordinalTransSet = $true) => ! [X0 : $i] : (((ordinal @ X0) = $true) => ! [X1 : $i] : (((in @ X1 @ X0) = $true) => ((subset @ X1 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.23/2.75 inference(fool_elimination,[status(thm)],[f942])).
% 8.23/2.75 thf(f942,plain,(
% 8.23/2.75 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => (binintTransitive => (transitivesetOp2 => (setunionTransitive => (ordinalMinLem1 => (ordinalTransSet => ! [X0 : $i] : ((ordinal @ X0) => ! [X1 : $i] : ((in @ X1 @ X0) => (subset @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.23/2.75 inference(rectify,[status(thm)],[f342])).
% 8.23/2.75 thf(f342,negated_conjecture,(
% 8.23/2.75 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => (binintTransitive => (transitivesetOp2 => (setunionTransitive => (ordinalMinLem1 => (ordinalTransSet => ! [X11 : $i] : ((ordinal @ X11) => ! [X3 : $i] : ((in @ X3 @ X11) => (subset @ X3 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.23/2.75 inference(negated_conjecture,[status(cth)],[f341])).
% 8.23/2.75 thf(f341,conjecture,(
% 8.23/2.75 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => (binintTransitive => (transitivesetOp2 => (setunionTransitive => (ordinalMinLem1 => (ordinalTransSet => ! [X11 : $i] : ((ordinal @ X11) => ! [X3 : $i] : ((in @ X3 @ X11) => (subset @ X3 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.23/2.75 file('/export/starexec/sandbox/tmp/tmp.PG6uUBnDB9/DTF2THF_9905.p',ordinalTransSet1)).
% 8.23/2.75 thf(f8598,plain,(
% 8.23/2.75 ((in @ sK142 @ sK141) != $true)),
% 8.23/2.75 inference(subsumption_resolution,[status(thm)],[f8595,f7031])).
% 8.23/2.75 thf(f7031,plain,(
% 8.23/2.75 ((transitiveset @ sK141) = $true)),
% 8.23/2.75 inference(binary_proxy_clausification,[status(thm)],[f7030])).
% 8.23/2.75 thf(f7030,plain,(
% 8.23/2.75 ($true = ((wellorderedByIn @ sK141) & (transitiveset @ sK141)))),
% 8.23/2.75 inference(beta_eta_normalization,[status(thm)],[f5621])).
% 8.23/2.75 thf(f5621,plain,(
% 8.23/2.75 ($true = ((^[Y0 : $i]: ((wellorderedByIn @ Y0) & (transitiveset @ Y0))) @ sK141))),
% 8.23/2.75 inference(definition_unfolding,[status(thm)],[f3828,f4852])).
% 8.23/2.75 thf(f4852,plain,(
% 8.23/2.75 (ordinal = (^[Y0 : $i]: ((wellorderedByIn @ Y0) & (transitiveset @ Y0))))),
% 8.23/2.75 inference(cnf_transformation,[status(thm)],[f883])).
% 8.23/2.75 thf(f883,plain,(
% 8.23/2.75 (ordinal = (^[Y0 : $i]: ((wellorderedByIn @ Y0) & (transitiveset @ Y0))))),
% 8.23/2.75 inference(fool_elimination,[status(thm)],[f882])).
% 8.23/2.75 thf(f882,plain,(
% 8.23/2.75 (ordinal = (^[X0 : $i] : ((transitiveset @ X0) & (wellorderedByIn @ X0))))),
% 8.23/2.75 inference(rectify,[status(thm)],[f338])).
% 8.23/2.75 thf(f338,axiom,(
% 8.23/2.75 (ordinal = (^[X1 : $i] : ((transitiveset @ X1) & (wellorderedByIn @ X1))))),
% 8.23/2.75 file('/export/starexec/sandbox/tmp/tmp.PG6uUBnDB9/DTF2THF_9905.p',ordinal)).
% 8.23/2.75 thf(f3828,plain,(
% 8.23/2.75 ($true = (ordinal @ sK141))),
% 8.23/2.75 inference(cnf_transformation,[status(thm)],[f1716])).
% 8.23/2.75 thf(f8595,plain,(
% 8.23/2.75 ((transitiveset @ sK141) != $true) | ((in @ sK142 @ sK141) != $true)),
% 8.23/2.75 inference(trivial_inequality_removal,[status(thm)],[f8592])).
% 8.23/2.75 thf(f8592,plain,(
% 8.23/2.75 ($true != $true) | ((transitiveset @ sK141) != $true) | ((in @ sK142 @ sK141) != $true)),
% 8.23/2.75 inference(superposition,[status(thm)],[f3826,f6895])).
% 8.23/2.75 thf(f6895,plain,(
% 8.23/2.75 ( ! [X2 : $i,X3 : $i] : (((subset @ X3 @ X2) = $true) | ((transitiveset @ X2) != $true) | ((in @ X3 @ X2) != $true)) )),
% 8.23/2.75 inference(trivial_inequality_removal,[status(thm)],[f5842])).
% 8.23/2.75 thf(f5842,plain,(
% 8.23/2.75 ( ! [X2 : $i,X3 : $i] : (((subset @ X3 @ X2) = $true) | ((transitiveset @ X2) != $true) | ((in @ X3 @ X2) != $true) | ($true != $true)) )),
% 8.23/2.75 inference(definition_unfolding,[status(thm)],[f4250,f3701])).
% 8.23/2.75 thf(f3701,plain,(
% 8.23/2.75 (transitivesetOp1 = $true)),
% 8.23/2.75 inference(cnf_transformation,[status(thm)],[f1716])).
% 8.23/2.75 thf(f4250,plain,(
% 8.23/2.75 ( ! [X2 : $i,X3 : $i] : (((transitiveset @ X2) != $true) | ((subset @ X3 @ X2) = $true) | ((in @ X3 @ X2) != $true) | (transitivesetOp1 != $true)) )),
% 8.23/2.75 inference(cnf_transformation,[status(thm)],[f2062])).
% 8.23/2.75 thf(f2062,plain,(
% 8.23/2.75 ((transitivesetOp1 = $true) | (((transitiveset @ sK345) = $true) & (((subset @ sK346 @ sK345) != $true) & ($true = (in @ sK346 @ sK345))))) & (! [X2 : $i] : (((transitiveset @ X2) != $true) | ! [X3 : $i] : (((subset @ X3 @ X2) = $true) | ((in @ X3 @ X2) != $true))) | (transitivesetOp1 != $true))),
% 8.23/2.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK345,sK346])],[f2059,f2061,f2060])).
% 8.23/2.75 thf(f2060,plain,(
% 8.23/2.75 ? [X0 : $i] : (((transitiveset @ X0) = $true) & ? [X1 : $i] : (((subset @ X1 @ X0) != $true) & ((in @ X1 @ X0) = $true))) => (((transitiveset @ sK345) = $true) & ? [X1 : $i] : (($true != (subset @ X1 @ sK345)) & ((in @ X1 @ sK345) = $true)))),
% 8.23/2.75 introduced(definition,[],[choice_axiom])).
% 8.23/2.75 thf(f2061,plain,(
% 8.23/2.75 ? [X1 : $i] : (($true != (subset @ X1 @ sK345)) & ((in @ X1 @ sK345) = $true)) => (((subset @ sK346 @ sK345) != $true) & ($true = (in @ sK346 @ sK345)))),
% 8.23/2.75 introduced(definition,[],[choice_axiom])).
% 8.23/2.75 thf(f2059,plain,(
% 8.23/2.75 ((transitivesetOp1 = $true) | ? [X0 : $i] : (((transitiveset @ X0) = $true) & ? [X1 : $i] : (((subset @ X1 @ X0) != $true) & ((in @ X1 @ X0) = $true)))) & (! [X2 : $i] : (((transitiveset @ X2) != $true) | ! [X3 : $i] : (((subset @ X3 @ X2) = $true) | ((in @ X3 @ X2) != $true))) | (transitivesetOp1 != $true))),
% 8.23/2.75 inference(rectify,[status(thm)],[f2058])).
% 8.23/2.75 thf(f2058,plain,(
% 8.23/2.75 ((transitivesetOp1 = $true) | ? [X0 : $i] : (((transitiveset @ X0) = $true) & ? [X1 : $i] : (((subset @ X1 @ X0) != $true) & ((in @ X1 @ X0) = $true)))) & (! [X0 : $i] : (((transitiveset @ X0) != $true) | ! [X1 : $i] : (((subset @ X1 @ X0) = $true) | ((in @ X1 @ X0) != $true))) | (transitivesetOp1 != $true))),
% 8.23/2.75 inference(nnf_transformation,[status(thm)],[f1306])).
% 8.23/2.75 thf(f1306,plain,(
% 8.23/2.75 (transitivesetOp1 = $true) <=> ! [X0 : $i] : (((transitiveset @ X0) != $true) | ! [X1 : $i] : (((subset @ X1 @ X0) = $true) | ((in @ X1 @ X0) != $true)))),
% 8.23/2.75 inference(ennf_transformation,[status(thm)],[f449])).
% 8.23/2.75 thf(f449,plain,(
% 8.23/2.75 (transitivesetOp1 = $true) <=> ! [X0 : $i] : (((transitiveset @ X0) = $true) => ! [X1 : $i] : (((in @ X1 @ X0) = $true) => ((subset @ X1 @ X0) = $true)))),
% 8.23/2.75 inference(fool_elimination,[status(thm)],[f448])).
% 8.23/2.75 thf(f448,plain,(
% 8.23/2.75 (! [X0 : $i] : ((transitiveset @ X0) => ! [X1 : $i] : ((in @ X1 @ X0) => (subset @ X1 @ X0))) = transitivesetOp1)),
% 8.23/2.75 inference(rectify,[status(thm)],[f332])).
% 8.23/2.75 thf(f332,axiom,(
% 8.23/2.75 (! [X11 : $i] : ((transitiveset @ X11) => ! [X3 : $i] : ((in @ X3 @ X11) => (subset @ X3 @ X11))) = transitivesetOp1)),
% 8.23/2.75 file('/export/starexec/sandbox/tmp/tmp.PG6uUBnDB9/DTF2THF_9905.p',transitivesetOp1)).
% 8.23/2.75 thf(f3826,plain,(
% 8.23/2.75 ((subset @ sK142 @ sK141) != $true)),
% 8.23/2.75 inference(cnf_transformation,[status(thm)],[f1716])).
% 8.23/2.75 % SZS output end Proof for DTF2THF_9905
% 8.23/2.75 % (10051)------------------------------
% 8.23/2.75 % (10051)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.23/2.75 % (10051)Termination reason: Refutation
% 8.23/2.75
% 8.23/2.75 % (10051)Memory used [KB]: 12025
% 8.23/2.75 % (10051)Time elapsed: 0.248 s
% 8.23/2.75 % (10051)Instructions burned: 489 (million)
% 8.23/2.75 % (10051)------------------------------
% 8.23/2.75 % (10051)------------------------------
% 8.23/2.75 % (10003)Success in time 0.796 s
%------------------------------------------------------------------------------