%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU817^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n018.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 09:37:27 AM UTC 2026
% Result : Theorem 6.58s 2.39s
% Output : Refutation 6.58s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 9
% Syntax : Number of formulae : 55 ( 20 unt; 0 typ; 0 def)
% Number of atoms : 5605 (1503 equ; 0 cnn)
% Maximal formula atoms : 337 ( 101 avg)
% Number of connectives : 2859 ( 85 ~; 53 |;1052 &; 300 @)
% ( 6 <=>;1363 =>; 0 <=; 0 <~>)
% Maximal formula depth : 343 ( 47 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 687 ( 684 usr; 343 con; 0-6 aty)
% Number of variables : 105 ( 5 ^; 70 !; 30 ?; 105 :)
% 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_412,type,
sP0: ( $i > $o ) > $i > $o ).
thf(func_def_414,type,
sP2: $i > $i > $i > $i > $o ).
thf(func_def_415,type,
sP3: $i > $i > $i > $o ).
thf(func_def_417,type,
sP5: $i > $i > $i > $o ).
thf(func_def_418,type,
sP6: $i > ( $i > $i ) > $i > $o ).
thf(func_def_420,type,
sP8: $i > $i > $i > $o ).
thf(func_def_421,type,
sP9: $i > $i > $o > $o ).
thf(func_def_422,type,
sP10: $i > $i > $i > $o ).
thf(func_def_423,type,
sP11: $i > $o ).
thf(func_def_424,type,
sP12: $i > $o ).
thf(func_def_427,type,
sP15: ( $i > $o ) > $o ).
thf(func_def_428,type,
sP16: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_429,type,
sP17: ( $i > $i > $o ) > $i > $o ).
thf(func_def_432,type,
sP20: $i > $i > $i > $o ).
thf(func_def_434,type,
sP22: $i > $i > $i > $o ).
thf(func_def_435,type,
sP23: $i > $i > $i > $o ).
thf(func_def_436,type,
sP24: $i > $i > $i > $i > $o ).
thf(func_def_437,type,
sP25: $i > $i > $i > $i > $o ).
thf(func_def_438,type,
sP26: ( $i > $o ) > $o ).
thf(func_def_439,type,
sP27: $i > ( $i > $o ) > $o ).
thf(func_def_440,type,
sP28: $i > ( $i > $o ) > $i > $o ).
thf(func_def_441,type,
sP29: $i > $i > $i > $i > $o ).
thf(func_def_442,type,
sP30: ( $i > $i ) > $i > $i > $o ).
thf(func_def_444,type,
sP32: ( $i > $o ) > $o ).
thf(func_def_447,type,
sP35: $i > $i > $i > $o ).
thf(func_def_448,type,
sP36: ( $i > $o ) > $i > $o ).
thf(func_def_449,type,
sP37: ( $i > $o ) > $i > $o ).
thf(func_def_450,type,
sP38: $i > $i > $i > $o > $o ).
thf(func_def_451,type,
sP39: $i > $i > $i > $i > $o ).
thf(func_def_453,type,
sP41: $i > $i > ( $i > $o ) > $o ).
thf(func_def_454,type,
sP42: $i > $i > $i > $i > $i > $o ).
thf(func_def_455,type,
sP43: $i > $i > $i > $o > $o ).
thf(func_def_456,type,
sP44: $o > $i > $i > $i > $o ).
thf(func_def_457,type,
sP45: $i > $i > $o > $o ).
thf(func_def_458,type,
sP46: $i > $i > $i > $o ).
thf(func_def_459,type,
sP47: $i > ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_460,type,
sP48: ( $i > $o ) > $i > $o ).
thf(func_def_461,type,
sP49: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_464,type,
sP52: $i > $i > $o ).
thf(func_def_466,type,
sP54: ( $i > $o ) > $i > $o ).
thf(func_def_467,type,
sP55: ( $i > $o ) > $i > $o ).
thf(func_def_468,type,
sP56: $i > $i > $i > $o ).
thf(func_def_469,type,
sP57: $i > $i > $i > $o ).
thf(func_def_470,type,
sP58: ( $i > $i ) > $i > $i > $o ).
thf(func_def_474,type,
sP62: $i > $i > $i > $i > $o ).
thf(func_def_475,type,
sP63: $o > $i > $i > $i > $i > $i > $o ).
thf(func_def_477,type,
sP65: $i > $i > $o ).
thf(func_def_479,type,
sP67: $i > $i > ( $i > $i ) > $o ).
thf(func_def_481,type,
sP69: $i > ( $i > $o ) > $o ).
thf(func_def_482,type,
sP70: $i > ( $i > $o ) > $i > $o ).
thf(func_def_483,type,
sP71: $i > ( $i > $o ) > $o ).
thf(func_def_485,type,
sP73: ( $i > $o ) > $i > $o ).
thf(func_def_488,type,
sP76: $i > $i > $i > $o ).
thf(func_def_489,type,
sP77: $i > $i > ( $i > $i > $o ) > $o ).
thf(func_def_490,type,
sP78: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_491,type,
sP79: ( $i > $o ) > $o ).
thf(func_def_492,type,
sP80: $i > $i > $i > $o ).
thf(func_def_493,type,
sP81: $i > $i > $o ).
thf(func_def_494,type,
sP82: $i > $i > $o ).
thf(func_def_495,type,
sP83: $i > $o ).
thf(func_def_496,type,
sP84: $i > $i > $o ).
thf(func_def_497,type,
sP85: $i > $i > $o ).
thf(func_def_498,type,
sP86: $i > $o > $i > $o ).
thf(func_def_499,type,
sP87: ( $i > $o ) > $i > $o ).
thf(func_def_501,type,
sP89: $i > $i > $i > $o ).
thf(func_def_502,type,
sP90: $i > ( $i > $i ) > $i > $o ).
thf(func_def_504,type,
sP92: $o > $i > $i > $o ).
thf(func_def_506,type,
sP94: $i > $i > $i > $o ).
thf(func_def_508,type,
sP96: ( $i > $o ) > $o ).
thf(func_def_509,type,
sP97: ( $i > $o ) > $i > $o ).
thf(func_def_511,type,
sP99: $i > ( $i > $o ) > $o ).
thf(func_def_514,type,
sP102: ( $i > $o ) > $i > $o ).
thf(func_def_515,type,
sP103: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_516,type,
sP104: $i > $o ).
thf(func_def_517,type,
sP105: $i > $i > $o ).
thf(func_def_521,type,
sK109: $i > $o ).
thf(func_def_542,type,
sK130: $i > $o ).
thf(func_def_544,type,
sK132: $i > ( $i > $o ) > $i ).
thf(func_def_545,type,
sK133: ( $i > $o ) > $i > $i ).
thf(func_def_546,type,
sK134: $i > $i > $i > $i ).
thf(func_def_547,type,
sK135: $i > $i > $i > $i > $i ).
thf(func_def_548,type,
sK136: $i > $i > $i > $i > $i ).
thf(func_def_549,type,
sK137: $i > $i > $i > $i > $i ).
thf(func_def_578,type,
sK166: $i > $o ).
thf(func_def_580,type,
sK168: $i > ( $i > $o ) > $i ).
thf(func_def_602,type,
sK190: $i > $i ).
thf(func_def_609,type,
sK197: $i > $i ).
thf(func_def_622,type,
sK210: $i > $o ).
thf(func_def_625,type,
sK213: $i > $i > $i > $i ).
thf(func_def_626,type,
sK214: $i > $i > $i > $i > $i ).
thf(func_def_644,type,
sK232: $i > $o ).
thf(func_def_650,type,
sK238: $i > $o ).
thf(func_def_652,type,
sK240: $i > $o ).
thf(func_def_662,type,
sK250: $i > $i ).
thf(func_def_664,type,
sK252: $i > ( $i > $i ) > $i > $i ).
thf(func_def_665,type,
sK253: $i > ( $i > $i ) > $i > $i ).
thf(func_def_684,type,
sK272: $i > $i ).
thf(func_def_706,type,
sK294: $i > $i > $i > $i ).
thf(func_def_707,type,
sK295: $i > $i > $i > $i ).
thf(func_def_713,type,
sK301: $i > $i > $i > $o ).
thf(func_def_717,type,
sK305: $i > $i ).
thf(func_def_718,type,
sK306: $i > $i ).
thf(func_def_728,type,
sK316: $i > $o ).
thf(func_def_729,type,
sK317: $i > $o ).
thf(func_def_730,type,
sK318: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_731,type,
sK319: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_743,type,
sK331: $i > $i ).
thf(func_def_744,type,
sK332: ( $i > $i ) > $i > $i > $i ).
thf(func_def_746,type,
sK334: $i > $i > $o ).
thf(func_def_751,type,
sK339: $i > $i > $o ).
thf(func_def_755,type,
sK343: $i > $o ).
thf(func_def_756,type,
sK344: ( $i > $o ) > $i ).
thf(func_def_771,type,
sK359: $i > ( $i > $o ) > $i ).
thf(func_def_772,type,
sK360: ( $i > $o ) > $i ).
thf(func_def_773,type,
sK361: $i > $o ).
thf(func_def_777,type,
sK365: $i > $i > $o ).
thf(func_def_779,type,
sK367: $i > ( $i > $i > $o ) > $i ).
thf(func_def_780,type,
sK368: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_781,type,
sK369: ( $i > $i > $o ) > $i > $i ).
thf(func_def_782,type,
sK370: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_792,type,
sK380: $i > $o ).
thf(func_def_793,type,
sK381: $i > $o ).
thf(func_def_794,type,
sK382: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_795,type,
sK383: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_796,type,
sK384: $i > $i > $i > $i ).
thf(func_def_802,type,
sK390: $i > $o ).
thf(func_def_809,type,
sK397: $i > $i > $i ).
thf(func_def_810,type,
sK398: $i > $i > $i > $i ).
thf(func_def_811,type,
sK399: $i > $i > $i > $i ).
thf(func_def_812,type,
sK400: $i > $i > $i > $i ).
thf(func_def_813,type,
sK401: $i > $i > $i > $i ).
thf(func_def_824,type,
sK412: $i > $i > $i > $i > $i ).
thf(func_def_825,type,
sK413: $i > $i > $i > $i > $i ).
thf(func_def_826,type,
sK414: $i > $i > $i > $i > $i ).
thf(func_def_827,type,
sK415: $i > $i > $i > $i > $i ).
thf(func_def_832,type,
sK420: ( $i > $o ) > $i ).
thf(func_def_833,type,
sK421: $i > ( $i > $o ) > $i ).
thf(func_def_834,type,
sK422: $i > $o ).
thf(func_def_837,type,
sK425: $i > ( $i > $o ) > $i > $i ).
thf(func_def_838,type,
sK426: $i > ( $i > $o ) > $i > $i ).
thf(func_def_839,type,
sK427: $i > ( $i > $o ) > $i ).
thf(func_def_842,type,
sK430: $i > $o ).
thf(func_def_843,type,
sK431: $i > $i > $i > $i > $i ).
thf(func_def_844,type,
sK432: $i > $i > $i > $i > $i ).
thf(func_def_877,type,
sK465: $i > $i ).
thf(func_def_878,type,
sK466: ( $i > $i ) > $i > $i > $i ).
thf(func_def_881,type,
sK469: $i > ( $i > $o ) > $i ).
thf(func_def_882,type,
sK470: ( $i > $o ) > $i ).
thf(func_def_883,type,
sK471: $i > $o ).
thf(func_def_887,type,
sK475: $i > $i > $i ).
thf(func_def_890,type,
sK478: $i > $i ).
thf(func_def_891,type,
sK479: ( $i > $i ) > $i > $i > $i ).
thf(func_def_894,type,
sK482: $i > $i ).
thf(func_def_896,type,
sK484: $i > $i > $i > $i ).
thf(func_def_912,type,
sK500: $i > $i > $o ).
thf(func_def_914,type,
sK502: $i > $o ).
thf(func_def_920,type,
sK508: ( $i > $o ) > $i > $i ).
thf(func_def_921,type,
sK509: ( $i > $o ) > $i > $i ).
thf(func_def_922,type,
sK510: ( $i > $o ) > $i > $i ).
thf(func_def_924,type,
sK512: $i > $o ).
thf(func_def_927,type,
sK515: $i > $i > $i > $i > $o ).
thf(func_def_939,type,
sK527: $i > $i > $i ).
thf(func_def_947,type,
sK535: $i > $i ).
thf(func_def_948,type,
sK536: ( $i > $i ) > $i > $i > $i ).
thf(func_def_968,type,
sK556: $i > $i > ( $i > $o ) > $i ).
thf(func_def_971,type,
sK559: $i > $o ).
thf(func_def_973,type,
sK561: $i > $i ).
thf(func_def_981,type,
sK569: $i > $i > $i > $i > $i > $i ).
thf(func_def_1005,type,
sK593: $i > $i > $i > $o ).
thf(func_def_1006,type,
sK594: $i > $i > $i > $i ).
thf(func_def_1020,type,
sK608: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_1021,type,
sK609: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_1024,type,
sK612: $i > $o ).
thf(func_def_1025,type,
sK613: $i > $o ).
thf(func_def_1026,type,
sK614: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1027,type,
sK615: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1028,type,
sK616: ( $i > $o ) > $i > $i ).
thf(func_def_1032,type,
sK620: $i > $o ).
thf(func_def_1039,type,
sK627: $i > $i ).
thf(func_def_1042,type,
sK630: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1048,type,
sK636: $i > $i > $i ).
thf(func_def_1077,type,
sK665: $i > $i > $i ).
thf(func_def_1078,type,
sK666: ( $i > $o ) > $i > $i ).
thf(func_def_1079,type,
sK667: ( $i > $o ) > $i > $i ).
thf(func_def_1080,type,
sK668: ( $i > $o ) > $i > $i ).
thf(func_def_1081,type,
sK669: $i > $o ).
thf(func_def_1083,type,
sK671: $i > $o ).
thf(func_def_1094,type,
sK682: $i > $i > $i > $i ).
thf(func_def_1095,type,
sK683: $i > $i > $i > $i ).
thf(func_def_1138,type,
sK726: $i > $o ).
thf(func_def_1141,type,
sK729: $i > $i ).
thf(func_def_1144,type,
sK732: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1145,type,
sK733: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1146,type,
sK734: $i > ( $i > $i ) > $i > $i > $i ).
thf(func_def_1149,type,
sK737: $i > $i > $i ).
thf(func_def_1151,type,
sK739: $i > $i ).
thf(func_def_1164,type,
sK752: $i > $i > $i > $i > $i ).
thf(func_def_1169,type,
sK757: $o > $i > $i > $i > $i > $i > $i ).
thf(func_def_1180,type,
sK768: $i > $i ).
thf(func_def_1186,type,
sK774: $i > $i > $i ).
thf(func_def_1187,type,
sK775: $i > $i > $i ).
thf(func_def_1195,type,
sK783: $i > $i ).
thf(func_def_1198,type,
sK786: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1199,type,
sK787: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1206,type,
sK794: $i > $i > $o ).
thf(func_def_1214,type,
sK802: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1215,type,
sK803: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1216,type,
sK804: $i > ( $i > $o ) > $i ).
thf(func_def_1219,type,
sK807: $i > $o ).
thf(func_def_1224,type,
sK812: $i > $o ).
thf(func_def_1225,type,
sK813: ( $i > $o ) > $i > $i ).
thf(func_def_1226,type,
sK814: $i > ( $i > $o ) > $i ).
thf(func_def_1227,type,
sK815: $i > $o ).
thf(func_def_1229,type,
sK817: $i > ( $i > $o ) > $i ).
thf(func_def_1230,type,
sK818: ( $i > $o ) > $i > $i ).
thf(func_def_1231,type,
sK819: $i > $i > $o ).
thf(func_def_1253,type,
sK841: $i > $i > $i ).
thf(func_def_1269,type,
sK857: $i > $i > $i > $i ).
thf(func_def_1270,type,
sK858: $i > $i > $i > $i > $i ).
thf(func_def_1278,type,
sK866: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1279,type,
sK867: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_1280,type,
sK868: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1281,type,
sK869: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1284,type,
sK872: $i > $i > $o ).
thf(func_def_1325,type,
sK913: ( $i > $o ) > $i ).
thf(func_def_1326,type,
sK914: $i > ( $i > $o ) > $i ).
thf(func_def_1327,type,
sK915: $i > $o ).
thf(func_def_1333,type,
sK921: $i > $i > $i ).
thf(func_def_1334,type,
sK922: $i > $i > $i ).
thf(func_def_1335,type,
sK923: $i > $i > $i ).
thf(func_def_1336,type,
sK924: $i > $i > $i ).
thf(func_def_1337,type,
sK925: $i > $i > $i ).
thf(func_def_1338,type,
sK926: $i > $i ).
thf(func_def_1339,type,
sK927: $i > $i ).
thf(func_def_1340,type,
sK928: $i > $i ).
thf(func_def_1341,type,
sK929: $i > $i ).
thf(func_def_1342,type,
sK930: $i > $i > $i ).
thf(func_def_1343,type,
sK931: $i > $i > $i ).
thf(func_def_1344,type,
sK932: $i > $i > $i > $i ).
thf(func_def_1345,type,
sK933: $i > $i > $i ).
thf(func_def_1346,type,
sK934: $i > $i > $i ).
thf(func_def_1347,type,
sK935: $i > $i > $i ).
thf(func_def_1348,type,
sK936: $i > $i > $i > $i ).
thf(func_def_1349,type,
sK937: $i > $i > $i > $i ).
thf(func_def_1351,type,
sK939: $i > $i ).
thf(func_def_1352,type,
sK940: $i > $o > $i > $i ).
thf(func_def_1356,type,
sK944: $i > $o ).
thf(func_def_1358,type,
sK946: $i > ( $i > $o ) > $i ).
thf(func_def_1359,type,
sK947: ( $i > $o ) > $i > $i ).
thf(func_def_1369,type,
sK957: $i > $i > $i > $i ).
thf(func_def_1370,type,
sK958: $i > $i > $i > $i ).
thf(func_def_1388,type,
sK976: $i > $o ).
thf(func_def_1393,type,
sK981: $i > $i ).
thf(func_def_1394,type,
sK982: $i > $i ).
thf(func_def_1395,type,
sK983: ( $i > $i ) > $i > $i ).
thf(func_def_1396,type,
sK984: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1423,type,
sK1011: $i > $o ).
thf(func_def_1431,type,
sK1019: $i > $o ).
thf(func_def_1452,type,
sK1040: ( $i > $o ) > $i ).
thf(func_def_1453,type,
sK1041: ( $i > $o ) > $i ).
thf(func_def_1454,type,
sK1042: $i > $o ).
thf(func_def_1472,type,
sK1060: $i > $o ).
thf(func_def_1474,type,
sK1062: $i > ( $i > $o ) > $i ).
thf(func_def_1475,type,
sK1063: ( $i > $o ) > $i > $i ).
thf(func_def_1487,type,
sK1075: $i > $o ).
thf(func_def_1488,type,
sK1076: ( $i > $o ) > $i > $i ).
thf(func_def_1489,type,
sK1077: $i > ( $i > $o ) > $i ).
thf(func_def_1509,type,
sK1097: $i > $o ).
thf(func_def_1511,type,
sK1099: $i > $i ).
thf(func_def_1513,type,
sK1101: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1525,type,
sK1113: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1526,type,
sK1114: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1527,type,
sK1115: ( $i > $o ) > $i > $i ).
thf(func_def_1531,type,
sK1119: $i > $o ).
thf(func_def_1548,type,
sK1136: $i > $i > $o ).
thf(func_def_1566,type,
sK1154: $i > $i > $o ).
thf(func_def_1571,type,
sK1159: $i > $i ).
thf(func_def_1573,type,
sK1161: $i > $i > $i ).
thf(func_def_1574,type,
sK1162: $i > $i ).
thf(func_def_1592,type,
sK1180: $i > $i > $i ).
thf(func_def_1601,type,
sK1189: $i > $i > $o ).
thf(func_def_1605,type,
ph1193:
!>[X0: $tType] : X0 ).
thf(f9908,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f9907,f6886]) ).
thf(f6886,plain,
( ( transitiveset @ sK963 )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f6885]) ).
thf(f6885,plain,
( ( ( wellorderedByIn @ sK963 )
& ( transitiveset @ sK963 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f6533]) ).
thf(f6533,plain,
( ( ^ [Y0: $i] :
( ( wellorderedByIn @ Y0 )
& ( transitiveset @ Y0 ) )
@ sK963 )
= $true ),
inference(definition_unfolding,[status(thm)],[f4971,f3811]) ).
thf(f3811,plain,
( ordinal
= ( ^ [Y0: $i] :
( ( wellorderedByIn @ Y0 )
& ( transitiveset @ Y0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f563]) ).
thf(f563,plain,
( ordinal
= ( ^ [Y0: $i] :
( ( wellorderedByIn @ Y0 )
& ( transitiveset @ Y0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f562]) ).
thf(f562,plain,
( ( ^ [X0: $i] :
( ( transitiveset @ X0 )
& ( wellorderedByIn @ X0 ) ) )
= ordinal ),
inference(rectify,[status(thm)],[f338]) ).
thf(f338,axiom,
( ( ^ [X1: $i] :
( ( transitiveset @ X1 )
& ( wellorderedByIn @ X1 ) ) )
= ordinal ),
file('/export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2THF_13596.p',ordinal) ).
thf(f4971,plain,
( ( ordinal @ sK963 )
= $true ),
inference(cnf_transformation,[status(thm)],[f3188]) ).
thf(f3188,plain,
( ( setextsub = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( notequalI1 = $true )
& ( woz13rule2 = $true )
& ( setunionsingleton = $true )
& ( singletonprop = $true )
& ( theeq = $true )
& ( breln1all1 = $true )
& ( emptyE1 = $true )
& ( upairinpowunion = $true )
& ( setunionTransitive = $true )
& ( ex1E1 = $true )
& ( emptyinPowerset = $true )
& ( complementTE1 = $true )
& ( upairsubunion = $true )
& ( exuI1 = $true )
& ( ifSingleton = $true )
& ( breln1invI = $true )
& ( cartprodmempair = $true )
& ( image1Ex = $true )
& ( symdiffIneg1 = $true )
& ( lamProp = $true )
& ( replAx = $true )
& ( ex1E2 = $true )
& ( notequalI2 = $true )
& ( binintersectSubset4 = $true )
& ( image1Equiv = $true )
& ( complementSubsetComplementIntersect = $true )
& ( breln1invprop = $true )
& ( iftrueProp1 = $true )
& ( binintersectTELcontra = $true )
& ( breln1SetBreln1 = $true )
& ( exuE1 = $true )
& ( woz2A = $true )
& ( inPowerset = $true )
& ( inCongP = $true )
& ( singletonsubset = $true )
& ( brelnall1 = $true )
& ( binintersectER = $true )
& ( app = $true )
& ( upairset2E = $true )
& ( emptysetAx = $true )
& ( cartprodpairin = $true )
& ( woz1_5 = $true )
& ( setbeta = $true )
& ( funcext = $true )
& ( breln1unionIL = $true )
& ( woz13rule1 = $true )
& ( setukpairinjR2 = $true )
& ( powersetTI1 = $true )
& ( cartprodpairmemER = $true )
& ( kpairp = $true )
& ( doubleComplementSub1 = $true )
& ( subsetI2 = $true )
& ( vacuousDall = $true )
& ( setukpairinjL = $true )
& ( image1E = $true )
& ( emptyI = $true )
& ( woz13rule0 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( funcext2 = $true )
& ( apProp = $true )
& ( setOfPairsIsBReln1 = $true )
& ( descrp = $true )
& ( breln1compE = $true )
& ( funcGraphProp4 = $true )
& ( setukpairIL = $true )
& ( setoftrueEq = $true )
& ( powersetT_lem = $true )
& ( funcextLem = $true )
& ( powersetsubset = $true )
& ( breln1unionIR = $true )
& ( emptyinunitempty = $true )
& ( binunionTIRcontra = $true )
& ( lam2lamEq = $true )
& ( setOfPairsIsBReln = $true )
& ( setunionI = $true )
& ( doubleComplementEq = $true )
& ( emptysetimpfalse = $true )
& ( noeltsimpempty = $true )
& ( descr__Cong = $true )
& ( beta2 = $true )
& ( binintersectTERcontra = $true )
& ( funcGraphProp3 = $true )
& ( kpairiskpair = $true )
& ( setextT = $true )
& ( breln1compI = $true )
& ( setunionsingleton1 = $true )
& ( image1I = $true )
& ( sepInPowerset = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodpairsurjEq = $true )
& ( exuE2 = $true )
& ( setadjoinSub = $true )
& ( powersetE = $true )
& ( setunionE = $true )
& ( image1Ex1 = $true )
& ( emptyset__Cong = $true )
& ( eta2 = $true )
& ( breln1unionprop = $true )
& ( omegaSAx = $true )
& ( complementTnotintersectT = $true )
& ( cartprodmempaircEq = $true )
& ( kfstpairEq = $true )
& ( lamp = $true )
& ( dsetconstrI = $true )
& ( contrasubsetT = $true )
& ( peano0notS = $true )
& ( demorgan2b = $true )
& ( powersetAx = $true )
& ( surjFuncSetFuncIn = $true )
& ( binunionEcases = $true )
& ( nonemptyI = $true )
& ( binunionTILcontra = $true )
& ( breln1unionI = $true )
& ( nonemptyE1 = $true )
& ( binunionRsub = $true )
& ( binunionTE = $true )
& ( cartprodsndpairEq = $true )
& ( breln1invE = $true )
& ( transitivesetOp2 = $true )
& ( transitivesetOp1 = $true )
& ( funcGraphProp1 = $true )
& ( setukpairinjL2 = $true )
& ( iftrue = $true )
& ( binunionIL = $true )
& ( binunionE = $true )
& ( setminusIRneg = $true )
& ( upairequniteq = $true )
& ( woz1_1 = $true )
& ( dpsetconstrER = $true )
& ( setext = $true )
& ( notinself = $true )
& ( sepSubset = $true )
& ( injFuncSetFuncIn = $true )
& ( binintersectSubset3 = $true )
& ( setadjoin__Cong = $true )
& ( woz1_4 = $true )
& ( dsetconstrEL = $true )
& ( omegaIndAx = $true )
& ( contraSubsetComplement = $true )
& ( disjointsetsI1 = $true )
& ( contrasubsetT2 = $true )
& ( cartprodsndin = $true )
& ( contrasubsetT1 = $true )
& ( demorgan2b2 = $true )
& ( ap2apEq1 = $true )
& ( subset2powerset = $true )
& ( symdiffE = $true )
& ( symdiffIneg2 = $true )
& ( complementT_lem = $true )
& ( binintersectLsub = $true )
& ( woz1_2 = $true )
& ( iffalseProp1 = $true )
& ( omegaSp = $true )
& ( setminusILneg = $true )
& ( emptysetsubset = $true )
& ( dpsetconstrEL1 = $true )
& ( notdexE = $true )
& ( iftrueorfalse = $true )
& ( singletonsswitch = $true )
& ( setminusEL = $true )
& ( injFuncSetFuncInj = $true )
& ( singletoninpowerset = $true )
& ( setminusERneg = $true )
& ( eqbreln = $true )
& ( quantDeMorgan3 = $true )
& ( ordinalMinLem1 = $true )
& ( setadjoinSub2 = $true )
& ( lam2p = $true )
& ( theprop = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( brelnall2 = $true )
& ( quantDeMorgan4 = $true )
& ( doubleComplementE1 = $true )
& ( notinemptyset = $true )
& ( subsetE2 = $true )
& ( setunion__Cong = $true )
& ( contrasubsetT3 = $true )
& ( setunionsingleton2 = $true )
& ( breln1compEex = $true )
& ( binintersectI = $true )
& ( exuEu = $true )
& ( complementTI1 = $true )
& ( setadjoinIL = $true )
& ( setminusLsub = $true )
& ( setadjoinE = $true )
& ( eqimpsubset1 = $true )
& ( prop2setI = $true )
& ( powersetE1 = $true )
& ( upairsetIR = $true )
& ( doubleComplementSub2 = $true )
& ( woz2W = $true )
& ( demorgan2 = $true )
& ( ksndpairEq = $true )
& ( exuE3e = $true )
& ( binintTransitive = $true )
& ( demorgan1a = $true )
& ( breln1compprop = $true )
& ( binunionTEcontra = $true )
& ( setminusI = $true )
& ( injFuncInInjFuncSet = $true )
& ( ubforcartprodlem2 = $true )
& ( exuI3 = $true )
& ( setextAx = $true )
& ( notsubsetI = $true )
& ( subbreln = $true )
& ( peanoSinj = $true )
& ( foundation2 = $true )
& ( demorgan2a1 = $true )
& ( emptysetE = $true )
& ( binintersectSubset2 = $true )
& ( breln1all2 = $true )
& ( uniqinunit = $true )
& ( subsetTrans = $true )
& ( setminusT_lem = $true )
& ( subsetTI = $true )
& ( binintersectRsub = $true )
& ( iffalse = $true )
& ( omega0Ax = $true )
& ( powersetTE1 = $true )
& ( demorgan1b = $true )
& ( powerset__Cong = $true )
& ( subsetRefl = $true )
& ( binunionIR = $true )
& ( emptyInPowerset = $true )
& ( cartprodpairmemEL = $true )
& ( choice2fnsingleton = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan2 = $true )
& ( exu__Cong = $true )
& ( subbreln1 = $true )
& ( ubforcartprodlem3 = $true )
& ( woz1_3 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( nonemptyI1 = $true )
& ( symdiffI2 = $true )
& ( demorgan2a = $true )
& ( cartprodfstin = $true )
& ( complementTcontraSubset = $true )
& ( kpairsurjEq = $true )
& ( dpsetconstrEL2 = $true )
& ( ex1I = $true )
& ( subsetE = $true )
& ( breln1unionE = $true )
& ( beta1 = $true )
& ( bs114d = $true )
& ( leftInvIsSurj = $true )
& ( binintersectEL = $true )
& ( omegaSclos = $true )
& ( binintersectSubset5 = $true )
& ( breln1unionEcases = $true )
& ( upairset2IR = $true )
& ( kfstsingleton = $true )
& ( foundationAx = $true )
& ( funcImageSingleton = $true )
& ( prop2set2propI = $true )
& ( in__Cong = $true )
& ( demorgan2a2 = $true )
& ( ap2apEq2 = $true )
& ( nonemptyImpWitness = $true )
& ( dpsetconstrERa = $true )
& ( funcGraphProp2 = $true )
& ( exuI2 = $true )
& ( eqinunit = $true )
& ( subsetI1 = $true )
& ( binunionT_lem = $true )
& ( surjCantorThm = $true )
& ( woz2Ex = $true )
& ( setunionAx = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( dsetconstrER = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( woz13rule4 = $true )
& ( ( ordinal @ sK963 )
= $true )
& ( ( in @ sK965 @ sK963 )
= $true )
& ( ( in @ sK964 @ sK965 )
= $true )
& ( $true
!= ( in @ sK964 @ sK963 ) )
& ( subPowSU = $true )
& ( dsetconstr__Cong = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR11 = $true )
& ( funcinfuncset = $true )
& ( upairsetE = $true )
& ( demorgan1 = $true )
& ( cartprodfstpairEq = $true )
& ( setminusER = $true )
& ( setadjoinIR = $true )
& ( setadjoinOr = $true )
& ( setminusSubset1 = $true )
& ( setunionE2 = $true )
& ( setminusELneg = $true )
& ( ifp = $true )
& ( eta1 = $true )
& ( secondinupair = $true )
& ( powersetI1 = $true )
& ( eqimpsubset2 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( exuE3u = $true )
& ( setukpairinjR12 = $true )
& ( binintersectT_lem = $true )
& ( notinsingleton = $true )
& ( singletoninpowunion = $true )
& ( doubleComplementI1 = $true )
& ( setadjoinAx = $true )
& ( ksndsingleton = $true )
& ( setukpairinjL1 = $true )
& ( breln1unionCommutes = $true )
& ( infuncsetfunc = $true )
& ( ap2p = $true )
& ( dpsetconstrI = $true )
& ( prop2setE = $true )
& ( cartprodmempair1 = $true )
& ( iffalseProp2 = $true )
& ( ubforcartprodlem1 = $true )
& ( woz13rule3 = $true )
& ( setminusSubset2 = $true )
& ( eqbreln1 = $true )
& ( notdallE = $true )
& ( binunionLsub = $true )
& ( dpsetconstrSub = $true )
& ( subsetemptysetimpeq = $true )
& ( inIntersectImpInUnion = $true )
& ( omega__Cong = $true )
& ( woz2B = $true )
& ( setukpairinjR = $true )
& ( surjFuncSetFuncSurj = $true )
& ( complementImpComplementIntersect = $true )
& ( singletonsuniq = $true )
& ( wellorderingAx = $true )
& ( setukpairinjR1 = $true )
& ( ex1I2 = $true )
& ( iftrueProp2 = $true )
& ( notinself2 = $true )
& ( setukpairIR = $true )
& ( powersetI = $true )
& ( upairsetIL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK963,sK964,sK965])],[f1311,f3187,f3186]) ).
thf(f3186,plain,
( ? [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
& ? [X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X1 @ X2 )
= $true )
& ( ( in @ X1 @ X0 )
!= $true ) ) )
=> ( ( ( ordinal @ sK963 )
= $true )
& ? [X2: $i,X1: $i] :
( ( $true
= ( in @ X2 @ sK963 ) )
& ( ( in @ X1 @ X2 )
= $true )
& ( ( in @ X1 @ sK963 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f3187,plain,
( ? [X2: $i,X1: $i] :
( ( $true
= ( in @ X2 @ sK963 ) )
& ( ( in @ X1 @ X2 )
= $true )
& ( ( in @ X1 @ sK963 )
!= $true ) )
=> ( ( ( in @ sK965 @ sK963 )
= $true )
& ( ( in @ sK964 @ sK965 )
= $true )
& ( $true
!= ( in @ sK964 @ sK963 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1311,plain,
( ( setextsub = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( notequalI1 = $true )
& ( woz13rule2 = $true )
& ( setunionsingleton = $true )
& ( singletonprop = $true )
& ( theeq = $true )
& ( breln1all1 = $true )
& ( emptyE1 = $true )
& ( upairinpowunion = $true )
& ( setunionTransitive = $true )
& ( ex1E1 = $true )
& ( emptyinPowerset = $true )
& ( complementTE1 = $true )
& ( upairsubunion = $true )
& ( exuI1 = $true )
& ( ifSingleton = $true )
& ( breln1invI = $true )
& ( cartprodmempair = $true )
& ( image1Ex = $true )
& ( symdiffIneg1 = $true )
& ( lamProp = $true )
& ( replAx = $true )
& ( ex1E2 = $true )
& ( notequalI2 = $true )
& ( binintersectSubset4 = $true )
& ( image1Equiv = $true )
& ( complementSubsetComplementIntersect = $true )
& ( breln1invprop = $true )
& ( iftrueProp1 = $true )
& ( binintersectTELcontra = $true )
& ( breln1SetBreln1 = $true )
& ( exuE1 = $true )
& ( woz2A = $true )
& ( inPowerset = $true )
& ( inCongP = $true )
& ( singletonsubset = $true )
& ( brelnall1 = $true )
& ( binintersectER = $true )
& ( app = $true )
& ( upairset2E = $true )
& ( emptysetAx = $true )
& ( cartprodpairin = $true )
& ( woz1_5 = $true )
& ( setbeta = $true )
& ( funcext = $true )
& ( breln1unionIL = $true )
& ( woz13rule1 = $true )
& ( setukpairinjR2 = $true )
& ( powersetTI1 = $true )
& ( cartprodpairmemER = $true )
& ( kpairp = $true )
& ( doubleComplementSub1 = $true )
& ( subsetI2 = $true )
& ( vacuousDall = $true )
& ( setukpairinjL = $true )
& ( image1E = $true )
& ( emptyI = $true )
& ( woz13rule0 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( funcext2 = $true )
& ( apProp = $true )
& ( setOfPairsIsBReln1 = $true )
& ( descrp = $true )
& ( breln1compE = $true )
& ( funcGraphProp4 = $true )
& ( setukpairIL = $true )
& ( setoftrueEq = $true )
& ( powersetT_lem = $true )
& ( funcextLem = $true )
& ( powersetsubset = $true )
& ( breln1unionIR = $true )
& ( emptyinunitempty = $true )
& ( binunionTIRcontra = $true )
& ( lam2lamEq = $true )
& ( setOfPairsIsBReln = $true )
& ( setunionI = $true )
& ( doubleComplementEq = $true )
& ( emptysetimpfalse = $true )
& ( noeltsimpempty = $true )
& ( descr__Cong = $true )
& ( beta2 = $true )
& ( binintersectTERcontra = $true )
& ( funcGraphProp3 = $true )
& ( kpairiskpair = $true )
& ( setextT = $true )
& ( breln1compI = $true )
& ( setunionsingleton1 = $true )
& ( image1I = $true )
& ( sepInPowerset = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodpairsurjEq = $true )
& ( exuE2 = $true )
& ( setadjoinSub = $true )
& ( powersetE = $true )
& ( setunionE = $true )
& ( image1Ex1 = $true )
& ( emptyset__Cong = $true )
& ( eta2 = $true )
& ( breln1unionprop = $true )
& ( omegaSAx = $true )
& ( complementTnotintersectT = $true )
& ( cartprodmempaircEq = $true )
& ( kfstpairEq = $true )
& ( lamp = $true )
& ( dsetconstrI = $true )
& ( contrasubsetT = $true )
& ( peano0notS = $true )
& ( demorgan2b = $true )
& ( powersetAx = $true )
& ( surjFuncSetFuncIn = $true )
& ( binunionEcases = $true )
& ( nonemptyI = $true )
& ( binunionTILcontra = $true )
& ( breln1unionI = $true )
& ( nonemptyE1 = $true )
& ( binunionRsub = $true )
& ( binunionTE = $true )
& ( cartprodsndpairEq = $true )
& ( breln1invE = $true )
& ( transitivesetOp2 = $true )
& ( transitivesetOp1 = $true )
& ( funcGraphProp1 = $true )
& ( setukpairinjL2 = $true )
& ( iftrue = $true )
& ( binunionIL = $true )
& ( binunionE = $true )
& ( setminusIRneg = $true )
& ( upairequniteq = $true )
& ( woz1_1 = $true )
& ( dpsetconstrER = $true )
& ( setext = $true )
& ( notinself = $true )
& ( sepSubset = $true )
& ( injFuncSetFuncIn = $true )
& ( binintersectSubset3 = $true )
& ( setadjoin__Cong = $true )
& ( woz1_4 = $true )
& ( dsetconstrEL = $true )
& ( omegaIndAx = $true )
& ( contraSubsetComplement = $true )
& ( disjointsetsI1 = $true )
& ( contrasubsetT2 = $true )
& ( cartprodsndin = $true )
& ( contrasubsetT1 = $true )
& ( demorgan2b2 = $true )
& ( ap2apEq1 = $true )
& ( subset2powerset = $true )
& ( symdiffE = $true )
& ( symdiffIneg2 = $true )
& ( complementT_lem = $true )
& ( binintersectLsub = $true )
& ( woz1_2 = $true )
& ( iffalseProp1 = $true )
& ( omegaSp = $true )
& ( setminusILneg = $true )
& ( emptysetsubset = $true )
& ( dpsetconstrEL1 = $true )
& ( notdexE = $true )
& ( iftrueorfalse = $true )
& ( singletonsswitch = $true )
& ( setminusEL = $true )
& ( injFuncSetFuncInj = $true )
& ( singletoninpowerset = $true )
& ( setminusERneg = $true )
& ( eqbreln = $true )
& ( quantDeMorgan3 = $true )
& ( ordinalMinLem1 = $true )
& ( setadjoinSub2 = $true )
& ( lam2p = $true )
& ( theprop = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( brelnall2 = $true )
& ( quantDeMorgan4 = $true )
& ( doubleComplementE1 = $true )
& ( notinemptyset = $true )
& ( subsetE2 = $true )
& ( setunion__Cong = $true )
& ( contrasubsetT3 = $true )
& ( setunionsingleton2 = $true )
& ( breln1compEex = $true )
& ( binintersectI = $true )
& ( exuEu = $true )
& ( complementTI1 = $true )
& ( setadjoinIL = $true )
& ( setminusLsub = $true )
& ( setadjoinE = $true )
& ( eqimpsubset1 = $true )
& ( prop2setI = $true )
& ( powersetE1 = $true )
& ( upairsetIR = $true )
& ( doubleComplementSub2 = $true )
& ( woz2W = $true )
& ( demorgan2 = $true )
& ( ksndpairEq = $true )
& ( exuE3e = $true )
& ( binintTransitive = $true )
& ( demorgan1a = $true )
& ( breln1compprop = $true )
& ( binunionTEcontra = $true )
& ( setminusI = $true )
& ( injFuncInInjFuncSet = $true )
& ( ubforcartprodlem2 = $true )
& ( exuI3 = $true )
& ( setextAx = $true )
& ( notsubsetI = $true )
& ( subbreln = $true )
& ( peanoSinj = $true )
& ( foundation2 = $true )
& ( demorgan2a1 = $true )
& ( emptysetE = $true )
& ( binintersectSubset2 = $true )
& ( breln1all2 = $true )
& ( uniqinunit = $true )
& ( subsetTrans = $true )
& ( setminusT_lem = $true )
& ( subsetTI = $true )
& ( binintersectRsub = $true )
& ( iffalse = $true )
& ( omega0Ax = $true )
& ( powersetTE1 = $true )
& ( demorgan1b = $true )
& ( powerset__Cong = $true )
& ( subsetRefl = $true )
& ( binunionIR = $true )
& ( emptyInPowerset = $true )
& ( cartprodpairmemEL = $true )
& ( choice2fnsingleton = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan2 = $true )
& ( exu__Cong = $true )
& ( subbreln1 = $true )
& ( ubforcartprodlem3 = $true )
& ( woz1_3 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( nonemptyI1 = $true )
& ( symdiffI2 = $true )
& ( demorgan2a = $true )
& ( cartprodfstin = $true )
& ( complementTcontraSubset = $true )
& ( kpairsurjEq = $true )
& ( dpsetconstrEL2 = $true )
& ( ex1I = $true )
& ( subsetE = $true )
& ( breln1unionE = $true )
& ( beta1 = $true )
& ( bs114d = $true )
& ( leftInvIsSurj = $true )
& ( binintersectEL = $true )
& ( omegaSclos = $true )
& ( binintersectSubset5 = $true )
& ( breln1unionEcases = $true )
& ( upairset2IR = $true )
& ( kfstsingleton = $true )
& ( foundationAx = $true )
& ( funcImageSingleton = $true )
& ( prop2set2propI = $true )
& ( in__Cong = $true )
& ( demorgan2a2 = $true )
& ( ap2apEq2 = $true )
& ( nonemptyImpWitness = $true )
& ( dpsetconstrERa = $true )
& ( funcGraphProp2 = $true )
& ( exuI2 = $true )
& ( eqinunit = $true )
& ( subsetI1 = $true )
& ( binunionT_lem = $true )
& ( surjCantorThm = $true )
& ( woz2Ex = $true )
& ( setunionAx = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( dsetconstrER = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( woz13rule4 = $true )
& ? [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
& ? [X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X1 @ X2 )
= $true )
& ( ( in @ X1 @ X0 )
!= $true ) ) )
& ( subPowSU = $true )
& ( dsetconstr__Cong = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR11 = $true )
& ( funcinfuncset = $true )
& ( upairsetE = $true )
& ( demorgan1 = $true )
& ( cartprodfstpairEq = $true )
& ( setminusER = $true )
& ( setadjoinIR = $true )
& ( setadjoinOr = $true )
& ( setminusSubset1 = $true )
& ( setunionE2 = $true )
& ( setminusELneg = $true )
& ( ifp = $true )
& ( eta1 = $true )
& ( secondinupair = $true )
& ( powersetI1 = $true )
& ( eqimpsubset2 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( exuE3u = $true )
& ( setukpairinjR12 = $true )
& ( binintersectT_lem = $true )
& ( notinsingleton = $true )
& ( singletoninpowunion = $true )
& ( doubleComplementI1 = $true )
& ( setadjoinAx = $true )
& ( ksndsingleton = $true )
& ( setukpairinjL1 = $true )
& ( breln1unionCommutes = $true )
& ( infuncsetfunc = $true )
& ( ap2p = $true )
& ( dpsetconstrI = $true )
& ( prop2setE = $true )
& ( cartprodmempair1 = $true )
& ( iffalseProp2 = $true )
& ( ubforcartprodlem1 = $true )
& ( woz13rule3 = $true )
& ( setminusSubset2 = $true )
& ( eqbreln1 = $true )
& ( notdallE = $true )
& ( binunionLsub = $true )
& ( dpsetconstrSub = $true )
& ( subsetemptysetimpeq = $true )
& ( inIntersectImpInUnion = $true )
& ( omega__Cong = $true )
& ( woz2B = $true )
& ( setukpairinjR = $true )
& ( surjFuncSetFuncSurj = $true )
& ( complementImpComplementIntersect = $true )
& ( singletonsuniq = $true )
& ( wellorderingAx = $true )
& ( setukpairinjR1 = $true )
& ( ex1I2 = $true )
& ( iftrueProp2 = $true )
& ( notinself2 = $true )
& ( setukpairIR = $true )
& ( powersetI = $true )
& ( upairsetIL = $true ) ),
inference(flattening,[status(thm)],[f1310]) ).
thf(f1310,plain,
( ? [X0: $i] :
( ? [X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
!= $true )
& ( ( in @ X1 @ X2 )
= $true )
& ( ( in @ X2 @ X0 )
= $true ) )
& ( ( ordinal @ X0 )
= $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)],[f363]) ).
thf(f363,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 )
=> ! [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
=> ! [X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ( ( ( in @ X1 @ X2 )
= $true )
=> ( ( in @ X1 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f362]) ).
thf(f362,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
=> ! [X0: $i] :
( ( ordinal @ X0 )
=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ( ( in @ X1 @ X2 )
=> ( in @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f341]) ).
thf(f341,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
=> ! [X11: $i] :
( ( ordinal @ X11 )
=> ! [X1: $i,X3: $i] :
( ( in @ X3 @ X11 )
=> ( ( in @ X1 @ X3 )
=> ( in @ X1 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f340]) ).
thf(f340,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
=> ! [X11: $i] :
( ( ordinal @ X11 )
=> ! [X1: $i,X3: $i] :
( ( in @ X3 @ X11 )
=> ( ( in @ X1 @ X3 )
=> ( in @ X1 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2THF_13596.p',ordinalTransSet) ).
thf(f9907,plain,
( ( transitiveset @ sK963 )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f9904,f4970]) ).
thf(f4970,plain,
( ( in @ sK965 @ sK963 )
= $true ),
inference(cnf_transformation,[status(thm)],[f3188]) ).
thf(f9904,plain,
( ( ( in @ sK965 @ sK963 )
!= $true )
| ( ( transitiveset @ sK963 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f9898]) ).
thf(f9898,plain,
( ( ( in @ sK965 @ sK963 )
!= $true )
| ( ( transitiveset @ sK963 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f9862,f7025]) ).
thf(f7025,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
| ( $true
!= ( transitiveset @ X2 ) )
| ( ( in @ X3 @ X2 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f6767]) ).
thf(f6767,plain,
! [X2: $i,X3: $i] :
( ( $true
!= ( transitiveset @ X2 ) )
| ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ X3 @ X2 )
= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f5533,f5124]) ).
thf(f5124,plain,
transitivesetOp1 = $true,
inference(cnf_transformation,[status(thm)],[f3188]) ).
thf(f5533,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( $true
!= ( transitiveset @ X2 ) )
| ( transitivesetOp1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f3526]) ).
thf(f3526,plain,
( ( ( transitivesetOp1 = $true )
| ( ( ( subset @ sK1174 @ sK1173 )
!= $true )
& ( ( in @ sK1174 @ sK1173 )
= $true )
& ( $true
= ( transitiveset @ sK1173 ) ) ) )
& ( ! [X2: $i] :
( ! [X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
| ( ( in @ X3 @ X2 )
!= $true ) )
| ( $true
!= ( transitiveset @ X2 ) ) )
| ( transitivesetOp1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1173,sK1174])],[f3523,f3525,f3524]) ).
thf(f3524,plain,
( ? [X0: $i] :
( ? [X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
& ( ( in @ X1 @ X0 )
= $true ) )
& ( $true
= ( transitiveset @ X0 ) ) )
=> ( ? [X1: $i] :
( ( $true
!= ( subset @ X1 @ sK1173 ) )
& ( ( in @ X1 @ sK1173 )
= $true ) )
& ( $true
= ( transitiveset @ sK1173 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f3525,plain,
( ? [X1: $i] :
( ( $true
!= ( subset @ X1 @ sK1173 ) )
& ( ( in @ X1 @ sK1173 )
= $true ) )
=> ( ( ( subset @ sK1174 @ sK1173 )
!= $true )
& ( ( in @ sK1174 @ sK1173 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f3523,plain,
( ( ( transitivesetOp1 = $true )
| ? [X0: $i] :
( ? [X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
& ( ( in @ X1 @ X0 )
= $true ) )
& ( $true
= ( transitiveset @ X0 ) ) ) )
& ( ! [X2: $i] :
( ! [X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
| ( ( in @ X3 @ X2 )
!= $true ) )
| ( $true
!= ( transitiveset @ X2 ) ) )
| ( transitivesetOp1 != $true ) ) ),
inference(rectify,[status(thm)],[f3522]) ).
thf(f3522,plain,
( ( ( transitivesetOp1 = $true )
| ? [X0: $i] :
( ? [X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
& ( ( in @ X1 @ X0 )
= $true ) )
& ( $true
= ( transitiveset @ X0 ) ) ) )
& ( ! [X0: $i] :
( ! [X1: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
| ( ( in @ X1 @ X0 )
!= $true ) )
| ( $true
!= ( transitiveset @ X0 ) ) )
| ( transitivesetOp1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1131]) ).
thf(f1131,plain,
( ( transitivesetOp1 = $true )
<=> ! [X0: $i] :
( ! [X1: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
| ( ( in @ X1 @ X0 )
!= $true ) )
| ( $true
!= ( transitiveset @ X0 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f976]) ).
thf(f976,plain,
( ( transitivesetOp1 = $true )
<=> ! [X0: $i] :
( ( $true
= ( transitiveset @ X0 ) )
=> ! [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
=> ( $true
= ( subset @ X1 @ X0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f975]) ).
thf(f975,plain,
( transitivesetOp1
= ( ! [X0: $i] :
( ( transitiveset @ X0 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( subset @ X1 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f332]) ).
thf(f332,axiom,
( transitivesetOp1
= ( ! [X11: $i] :
( ( transitiveset @ X11 )
=> ! [X3: $i] :
( ( in @ X3 @ X11 )
=> ( subset @ X3 @ X11 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2THF_13596.p',transitivesetOp1) ).
thf(f9862,plain,
( $true
!= ( subset @ sK965 @ sK963 ) ),
inference(trivial_inequality_removal,[status(thm)],[f9850]) ).
thf(f9850,plain,
( ( $true != $true )
| ( $true
!= ( subset @ sK965 @ sK963 ) ) ),
inference(superposition,[status(thm)],[f8933,f4969]) ).
thf(f4969,plain,
( ( in @ sK964 @ sK965 )
= $true ),
inference(cnf_transformation,[status(thm)],[f3188]) ).
thf(f8933,plain,
! [X0: $i] :
( ( $true
!= ( in @ sK964 @ X0 ) )
| ( ( subset @ X0 @ sK963 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f8926]) ).
thf(f8926,plain,
! [X0: $i] :
( ( ( subset @ X0 @ sK963 )
!= $true )
| ( $true != $true )
| ( $true
!= ( in @ sK964 @ X0 ) ) ),
inference(superposition,[status(thm)],[f4968,f6841]) ).
thf(f6841,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( $true
!= ( subset @ X2 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f6551]) ).
thf(f6551,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true != $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( $true
!= ( subset @ X2 @ X0 ) )
| ( ( in @ X1 @ X0 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f5266,f5040]) ).
thf(f5040,plain,
notsubsetI = $true,
inference(cnf_transformation,[status(thm)],[f3188]) ).
thf(f5266,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( subset @ X2 @ X0 ) )
| ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( notsubsetI != $true ) ),
inference(cnf_transformation,[status(thm)],[f3210]) ).
thf(f3210,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X0 ) )
| ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
| ( notsubsetI != $true ) )
& ( ( notsubsetI = $true )
| ( ( ( subset @ sK979 @ sK977 )
= $true )
& ( ( in @ sK978 @ sK977 )
!= $true )
& ( ( in @ sK978 @ sK979 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK977,sK978,sK979])],[f3208,f3209]) ).
thf(f3209,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X3 )
= $true )
& ( ( in @ X4 @ X3 )
!= $true )
& ( $true
= ( in @ X4 @ X5 ) ) )
=> ( ( ( subset @ sK979 @ sK977 )
= $true )
& ( ( in @ sK978 @ sK977 )
!= $true )
& ( ( in @ sK978 @ sK979 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f3208,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X0 ) )
| ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
| ( notsubsetI != $true ) )
& ( ( notsubsetI = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X3 )
= $true )
& ( ( in @ X4 @ X3 )
!= $true )
& ( $true
= ( in @ X4 @ X5 ) ) ) ) ),
inference(rectify,[status(thm)],[f3207]) ).
thf(f3207,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X0 ) )
| ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
| ( notsubsetI != $true ) )
& ( ( notsubsetI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X2 @ X0 ) )
& ( ( in @ X1 @ X0 )
!= $true )
& ( ( in @ X1 @ X2 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1321]) ).
thf(f1321,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X0 ) )
| ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
<=> ( notsubsetI = $true ) ),
inference(flattening,[status(thm)],[f1320]) ).
thf(f1320,plain,
( ( notsubsetI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X0 ) )
| ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f1040]) ).
thf(f1040,plain,
( ( notsubsetI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( ( in @ X1 @ X0 )
!= $true )
=> ( $true
!= ( subset @ X2 @ X0 ) ) ) ) ),
inference(flattening,[status(thm)],[f475]) ).
thf(f475,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( ( in @ X1 @ X0 )
!= $true )
=> ( $true
!= ( subset @ X2 @ X0 ) ) ) )
<=> ( notsubsetI = $true ) ),
inference(fool_elimination,[status(thm)],[f474]) ).
thf(f474,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
=> ( ~ ( in @ X1 @ X0 )
=> ~ ( subset @ X2 @ X0 ) ) )
= notsubsetI ),
inference(rectify,[status(thm)],[f85]) ).
thf(f85,axiom,
! [X4: $i,X1: $i,X3: $i] :
( ( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ X4 )
=> ~ ( subset @ X3 @ X4 ) ) )
= notsubsetI ),
file('/export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2THF_13596.p',notsubsetI) ).
thf(f4968,plain,
( $true
!= ( in @ sK964 @ sK963 ) ),
inference(cnf_transformation,[status(thm)],[f3188]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU817^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n018.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:34:46 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.48 ---- Original DTF file ---
% 0.31/0.48 thf(spec,logic,$$dhol).
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 % File : SEU817^1 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.48 % Domain : Set Theory
% 0.31/0.48 % Problem : Ordinals
% 0.31/0.48 % Version : Especial.
% 0.31/0.48 % English : (! X:i.ordinal X -> (! x:i.! A:i.in A X -> in x A -> in x X))
% 0.31/0.48
% 0.31/0.48 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.48 % Source : [Bro08]
% 0.31/0.48 % Names : ZFC319g [Bro08]
% 0.31/0.48
% 0.31/0.48 % Status : Theorem
% 0.31/0.48 % Rating : 0.22 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, 0.44 v7.2.0, 0.38 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
% 0.31/0.48 % Syntax : Number of formulae : 736 ( 339 unt; 396 typ; 339 def)
% 0.31/0.48 % Number of atoms : 2095 ( 486 equ; 0 cnn)
% 0.31/0.48 % Maximal formula atoms : 337 ( 6 avg)
% 0.31/0.48 % Number of connectives : 4541 ( 78 ~; 17 |; 60 &;3293 @)
% 0.31/0.48 % ( 17 <=>;1076 =>; 0 <=; 0 <~>)
% 0.31/0.48 % Maximal formula depth : 342 ( 2 avg)
% 0.31/0.48 % Number of types : 2 ( 0 usr)
% 0.31/0.48 % Number of type conns : 194 ( 194 >; 0 *; 0 +; 0 <<)
% 0.31/0.48 % Number of symbols : 399 ( 396 usr; 337 con; 0-4 aty)
% 0.31/0.48 % Number of variables : 1179 ( 79 ^;1055 !; 45 ?;1179 :)
% 0.31/0.48 % SPC : TH0_THM_EQU_NAR
% 0.31/0.48
% 0.31/0.48 % Comments : http://mathgate.info/detsetitem.php?id=513
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 thf(in_type,type,
% 0.31/0.48 in: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(exu_type,type,
% 0.31/0.48 exu: ( $i > $o ) > $o ).
% 0.31/0.48
% 0.31/0.48 thf(exu,definition,
% 0.31/0.48 ( exu
% 0.31/0.48 = ( ^ [Xphi: $i > $o] :
% 0.31/0.48 ? [Xx: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 & ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setextAx_type,type,
% 0.31/0.48 setextAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setextAx,definition,
% 0.31/0.48 ( setextAx
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 <=> ( in @ Xx @ B ) )
% 0.31/0.48 => ( A = B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyset_type,type,
% 0.31/0.48 emptyset: $i ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetAx_type,type,
% 0.31/0.48 emptysetAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetAx,definition,
% 0.31/0.48 ( emptysetAx
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoin_type,type,
% 0.31/0.48 setadjoin: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinAx_type,type,
% 0.31/0.48 setadjoinAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinAx,definition,
% 0.31/0.48 ( setadjoinAx
% 0.31/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.48 <=> ( ( Xy = Xx )
% 0.31/0.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powerset_type,type,
% 0.31/0.48 powerset: $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(powersetAx_type,type,
% 0.31/0.48 powersetAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetAx,definition,
% 0.31/0.48 ( powersetAx
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.48 <=> ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunion_type,type,
% 0.31/0.48 setunion: $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(setunionAx_type,type,
% 0.31/0.48 setunionAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionAx,definition,
% 0.31/0.48 ( setunionAx
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.48 <=> ? [B: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(omega_type,type,
% 0.31/0.48 omega: $i ).
% 0.31/0.48
% 0.31/0.48 thf(omega0Ax_type,type,
% 0.31/0.48 omega0Ax: $o ).
% 0.31/0.48
% 0.31/0.48 thf(omega0Ax,definition,
% 0.31/0.48 ( omega0Ax
% 0.31/0.48 = ( in @ emptyset @ omega ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(omegaSAx_type,type,
% 0.31/0.48 omegaSAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(omegaSAx,definition,
% 0.31/0.48 ( omegaSAx
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ omega )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(omegaIndAx_type,type,
% 0.31/0.48 omegaIndAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(omegaIndAx,definition,
% 0.31/0.48 ( omegaIndAx
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ( ( in @ emptyset @ A )
% 0.31/0.48 & ! [Xx: $i] :
% 0.31/0.48 ( ( ( in @ Xx @ omega )
% 0.31/0.48 & ( in @ Xx @ A ) )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ omega )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(replAx_type,type,
% 0.31/0.48 replAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(replAx,definition,
% 0.31/0.48 ( replAx
% 0.31/0.48 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( exu
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.48 => ? [B: $i] :
% 0.31/0.48 ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 <=> ? [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(foundationAx_type,type,
% 0.31/0.48 foundationAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(foundationAx,definition,
% 0.31/0.48 ( foundationAx
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.48 => ? [B: $i] :
% 0.31/0.48 ( ( in @ B @ A )
% 0.31/0.48 & ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(wellorderingAx_type,type,
% 0.31/0.48 wellorderingAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(wellorderingAx,definition,
% 0.31/0.48 ( wellorderingAx
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ? [B: $i] :
% 0.31/0.48 ( ! [C: $i] :
% 0.31/0.48 ( ( in @ C @ B )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ C )
% 0.31/0.48 => ( in @ Xx @ A ) ) )
% 0.31/0.48 & ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( ( in @ Xx @ A )
% 0.31/0.48 & ( in @ Xy @ A ) )
% 0.31/0.48 => ( ! [C: $i] :
% 0.31/0.48 ( ( in @ C @ B )
% 0.31/0.48 => ( ( in @ Xx @ C )
% 0.31/0.48 <=> ( in @ Xy @ C ) ) )
% 0.31/0.48 => ( Xx = Xy ) ) )
% 0.31/0.48 & ! [C: $i,D: $i] :
% 0.31/0.48 ( ( ( in @ C @ B )
% 0.31/0.48 & ( in @ D @ B ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ C )
% 0.31/0.48 => ( in @ Xx @ D ) )
% 0.31/0.48 | ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ D )
% 0.31/0.48 => ( in @ Xx @ C ) ) ) )
% 0.31/0.48 & ! [C: $i] :
% 0.31/0.48 ( ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ C )
% 0.31/0.48 => ( in @ Xx @ A ) )
% 0.31/0.48 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/0.48 => ? [D: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ D @ B )
% 0.31/0.48 & ( in @ Xx @ C )
% 0.31/0.48 & ~ ? [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ D )
% 0.31/0.48 & ( in @ Xy @ C ) )
% 0.31/0.48 & ! [E: $i] :
% 0.31/0.48 ( ( in @ E @ B )
% 0.31/0.48 => ( ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ E )
% 0.31/0.48 => ( in @ Xy @ D ) )
% 0.31/0.48 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(descr_type,type,
% 0.31/0.48 descr: ( $i > $o ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(descrp_type,type,
% 0.31/0.48 descrp: $o ).
% 0.31/0.48
% 0.31/0.48 thf(descrp,definition,
% 0.31/0.48 ( descrp
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ( Xphi
% 0.31/0.48 @ ( descr
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstr_type,type,
% 0.31/0.48 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrI_type,type,
% 0.31/0.48 dsetconstrI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrI,definition,
% 0.31/0.48 ( dsetconstrI
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrEL_type,type,
% 0.31/0.48 dsetconstrEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrEL,definition,
% 0.31/0.48 ( dsetconstrEL
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrER_type,type,
% 0.31/0.48 dsetconstrER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrER,definition,
% 0.31/0.48 ( dsetconstrER
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.48 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuE1_type,type,
% 0.31/0.48 exuE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuE1,definition,
% 0.31/0.48 ( exuE1
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 & ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(prop2set_type,type,
% 0.31/0.48 prop2set: $o > $i ).
% 0.31/0.48
% 0.31/0.48 thf(prop2setE_type,type,
% 0.31/0.48 prop2setE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(prop2setE,definition,
% 0.31/0.48 ( prop2setE
% 0.31/0.48 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/0.48 => Xphi ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetE_type,type,
% 0.31/0.48 emptysetE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptysetE,definition,
% 0.31/0.49 ( emptysetE
% 0.31/0.49 = ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ emptyset )
% 0.31/0.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(emptysetimpfalse_type,type,
% 0.31/0.49 emptysetimpfalse: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptysetimpfalse,definition,
% 0.31/0.49 ( emptysetimpfalse
% 0.31/0.49 = ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ emptyset )
% 0.31/0.49 => $false ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(notinemptyset_type,type,
% 0.31/0.49 notinemptyset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(notinemptyset,definition,
% 0.31/0.49 ( notinemptyset
% 0.31/0.49 = ( ! [Xx: $i] :
% 0.31/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(exuE3e_type,type,
% 0.31/0.49 exuE3e: $o ).
% 0.31/0.49
% 0.31/0.49 thf(exuE3e,definition,
% 0.31/0.49 ( exuE3e
% 0.31/0.49 = ( ! [Xphi: $i > $o] :
% 0.31/0.49 ( ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setext_type,type,
% 0.31/0.49 setext: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setext,definition,
% 0.31/0.49 ( setext
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ Xx @ B ) )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ B )
% 0.31/0.49 => ( in @ Xx @ A ) )
% 0.31/0.49 => ( A = B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(emptyI_type,type,
% 0.31/0.49 emptyI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptyI,definition,
% 0.31/0.49 ( emptyI
% 0.31/0.49 = ( ! [A: $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ~ ( in @ Xx @ A )
% 0.31/0.49 => ( A = emptyset ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(noeltsimpempty_type,type,
% 0.31/0.49 noeltsimpempty: $o ).
% 0.31/0.49
% 0.31/0.49 thf(noeltsimpempty,definition,
% 0.31/0.49 ( noeltsimpempty
% 0.31/0.49 = ( ! [A: $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ~ ( in @ Xx @ A )
% 0.31/0.49 => ( A = emptyset ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setbeta_type,type,
% 0.31/0.49 setbeta: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setbeta,definition,
% 0.31/0.49 ( setbeta
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx
% 0.31/0.49 @ ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(nonempty_type,type,
% 0.31/0.49 nonempty: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(nonempty,definition,
% 0.31/0.49 ( nonempty
% 0.31/0.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(nonemptyE1_type,type,
% 0.31/0.49 nonemptyE1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(nonemptyE1,definition,
% 0.31/0.49 ( nonemptyE1
% 0.31/0.49 = ( ! [A: $i] :
% 0.31/0.49 ( ( nonempty @ A )
% 0.31/0.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(nonemptyI_type,type,
% 0.31/0.49 nonemptyI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(nonemptyI,definition,
% 0.31/0.49 ( nonemptyI
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( Xphi @ Xx )
% 0.31/0.49 => ( nonempty
% 0.31/0.49 @ ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(nonemptyI1_type,type,
% 0.31/0.49 nonemptyI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(nonemptyI1,definition,
% 0.31/0.49 ( nonemptyI1
% 0.31/0.49 = ( ! [A: $i] :
% 0.31/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.49 => ( nonempty @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinIL_type,type,
% 0.31/0.49 setadjoinIL: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinIL,definition,
% 0.31/0.49 ( setadjoinIL
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(emptyinunitempty_type,type,
% 0.31/0.49 emptyinunitempty: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptyinunitempty,definition,
% 0.31/0.49 ( emptyinunitempty
% 0.31/0.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinIR_type,type,
% 0.31/0.49 setadjoinIR: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinIR,definition,
% 0.31/0.49 ( setadjoinIR
% 0.31/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinE_type,type,
% 0.31/0.49 setadjoinE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinE,definition,
% 0.31/0.49 ( setadjoinE
% 0.31/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.49 => ! [Xphi: $o] :
% 0.31/0.49 ( ( ( Xy = Xx )
% 0.31/0.49 => Xphi )
% 0.31/0.49 => ( ( ( in @ Xy @ A )
% 0.31/0.49 => Xphi )
% 0.31/0.49 => Xphi ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinOr_type,type,
% 0.31/0.49 setadjoinOr: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinOr,definition,
% 0.31/0.49 ( setadjoinOr
% 0.31/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.49 => ( ( Xy = Xx )
% 0.31/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setoftrueEq_type,type,
% 0.31/0.49 setoftrueEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setoftrueEq,definition,
% 0.31/0.49 ( setoftrueEq
% 0.31/0.49 = ( ! [A: $i] :
% 0.31/0.49 ( ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xx: $i] : $true )
% 0.31/0.49 = A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powersetI_type,type,
% 0.31/0.49 powersetI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powersetI,definition,
% 0.31/0.49 ( powersetI
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ B )
% 0.31/0.49 => ( in @ Xx @ A ) )
% 0.31/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(emptyinPowerset_type,type,
% 0.31/0.49 emptyinPowerset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptyinPowerset,definition,
% 0.31/0.49 ( emptyinPowerset
% 0.31/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(emptyInPowerset_type,type,
% 0.31/0.49 emptyInPowerset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptyInPowerset,definition,
% 0.31/0.49 ( emptyInPowerset
% 0.31/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powersetE_type,type,
% 0.31/0.49 powersetE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powersetE,definition,
% 0.31/0.49 ( powersetE
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.49 => ( ( in @ Xx @ B )
% 0.31/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setunionI_type,type,
% 0.31/0.49 setunionI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setunionI,definition,
% 0.31/0.49 ( setunionI
% 0.31/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.49 ( ( in @ Xx @ B )
% 0.31/0.49 => ( ( in @ B @ A )
% 0.31/0.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setunionE_type,type,
% 0.31/0.49 setunionE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setunionE,definition,
% 0.31/0.49 ( setunionE
% 0.31/0.49 = ( ! [A: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.49 => ! [Xphi: $o] :
% 0.31/0.49 ( ! [B: $i] :
% 0.31/0.49 ( ( in @ Xx @ B )
% 0.31/0.49 => ( ( in @ B @ A )
% 0.31/0.49 => Xphi ) )
% 0.31/0.49 => Xphi ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subPowSU_type,type,
% 0.31/0.49 subPowSU: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subPowSU,definition,
% 0.31/0.49 ( subPowSU
% 0.31/0.49 = ( ! [A: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(exuE2_type,type,
% 0.31/0.49 exuE2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(exuE2,definition,
% 0.31/0.49 ( exuE2
% 0.31/0.49 = ( ! [Xphi: $i > $o] :
% 0.31/0.49 ( ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 => ? [Xx: $i] :
% 0.31/0.49 ! [Xy: $i] :
% 0.31/0.49 ( ( Xphi @ Xy )
% 0.31/0.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(nonemptyImpWitness_type,type,
% 0.31/0.49 nonemptyImpWitness: $o ).
% 0.31/0.49
% 0.31/0.49 thf(nonemptyImpWitness,definition,
% 0.31/0.49 ( nonemptyImpWitness
% 0.31/0.49 = ( ! [A: $i] :
% 0.31/0.49 ( ( nonempty @ A )
% 0.31/0.49 => ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & $true ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(uniqinunit_type,type,
% 0.31/0.49 uniqinunit: $o ).
% 0.31/0.49
% 0.31/0.49 thf(uniqinunit,definition,
% 0.31/0.49 ( uniqinunit
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(notinsingleton_type,type,
% 0.31/0.49 notinsingleton: $o ).
% 0.31/0.49
% 0.31/0.49 thf(notinsingleton,definition,
% 0.31/0.49 ( notinsingleton
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx != Xy )
% 0.31/0.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(eqinunit_type,type,
% 0.31/0.49 eqinunit: $o ).
% 0.31/0.49
% 0.31/0.49 thf(eqinunit,definition,
% 0.31/0.49 ( eqinunit
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx = Xy )
% 0.31/0.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(singletonsswitch_type,type,
% 0.31/0.49 singletonsswitch: $o ).
% 0.31/0.49
% 0.31/0.49 thf(singletonsswitch,definition,
% 0.31/0.49 ( singletonsswitch
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(upairsetE_type,type,
% 0.31/0.49 upairsetE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairsetE,definition,
% 0.31/0.49 ( upairsetE
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.49 => ( ( Xz = Xx )
% 0.31/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(upairsetIL_type,type,
% 0.31/0.49 upairsetIL: $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairsetIL,definition,
% 0.31/0.49 ( upairsetIL
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(upairsetIR_type,type,
% 0.31/0.49 upairsetIR: $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairsetIR,definition,
% 0.31/0.49 ( upairsetIR
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(emptyE1_type,type,
% 0.31/0.49 emptyE1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptyE1,definition,
% 0.31/0.49 ( emptyE1
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ( Xphi @ Xx ) )
% 0.31/0.49 => ( ( ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 = emptyset )
% 0.31/0.49 => $false ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(vacuousDall_type,type,
% 0.31/0.49 vacuousDall: $o ).
% 0.31/0.49
% 0.31/0.49 thf(vacuousDall,definition,
% 0.31/0.49 ( vacuousDall
% 0.31/0.49 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ emptyset )
% 0.31/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(quantDeMorgan1_type,type,
% 0.31/0.49 quantDeMorgan1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(quantDeMorgan1,definition,
% 0.31/0.49 ( quantDeMorgan1
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ~ ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( Xphi @ Xx ) )
% 0.31/0.49 => ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(quantDeMorgan2_type,type,
% 0.31/0.49 quantDeMorgan2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(quantDeMorgan2,definition,
% 0.31/0.49 ( quantDeMorgan2
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ~ ( Xphi @ Xx ) )
% 0.31/0.49 => ~ ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(quantDeMorgan3_type,type,
% 0.31/0.49 quantDeMorgan3: $o ).
% 0.31/0.49
% 0.31/0.49 thf(quantDeMorgan3,definition,
% 0.31/0.49 ( quantDeMorgan3
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ~ ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ( Xphi @ Xx ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(quantDeMorgan4_type,type,
% 0.31/0.49 quantDeMorgan4: $o ).
% 0.31/0.49
% 0.31/0.49 thf(quantDeMorgan4,definition,
% 0.31/0.49 ( quantDeMorgan4
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ~ ( Xphi @ Xx ) )
% 0.31/0.49 => ~ ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(prop2setI_type,type,
% 0.31/0.49 prop2setI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(prop2setI,definition,
% 0.31/0.49 ( prop2setI
% 0.31/0.49 = ( ! [Xphi: $o] :
% 0.31/0.49 ( Xphi
% 0.31/0.49 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(set2prop_type,type,
% 0.31/0.49 set2prop: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(prop2set2propI_type,type,
% 0.31/0.49 prop2set2propI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(prop2set2propI,definition,
% 0.31/0.49 ( prop2set2propI
% 0.31/0.49 = ( ! [Xphi: $o] :
% 0.31/0.49 ( Xphi
% 0.31/0.49 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(notdexE_type,type,
% 0.31/0.49 notdexE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(notdexE,definition,
% 0.31/0.49 ( notdexE
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ~ ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ( Xphi @ Xx ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(notdallE_type,type,
% 0.31/0.49 notdallE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(notdallE,definition,
% 0.31/0.49 ( notdallE
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ~ ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( Xphi @ Xx ) )
% 0.31/0.49 => ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(exuI1_type,type,
% 0.31/0.49 exuI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(exuI1,definition,
% 0.31/0.49 ( exuI1
% 0.31/0.49 = ( ! [Xphi: $i > $o] :
% 0.31/0.49 ( ? [Xx: $i] :
% 0.31/0.49 ( ( Xphi @ Xx )
% 0.31/0.49 & ! [Xy: $i] :
% 0.31/0.49 ( ( Xphi @ Xy )
% 0.31/0.49 => ( Xx = Xy ) ) )
% 0.31/0.49 => ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(exuI3_type,type,
% 0.31/0.49 exuI3: $o ).
% 0.31/0.49
% 0.31/0.49 thf(exuI3,definition,
% 0.31/0.49 ( exuI3
% 0.31/0.49 = ( ! [Xphi: $i > $o] :
% 0.31/0.49 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.31/0.49 => ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xphi @ Xx )
% 0.31/0.49 => ( ( Xphi @ Xy )
% 0.31/0.49 => ( Xx = Xy ) ) )
% 0.31/0.49 => ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(exuI2_type,type,
% 0.31/0.49 exuI2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(exuI2,definition,
% 0.31/0.49 ( exuI2
% 0.31/0.49 = ( ! [Xphi: $i > $o] :
% 0.31/0.49 ( ? [Xx: $i] :
% 0.31/0.49 ! [Xy: $i] :
% 0.31/0.49 ( ( Xphi @ Xy )
% 0.31/0.49 <=> ( Xy = Xx ) )
% 0.31/0.49 => ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(inCongP_type,type,
% 0.31/0.49 inCongP: $o ).
% 0.31/0.49
% 0.31/0.49 thf(inCongP,definition,
% 0.31/0.49 ( inCongP
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx = Xy )
% 0.31/0.49 => ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(in__Cong_type,type,
% 0.31/0.49 in__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(in__Cong,definition,
% 0.31/0.49 ( in__Cong
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx = Xy )
% 0.31/0.49 => ( ( in @ Xx @ A )
% 0.31/0.49 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(exuE3u_type,type,
% 0.31/0.49 exuE3u: $o ).
% 0.31/0.49
% 0.31/0.49 thf(exuE3u,definition,
% 0.31/0.49 ( exuE3u
% 0.31/0.49 = ( ! [Xphi: $i > $o] :
% 0.31/0.49 ( ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 => ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xphi @ Xx )
% 0.31/0.49 => ( ( Xphi @ Xy )
% 0.31/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(exu__Cong_type,type,
% 0.31/0.49 exu__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(exu__Cong,definition,
% 0.31/0.49 ( exu__Cong
% 0.31/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx = Xy )
% 0.31/0.49 => ( ( Xphi @ Xx )
% 0.31/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.49 => ( ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 <=> ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(emptyset__Cong_type,type,
% 0.31/0.49 emptyset__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptyset__Cong,definition,
% 0.31/0.49 ( emptyset__Cong
% 0.31/0.49 = ( emptyset = emptyset ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoin__Cong_type,type,
% 0.31/0.49 setadjoin__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoin__Cong,definition,
% 0.31/0.49 ( setadjoin__Cong
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx = Xy )
% 0.31/0.49 => ! [Xz: $i,Xu: $i] :
% 0.31/0.49 ( ( Xz = Xu )
% 0.31/0.49 => ( ( setadjoin @ Xx @ Xz )
% 0.31/0.49 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powerset__Cong_type,type,
% 0.31/0.49 powerset__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powerset__Cong,definition,
% 0.31/0.49 ( powerset__Cong
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ( ( powerset @ A )
% 0.31/0.49 = ( powerset @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setunion__Cong_type,type,
% 0.31/0.49 setunion__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setunion__Cong,definition,
% 0.31/0.49 ( setunion__Cong
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ( ( setunion @ A )
% 0.31/0.49 = ( setunion @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(omega__Cong_type,type,
% 0.31/0.49 omega__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(omega__Cong,definition,
% 0.31/0.49 ( omega__Cong
% 0.31/0.49 = ( omega = omega ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(exuEu_type,type,
% 0.31/0.49 exuEu: $o ).
% 0.31/0.49
% 0.31/0.49 thf(exuEu,definition,
% 0.31/0.49 ( exuEu
% 0.31/0.49 = ( ! [Xphi: $i > $o] :
% 0.31/0.49 ( ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 => ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xphi @ Xx )
% 0.31/0.49 => ( ( Xphi @ Xy )
% 0.31/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(descr__Cong_type,type,
% 0.31/0.49 descr__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(descr__Cong,definition,
% 0.31/0.49 ( descr__Cong
% 0.31/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx = Xy )
% 0.31/0.49 => ( ( Xphi @ Xx )
% 0.31/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.49 => ( ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 => ( ( exu
% 0.31/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/0.49 => ( ( descr
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 = ( descr
% 0.31/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(dsetconstr__Cong_type,type,
% 0.31/0.49 dsetconstr__Cong: $o ).
% 0.31/0.49
% 0.31/0.49 thf(dsetconstr__Cong,definition,
% 0.31/0.49 ( dsetconstr__Cong
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( Xx = Xy )
% 0.31/0.49 => ( ( Xphi @ Xx )
% 0.31/0.49 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/0.49 => ( ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 = ( dsetconstr @ B
% 0.31/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subset_type,type,
% 0.31/0.49 subset: $i > $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(disjoint_type,type,
% 0.31/0.49 disjoint: $i > $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(setsmeet_type,type,
% 0.31/0.49 setsmeet: $i > $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetI1_type,type,
% 0.31/0.49 subsetI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetI1,definition,
% 0.31/0.49 ( subsetI1
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ Xx @ B ) )
% 0.31/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(eqimpsubset2_type,type,
% 0.31/0.49 eqimpsubset2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(eqimpsubset2,definition,
% 0.31/0.49 ( eqimpsubset2
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(eqimpsubset1_type,type,
% 0.31/0.49 eqimpsubset1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(eqimpsubset1,definition,
% 0.31/0.49 ( eqimpsubset1
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( A = B )
% 0.31/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subsetI2_type,type,
% 0.31/0.49 subsetI2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetI2,definition,
% 0.31/0.49 ( subsetI2
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ Xx @ B ) )
% 0.31/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(emptysetsubset_type,type,
% 0.31/0.49 emptysetsubset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(emptysetsubset,definition,
% 0.31/0.49 ( emptysetsubset
% 0.31/0.49 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subsetE_type,type,
% 0.31/0.49 subsetE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetE,definition,
% 0.31/0.49 ( subsetE
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subsetE2_type,type,
% 0.31/0.49 subsetE2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetE2,definition,
% 0.31/0.49 ( subsetE2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( ~ ( in @ Xx @ B )
% 0.31/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(notsubsetI_type,type,
% 0.31/0.49 notsubsetI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(notsubsetI,definition,
% 0.31/0.49 ( notsubsetI
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ B )
% 0.31/0.49 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(notequalI1_type,type,
% 0.31/0.49 notequalI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(notequalI1,definition,
% 0.31/0.49 ( notequalI1
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ~ ( subset @ A @ B )
% 0.31/0.49 => ( A != B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(notequalI2_type,type,
% 0.31/0.49 notequalI2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(notequalI2,definition,
% 0.31/0.49 ( notequalI2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ B )
% 0.31/0.49 => ( A != B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subsetRefl_type,type,
% 0.31/0.49 subsetRefl: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetRefl,definition,
% 0.31/0.49 ( subsetRefl
% 0.31/0.49 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subsetTrans_type,type,
% 0.31/0.49 subsetTrans: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetTrans,definition,
% 0.31/0.49 ( subsetTrans
% 0.31/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( ( subset @ B @ C )
% 0.31/0.49 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinSub_type,type,
% 0.31/0.49 setadjoinSub: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinSub,definition,
% 0.31/0.49 ( setadjoinSub
% 0.31/0.49 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinSub2_type,type,
% 0.31/0.49 setadjoinSub2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setadjoinSub2,definition,
% 0.31/0.49 ( setadjoinSub2
% 0.31/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subset2powerset_type,type,
% 0.31/0.49 subset2powerset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subset2powerset,definition,
% 0.31/0.49 ( subset2powerset
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setextsub_type,type,
% 0.31/0.49 setextsub: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setextsub,definition,
% 0.31/0.49 ( setextsub
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( ( subset @ B @ A )
% 0.31/0.49 => ( A = B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subsetemptysetimpeq_type,type,
% 0.31/0.49 subsetemptysetimpeq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetemptysetimpeq,definition,
% 0.31/0.49 ( subsetemptysetimpeq
% 0.31/0.49 = ( ! [A: $i] :
% 0.31/0.49 ( ( subset @ A @ emptyset )
% 0.31/0.49 => ( A = emptyset ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powersetI1_type,type,
% 0.31/0.49 powersetI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powersetI1,definition,
% 0.31/0.49 ( powersetI1
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( subset @ B @ A )
% 0.31/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powersetE1_type,type,
% 0.31/0.49 powersetE1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powersetE1,definition,
% 0.31/0.49 ( powersetE1
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(inPowerset_type,type,
% 0.31/0.49 inPowerset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(inPowerset,definition,
% 0.31/0.49 ( inPowerset
% 0.31/0.49 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powersetsubset_type,type,
% 0.31/0.49 powersetsubset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powersetsubset,definition,
% 0.31/0.49 ( powersetsubset
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sepInPowerset_type,type,
% 0.31/0.49 sepInPowerset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(sepInPowerset,definition,
% 0.31/0.49 ( sepInPowerset
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( in
% 0.31/0.49 @ ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 @ ( powerset @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sepSubset_type,type,
% 0.31/0.49 sepSubset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(sepSubset,definition,
% 0.31/0.49 ( sepSubset
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( subset
% 0.31/0.49 @ ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunion_type,type,
% 0.31/0.49 binunion: $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(binunionIL_type,type,
% 0.31/0.49 binunionIL: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionIL,definition,
% 0.31/0.49 ( binunionIL
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(upairset2IR_type,type,
% 0.31/0.49 upairset2IR: $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairset2IR,definition,
% 0.31/0.49 ( upairset2IR
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionIR_type,type,
% 0.31/0.49 binunionIR: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionIR,definition,
% 0.31/0.49 ( binunionIR
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ B )
% 0.31/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionEcases_type,type,
% 0.31/0.49 binunionEcases: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionEcases,definition,
% 0.31/0.49 ( binunionEcases
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.49 => ( ( ( in @ Xx @ A )
% 0.31/0.49 => Xphi )
% 0.31/0.49 => ( ( ( in @ Xx @ B )
% 0.31/0.49 => Xphi )
% 0.31/0.49 => Xphi ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionE_type,type,
% 0.31/0.49 binunionE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionE,definition,
% 0.31/0.49 ( binunionE
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.49 => ( ( in @ Xx @ A )
% 0.31/0.49 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionLsub_type,type,
% 0.31/0.49 binunionLsub: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionLsub,definition,
% 0.31/0.49 ( binunionLsub
% 0.31/0.49 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionRsub_type,type,
% 0.31/0.49 binunionRsub: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionRsub,definition,
% 0.31/0.49 ( binunionRsub
% 0.31/0.49 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersect_type,type,
% 0.31/0.49 binintersect: $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectI_type,type,
% 0.31/0.49 binintersectI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectI,definition,
% 0.31/0.49 ( binintersectI
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ B )
% 0.31/0.49 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset5_type,type,
% 0.31/0.49 binintersectSubset5: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset5,definition,
% 0.31/0.49 ( binintersectSubset5
% 0.31/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.49 ( ( subset @ C @ A )
% 0.31/0.49 => ( ( subset @ C @ B )
% 0.31/0.49 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectEL_type,type,
% 0.31/0.49 binintersectEL: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectEL,definition,
% 0.31/0.49 ( binintersectEL
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectLsub_type,type,
% 0.31/0.49 binintersectLsub: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectLsub,definition,
% 0.31/0.49 ( binintersectLsub
% 0.31/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset2_type,type,
% 0.31/0.49 binintersectSubset2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset2,definition,
% 0.31/0.49 ( binintersectSubset2
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( ( binintersect @ A @ B )
% 0.31/0.49 = A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset3_type,type,
% 0.31/0.49 binintersectSubset3: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset3,definition,
% 0.31/0.49 ( binintersectSubset3
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( ( binintersect @ A @ B )
% 0.31/0.49 = B )
% 0.31/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectER_type,type,
% 0.31/0.49 binintersectER: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectER,definition,
% 0.31/0.49 ( binintersectER
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.49 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(disjointsetsI1_type,type,
% 0.31/0.49 disjointsetsI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(disjointsetsI1,definition,
% 0.31/0.49 ( disjointsetsI1
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ~ ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ( in @ Xx @ B ) )
% 0.31/0.49 => ( ( binintersect @ A @ B )
% 0.31/0.49 = emptyset ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectRsub_type,type,
% 0.31/0.49 binintersectRsub: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectRsub,definition,
% 0.31/0.49 ( binintersectRsub
% 0.31/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset4_type,type,
% 0.31/0.49 binintersectSubset4: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset4,definition,
% 0.31/0.49 ( binintersectSubset4
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( subset @ B @ A )
% 0.31/0.49 => ( ( binintersect @ A @ B )
% 0.31/0.49 = B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset1_type,type,
% 0.31/0.49 binintersectSubset1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectSubset1,definition,
% 0.31/0.49 ( binintersectSubset1
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( ( binintersect @ A @ B )
% 0.31/0.49 = A )
% 0.31/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(bs114d_type,type,
% 0.31/0.49 bs114d: $o ).
% 0.31/0.49
% 0.31/0.49 thf(bs114d,definition,
% 0.31/0.49 ( bs114d
% 0.31/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.49 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/0.49 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(regular_type,type,
% 0.31/0.49 regular: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminus_type,type,
% 0.31/0.49 setminus: $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(setminusI_type,type,
% 0.31/0.49 setminusI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusI,definition,
% 0.31/0.49 ( setminusI
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ B )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusEL_type,type,
% 0.31/0.49 setminusEL: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusEL,definition,
% 0.31/0.49 ( setminusEL
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusER_type,type,
% 0.31/0.49 setminusER: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusER,definition,
% 0.31/0.49 ( setminusER
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.49 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusSubset2_type,type,
% 0.31/0.49 setminusSubset2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusSubset2,definition,
% 0.31/0.49 ( setminusSubset2
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( subset @ A @ B )
% 0.31/0.49 => ( ( setminus @ A @ B )
% 0.31/0.49 = emptyset ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusERneg_type,type,
% 0.31/0.49 setminusERneg: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusERneg,definition,
% 0.31/0.49 ( setminusERneg
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.49 => ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusELneg_type,type,
% 0.31/0.49 setminusELneg: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusELneg,definition,
% 0.31/0.49 ( setminusELneg
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.49 => ( ~ ( in @ Xx @ B )
% 0.31/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusILneg_type,type,
% 0.31/0.49 setminusILneg: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusILneg,definition,
% 0.31/0.49 ( setminusILneg
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ~ ( in @ Xx @ A )
% 0.31/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusIRneg_type,type,
% 0.31/0.49 setminusIRneg: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusIRneg,definition,
% 0.31/0.49 ( setminusIRneg
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ B )
% 0.31/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusLsub_type,type,
% 0.31/0.49 setminusLsub: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusLsub,definition,
% 0.31/0.49 ( setminusLsub
% 0.31/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusSubset1_type,type,
% 0.31/0.49 setminusSubset1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusSubset1,definition,
% 0.31/0.49 ( setminusSubset1
% 0.31/0.49 = ( ! [A: $i,B: $i] :
% 0.31/0.49 ( ( ( setminus @ A @ B )
% 0.31/0.49 = emptyset )
% 0.31/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(symdiff_type,type,
% 0.31/0.49 symdiff: $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffE_type,type,
% 0.31/0.49 symdiffE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffE,definition,
% 0.31/0.49 ( symdiffE
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/0.49 => ! [Xphi: $o] :
% 0.31/0.49 ( ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ B )
% 0.31/0.49 => Xphi ) )
% 0.31/0.49 => ( ( ~ ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ B )
% 0.31/0.49 => Xphi ) )
% 0.31/0.49 => Xphi ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffI1_type,type,
% 0.31/0.49 symdiffI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffI1,definition,
% 0.31/0.49 ( symdiffI1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ B )
% 0.31/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffI2_type,type,
% 0.31/0.49 symdiffI2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffI2,definition,
% 0.31/0.49 ( symdiffI2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ~ ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ B )
% 0.31/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffIneg1_type,type,
% 0.31/0.49 symdiffIneg1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffIneg1,definition,
% 0.31/0.49 ( symdiffIneg1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ B )
% 0.31/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffIneg2_type,type,
% 0.31/0.49 symdiffIneg2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(symdiffIneg2,definition,
% 0.31/0.49 ( symdiffIneg2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ~ ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ B )
% 0.31/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(iskpair_type,type,
% 0.31/0.49 iskpair: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(secondinupair_type,type,
% 0.31/0.49 secondinupair: $o ).
% 0.31/0.49
% 0.31/0.49 thf(secondinupair,definition,
% 0.31/0.49 ( secondinupair
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairIL_type,type,
% 0.31/0.49 setukpairIL: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairIL,definition,
% 0.31/0.49 ( setukpairIL
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairIR_type,type,
% 0.31/0.49 setukpairIR: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairIR,definition,
% 0.31/0.49 ( setukpairIR
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(kpairiskpair_type,type,
% 0.31/0.49 kpairiskpair: $o ).
% 0.31/0.49
% 0.31/0.49 thf(kpairiskpair,definition,
% 0.31/0.49 ( kpairiskpair
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(kpair_type,type,
% 0.31/0.49 kpair: $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(kpairp_type,type,
% 0.31/0.49 kpairp: $o ).
% 0.31/0.49
% 0.31/0.49 thf(kpairp,definition,
% 0.31/0.49 ( kpairp
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprod_type,type,
% 0.31/0.49 cartprod: $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(singletonsubset_type,type,
% 0.31/0.49 singletonsubset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(singletonsubset,definition,
% 0.31/0.49 ( singletonsubset
% 0.31/0.49 = ( ! [A: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(singletoninpowerset_type,type,
% 0.31/0.49 singletoninpowerset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(singletoninpowerset,definition,
% 0.31/0.49 ( singletoninpowerset
% 0.31/0.49 = ( ! [A: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(singletoninpowunion_type,type,
% 0.31/0.49 singletoninpowunion: $o ).
% 0.31/0.49
% 0.31/0.49 thf(singletoninpowunion,definition,
% 0.31/0.49 ( singletoninpowunion
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(upairset2E_type,type,
% 0.31/0.49 upairset2E: $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairset2E,definition,
% 0.31/0.49 ( upairset2E
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.49 => ( ( Xz = Xx )
% 0.31/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(upairsubunion_type,type,
% 0.31/0.49 upairsubunion: $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairsubunion,definition,
% 0.31/0.49 ( upairsubunion
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(upairinpowunion_type,type,
% 0.31/0.49 upairinpowunion: $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairinpowunion,definition,
% 0.31/0.49 ( upairinpowunion
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ubforcartprodlem1_type,type,
% 0.31/0.49 ubforcartprodlem1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ubforcartprodlem1,definition,
% 0.31/0.49 ( ubforcartprodlem1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ubforcartprodlem2_type,type,
% 0.31/0.49 ubforcartprodlem2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ubforcartprodlem2,definition,
% 0.31/0.49 ( ubforcartprodlem2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ubforcartprodlem3_type,type,
% 0.31/0.49 ubforcartprodlem3: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ubforcartprodlem3,definition,
% 0.31/0.49 ( ubforcartprodlem3
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodpairin_type,type,
% 0.31/0.49 cartprodpairin: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodpairin,definition,
% 0.31/0.49 ( cartprodpairin
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodmempair1_type,type,
% 0.31/0.49 cartprodmempair1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodmempair1,definition,
% 0.31/0.49 ( cartprodmempair1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.49 => ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ? [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 & ( Xu
% 0.31/0.49 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodmempair_type,type,
% 0.31/0.49 cartprodmempair: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodmempair,definition,
% 0.31/0.49 ( cartprodmempair
% 0.31/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.49 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setunionE2_type,type,
% 0.31/0.49 setunionE2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setunionE2,definition,
% 0.31/0.49 ( setunionE2
% 0.31/0.49 = ( ! [A: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.49 => ? [X: $i] :
% 0.31/0.49 ( ( in @ X @ A )
% 0.31/0.49 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setunionsingleton1_type,type,
% 0.31/0.49 setunionsingleton1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setunionsingleton1,definition,
% 0.31/0.49 ( setunionsingleton1
% 0.31/0.49 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setunionsingleton2_type,type,
% 0.31/0.49 setunionsingleton2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setunionsingleton2,definition,
% 0.31/0.49 ( setunionsingleton2
% 0.31/0.49 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setunionsingleton_type,type,
% 0.31/0.49 setunionsingleton: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setunionsingleton,definition,
% 0.31/0.49 ( setunionsingleton
% 0.31/0.49 = ( ! [Xx: $i] :
% 0.31/0.49 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/0.49 = Xx ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(singleton_type,type,
% 0.31/0.49 singleton: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(singletonprop_type,type,
% 0.31/0.49 singletonprop: $o ).
% 0.31/0.49
% 0.31/0.49 thf(singletonprop,definition,
% 0.31/0.49 ( singletonprop
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( ( Xphi @ Xx )
% 0.31/0.49 => ( ( Xphi @ Xy )
% 0.31/0.49 => ( Xx = Xy ) ) ) ) )
% 0.31/0.49 => ( ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ( Xphi @ Xx ) )
% 0.31/0.49 => ( singleton
% 0.31/0.49 @ ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ex1_type,type,
% 0.31/0.49 ex1: $i > ( $i > $o ) > $o ).
% 0.31/0.49
% 0.31/0.49 thf(ex1E1_type,type,
% 0.31/0.49 ex1E1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ex1E1,definition,
% 0.31/0.49 ( ex1E1
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ( ex1 @ A
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 => ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ex1I_type,type,
% 0.31/0.49 ex1I: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ex1I,definition,
% 0.31/0.49 ( ex1I
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( Xphi @ Xx )
% 0.31/0.49 => ( ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( ( Xphi @ Xy )
% 0.31/0.49 => ( Xy = Xx ) ) )
% 0.31/0.49 => ( ex1 @ A
% 0.31/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ex1I2_type,type,
% 0.31/0.49 ex1I2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ex1I2,definition,
% 0.31/0.49 ( ex1I2
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( ( Xphi @ Xx )
% 0.31/0.49 => ( ( Xphi @ Xy )
% 0.31/0.49 => ( Xx = Xy ) ) ) ) )
% 0.31/0.49 => ( ? [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 & ( Xphi @ Xx ) )
% 0.31/0.49 => ( ex1 @ A
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(singletonsuniq_type,type,
% 0.31/0.49 singletonsuniq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(singletonsuniq,definition,
% 0.31/0.49 ( singletonsuniq
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/0.49 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(atmost1p_type,type,
% 0.31/0.49 atmost1p: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(atleast2p_type,type,
% 0.31/0.49 atleast2p: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(atmost2p_type,type,
% 0.31/0.49 atmost2p: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairsetp_type,type,
% 0.31/0.49 upairsetp: $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjL1_type,type,
% 0.31/0.49 setukpairinjL1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjL1,definition,
% 0.31/0.49 ( setukpairinjL1
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.49 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(kfstsingleton_type,type,
% 0.31/0.49 kfstsingleton: $o ).
% 0.31/0.49
% 0.31/0.49 thf(kfstsingleton,definition,
% 0.31/0.49 ( kfstsingleton
% 0.31/0.49 = ( ! [Xu: $i] :
% 0.31/0.49 ( ( iskpair @ Xu )
% 0.31/0.49 => ( singleton
% 0.31/0.49 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.49 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(theprop_type,type,
% 0.31/0.49 theprop: $o ).
% 0.31/0.49
% 0.31/0.49 thf(theprop,definition,
% 0.31/0.49 ( theprop
% 0.31/0.49 = ( ! [X: $i] :
% 0.31/0.49 ( ( singleton @ X )
% 0.31/0.49 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(kfst_type,type,
% 0.31/0.49 kfst: $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(kfstpairEq_type,type,
% 0.31/0.49 kfstpairEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(kfstpairEq,definition,
% 0.31/0.49 ( kfstpairEq
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.49 = Xx ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodfstin_type,type,
% 0.31/0.49 cartprodfstin: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodfstin,definition,
% 0.31/0.49 ( cartprodfstin
% 0.31/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.49 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjL2_type,type,
% 0.31/0.49 setukpairinjL2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjL2,definition,
% 0.31/0.49 ( setukpairinjL2
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjL_type,type,
% 0.31/0.49 setukpairinjL: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjL,definition,
% 0.31/0.49 ( setukpairinjL
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.49 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.49 = ( kpair @ Xz @ Xu ) )
% 0.31/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR11_type,type,
% 0.31/0.49 setukpairinjR11: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR11,definition,
% 0.31/0.49 ( setukpairinjR11
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx = Xy )
% 0.31/0.49 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.49 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR12_type,type,
% 0.31/0.49 setukpairinjR12: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR12,definition,
% 0.31/0.49 ( setukpairinjR12
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xx = Xy )
% 0.31/0.49 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.49 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR1_type,type,
% 0.31/0.49 setukpairinjR1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR1,definition,
% 0.31/0.49 ( setukpairinjR1
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.49 => ( ( Xz = Xu )
% 0.31/0.49 => ( Xy = Xu ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(upairequniteq_type,type,
% 0.31/0.49 upairequniteq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(upairequniteq,definition,
% 0.31/0.49 ( upairequniteq
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.49 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.49 = ( setadjoin @ Xz @ emptyset ) )
% 0.31/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR2_type,type,
% 0.31/0.49 setukpairinjR2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR2,definition,
% 0.31/0.49 ( setukpairinjR2
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.49 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR_type,type,
% 0.31/0.49 setukpairinjR: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setukpairinjR,definition,
% 0.31/0.49 ( setukpairinjR
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.49 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.49 = ( kpair @ Xz @ Xu ) )
% 0.31/0.49 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ksndsingleton_type,type,
% 0.31/0.49 ksndsingleton: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ksndsingleton,definition,
% 0.31/0.49 ( ksndsingleton
% 0.31/0.49 = ( ! [Xu: $i] :
% 0.31/0.49 ( ( iskpair @ Xu )
% 0.31/0.49 => ( singleton
% 0.31/0.49 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.49 @ ^ [Xx: $i] :
% 0.31/0.49 ( Xu
% 0.31/0.49 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ksnd_type,type,
% 0.31/0.49 ksnd: $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(ksndpairEq_type,type,
% 0.31/0.49 ksndpairEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ksndpairEq,definition,
% 0.31/0.49 ( ksndpairEq
% 0.31/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.49 = Xy ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(kpairsurjEq_type,type,
% 0.31/0.49 kpairsurjEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(kpairsurjEq,definition,
% 0.31/0.49 ( kpairsurjEq
% 0.31/0.49 = ( ! [Xu: $i] :
% 0.31/0.49 ( ( iskpair @ Xu )
% 0.31/0.49 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.49 = Xu ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodsndin_type,type,
% 0.31/0.49 cartprodsndin: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodsndin,definition,
% 0.31/0.49 ( cartprodsndin
% 0.31/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.49 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodpairmemEL_type,type,
% 0.31/0.49 cartprodpairmemEL: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodpairmemEL,definition,
% 0.31/0.49 ( cartprodpairmemEL
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodpairmemER_type,type,
% 0.31/0.49 cartprodpairmemER: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodpairmemER,definition,
% 0.31/0.49 ( cartprodpairmemER
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.49 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodmempaircEq_type,type,
% 0.31/0.49 cartprodmempaircEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodmempaircEq,definition,
% 0.31/0.49 ( cartprodmempaircEq
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( kpair @ Xx @ Xy )
% 0.31/0.49 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodfstpairEq_type,type,
% 0.31/0.49 cartprodfstpairEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodfstpairEq,definition,
% 0.31/0.49 ( cartprodfstpairEq
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.49 = Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodsndpairEq_type,type,
% 0.31/0.49 cartprodsndpairEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodsndpairEq,definition,
% 0.31/0.49 ( cartprodsndpairEq
% 0.31/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.49 = Xy ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodpairsurjEq_type,type,
% 0.31/0.49 cartprodpairsurjEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(cartprodpairsurjEq,definition,
% 0.31/0.49 ( cartprodpairsurjEq
% 0.31/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.49 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.49 = Xu ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(breln_type,type,
% 0.31/0.49 breln: $i > $i > $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstr_type,type,
% 0.31/0.49 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrI_type,type,
% 0.31/0.49 dpsetconstrI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrI,definition,
% 0.31/0.49 ( dpsetconstrI
% 0.31/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( Xphi @ Xx @ Xy )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.49 @ ( dpsetconstr @ A @ B
% 0.31/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrSub_type,type,
% 0.31/0.49 dpsetconstrSub: $o ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrSub,definition,
% 0.31/0.49 ( dpsetconstrSub
% 0.31/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.49 ( subset
% 0.31/0.49 @ ( dpsetconstr @ A @ B
% 0.31/0.49 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.31/0.49 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setOfPairsIsBReln_type,type,
% 0.31/0.49 setOfPairsIsBReln: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setOfPairsIsBReln,definition,
% 0.31/0.49 ( setOfPairsIsBReln
% 0.31/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.49 ( breln @ A @ B
% 0.31/0.49 @ ( dpsetconstr @ A @ B
% 0.31/0.49 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrERa_type,type,
% 0.31/0.49 dpsetconstrERa: $o ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrERa,definition,
% 0.31/0.49 ( dpsetconstrERa
% 0.31/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.49 @ ( dpsetconstr @ A @ B
% 0.31/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.49 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrEL1_type,type,
% 0.31/0.49 dpsetconstrEL1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrEL1,definition,
% 0.31/0.49 ( dpsetconstrEL1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.49 @ ( dpsetconstr @ A @ B
% 0.31/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrEL2_type,type,
% 0.31/0.49 dpsetconstrEL2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrEL2,definition,
% 0.31/0.49 ( dpsetconstrEL2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.49 @ ( dpsetconstr @ A @ B
% 0.31/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.49 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrER_type,type,
% 0.31/0.49 dpsetconstrER: $o ).
% 0.31/0.49
% 0.31/0.49 thf(dpsetconstrER,definition,
% 0.31/0.49 ( dpsetconstrER
% 0.31/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.49 @ ( dpsetconstr @ A @ B
% 0.31/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.49 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(func_type,type,
% 0.31/0.49 func: $i > $i > $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcSet_type,type,
% 0.31/0.49 funcSet: $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(funcImageSingleton_type,type,
% 0.31/0.49 funcImageSingleton: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcImageSingleton,definition,
% 0.31/0.49 ( funcImageSingleton
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( singleton
% 0.31/0.49 @ ( dsetconstr @ B
% 0.31/0.49 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(apProp_type,type,
% 0.31/0.49 apProp: $o ).
% 0.31/0.49
% 0.31/0.49 thf(apProp,definition,
% 0.31/0.49 ( apProp
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in
% 0.31/0.49 @ ( setunion
% 0.31/0.49 @ ( dsetconstr @ B
% 0.31/0.49 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.31/0.49 @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ap_type,type,
% 0.31/0.49 ap: $i > $i > $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(app_type,type,
% 0.31/0.49 app: $o ).
% 0.31/0.49
% 0.31/0.49 thf(app,definition,
% 0.31/0.49 ( app
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(infuncsetfunc_type,type,
% 0.31/0.49 infuncsetfunc: $o ).
% 0.31/0.49
% 0.31/0.49 thf(infuncsetfunc,definition,
% 0.31/0.49 ( infuncsetfunc
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.49 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ap2p_type,type,
% 0.31/0.49 ap2p: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ap2p,definition,
% 0.31/0.49 ( ap2p
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(funcinfuncset_type,type,
% 0.31/0.49 funcinfuncset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcinfuncset,definition,
% 0.31/0.49 ( funcinfuncset
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(lamProp_type,type,
% 0.31/0.49 lamProp: $o ).
% 0.31/0.49
% 0.31/0.49 thf(lamProp,definition,
% 0.31/0.49 ( lamProp
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.49 => ( func @ A @ B
% 0.31/0.49 @ ( dpsetconstr @ A @ B
% 0.31/0.49 @ ^ [Xx: $i,Xy: $i] :
% 0.31/0.49 ( ( Xf @ Xx )
% 0.31/0.49 = Xy ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(lam_type,type,
% 0.31/0.49 lam: $i > $i > ( $i > $i ) > $i ).
% 0.31/0.49
% 0.31/0.49 thf(lamp_type,type,
% 0.31/0.49 lamp: $o ).
% 0.31/0.49
% 0.31/0.49 thf(lamp,definition,
% 0.31/0.49 ( lamp
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.49 => ( func @ A @ B
% 0.31/0.49 @ ( lam @ A @ B
% 0.31/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(lam2p_type,type,
% 0.31/0.49 lam2p: $o ).
% 0.31/0.49
% 0.31/0.49 thf(lam2p,definition,
% 0.31/0.49 ( lam2p
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.49 => ( in
% 0.31/0.49 @ ( lam @ A @ B
% 0.31/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/0.49 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(brelnall1_type,type,
% 0.31/0.49 brelnall1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(brelnall1,definition,
% 0.31/0.49 ( brelnall1
% 0.31/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.49 ( ( breln @ A @ B @ R )
% 0.31/0.49 => ! [Xphi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.49 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ R )
% 0.31/0.49 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(brelnall2_type,type,
% 0.31/0.49 brelnall2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(brelnall2,definition,
% 0.31/0.49 ( brelnall2
% 0.31/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.49 ( ( breln @ A @ B @ R )
% 0.31/0.49 => ! [Xphi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.49 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ R )
% 0.31/0.49 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ex1E2_type,type,
% 0.31/0.49 ex1E2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ex1E2,definition,
% 0.31/0.49 ( ex1E2
% 0.31/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.49 ( ( ex1 @ A
% 0.31/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( ( Xphi @ Xx )
% 0.31/0.49 => ( ( Xphi @ Xy )
% 0.31/0.49 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(funcGraphProp1_type,type,
% 0.31/0.49 funcGraphProp1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcGraphProp1,definition,
% 0.31/0.49 ( funcGraphProp1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(funcGraphProp3_type,type,
% 0.31/0.49 funcGraphProp3: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcGraphProp3,definition,
% 0.31/0.49 ( funcGraphProp3
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(funcGraphProp2_type,type,
% 0.31/0.49 funcGraphProp2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcGraphProp2,definition,
% 0.31/0.49 ( funcGraphProp2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.49 = Xy ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(funcextLem_type,type,
% 0.31/0.49 funcextLem: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcextLem,definition,
% 0.31/0.49 ( funcextLem
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ! [Xg: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xg )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.49 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(funcGraphProp4_type,type,
% 0.31/0.49 funcGraphProp4: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcGraphProp4,definition,
% 0.31/0.49 ( funcGraphProp4
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.49 = Xy ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subbreln_type,type,
% 0.31/0.49 subbreln: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subbreln,definition,
% 0.31/0.49 ( subbreln
% 0.31/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.49 ( ( breln @ A @ B @ R )
% 0.31/0.49 => ! [S: $i] :
% 0.31/0.49 ( ( breln @ A @ B @ S )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.49 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(eqbreln_type,type,
% 0.31/0.49 eqbreln: $o ).
% 0.31/0.49
% 0.31/0.49 thf(eqbreln,definition,
% 0.31/0.49 ( eqbreln
% 0.31/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.49 ( ( breln @ A @ B @ R )
% 0.31/0.49 => ! [S: $i] :
% 0.31/0.49 ( ( breln @ A @ B @ S )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ B )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.31/0.49 => ( R = S ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(funcext_type,type,
% 0.31/0.49 funcext: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcext,definition,
% 0.31/0.49 ( funcext
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ! [Xg: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xg )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.49 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.49 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(funcext2_type,type,
% 0.31/0.49 funcext2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(funcext2,definition,
% 0.31/0.49 ( funcext2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.49 => ! [Xg: $i] :
% 0.31/0.49 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.49 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.49 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ap2apEq1_type,type,
% 0.31/0.49 ap2apEq1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ap2apEq1,definition,
% 0.31/0.49 ( ap2apEq1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.49 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ap2apEq2_type,type,
% 0.31/0.49 ap2apEq2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ap2apEq2,definition,
% 0.31/0.49 ( ap2apEq2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.49 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(beta1_type,type,
% 0.31/0.49 beta1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(beta1,definition,
% 0.31/0.49 ( beta1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( ap @ A @ B
% 0.31/0.49 @ ( lam @ A @ B
% 0.31/0.49 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/0.49 @ Xx )
% 0.31/0.49 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(eta1_type,type,
% 0.31/0.49 eta1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(eta1,definition,
% 0.31/0.49 ( eta1
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( func @ A @ B @ Xf )
% 0.31/0.49 => ( ( lam @ A @ B
% 0.31/0.49 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/0.49 = Xf ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(lam2lamEq_type,type,
% 0.31/0.49 lam2lamEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(lam2lamEq,definition,
% 0.31/0.49 ( lam2lamEq
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.49 => ( ( lam @ A @ B
% 0.31/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/0.49 = ( lam @ A @ B
% 0.31/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(beta2_type,type,
% 0.31/0.49 beta2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(beta2,definition,
% 0.31/0.49 ( beta2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( ap @ A @ B
% 0.31/0.49 @ ( lam @ A @ B
% 0.31/0.49 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/0.49 @ Xx )
% 0.31/0.49 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(eta2_type,type,
% 0.31/0.49 eta2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(eta2,definition,
% 0.31/0.49 ( eta2
% 0.31/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.49 => ( ( lam @ A @ B
% 0.31/0.49 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/0.49 = Xf ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(iffalseProp1_type,type,
% 0.31/0.49 iffalseProp1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(iffalseProp1,definition,
% 0.31/0.49 ( iffalseProp1
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( ~ Xphi
% 0.31/0.49 => ( in @ Xy
% 0.31/0.49 @ ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xz: $i] :
% 0.31/0.49 ( ( Xphi
% 0.31/0.49 & ( Xz = Xx ) )
% 0.31/0.49 | ( ~ Xphi
% 0.31/0.49 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(iffalseProp2_type,type,
% 0.31/0.49 iffalseProp2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(iffalseProp2,definition,
% 0.31/0.49 ( iffalseProp2
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( ~ Xphi
% 0.31/0.49 => ( ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xz: $i] :
% 0.31/0.49 ( ( Xphi
% 0.31/0.49 & ( Xz = Xx ) )
% 0.31/0.49 | ( ~ Xphi
% 0.31/0.49 & ( Xz = Xy ) ) ) )
% 0.31/0.49 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(iftrueProp1_type,type,
% 0.31/0.49 iftrueProp1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(iftrueProp1,definition,
% 0.31/0.49 ( iftrueProp1
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( Xphi
% 0.31/0.49 => ( in @ Xx
% 0.31/0.49 @ ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xz: $i] :
% 0.31/0.49 ( ( Xphi
% 0.31/0.49 & ( Xz = Xx ) )
% 0.31/0.49 | ( ~ Xphi
% 0.31/0.49 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(iftrueProp2_type,type,
% 0.31/0.49 iftrueProp2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(iftrueProp2,definition,
% 0.31/0.49 ( iftrueProp2
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( Xphi
% 0.31/0.49 => ( ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xz: $i] :
% 0.31/0.49 ( ( Xphi
% 0.31/0.49 & ( Xz = Xx ) )
% 0.31/0.49 | ( ~ Xphi
% 0.31/0.49 & ( Xz = Xy ) ) ) )
% 0.31/0.49 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(ifSingleton_type,type,
% 0.31/0.49 ifSingleton: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ifSingleton,definition,
% 0.31/0.49 ( ifSingleton
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( singleton
% 0.31/0.49 @ ( dsetconstr @ A
% 0.31/0.49 @ ^ [Xz: $i] :
% 0.31/0.49 ( ( Xphi
% 0.31/0.49 & ( Xz = Xx ) )
% 0.31/0.49 | ( ~ Xphi
% 0.31/0.49 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(if_type,type,
% 0.31/0.49 if: $i > $o > $i > $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(ifp_type,type,
% 0.31/0.49 ifp: $o ).
% 0.31/0.49
% 0.31/0.49 thf(ifp,definition,
% 0.31/0.49 ( ifp
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(theeq_type,type,
% 0.31/0.49 theeq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(theeq,definition,
% 0.31/0.49 ( theeq
% 0.31/0.49 = ( ! [X: $i] :
% 0.31/0.49 ( ( singleton @ X )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ X )
% 0.31/0.49 => ( ( setunion @ X )
% 0.31/0.49 = Xx ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(iftrue_type,type,
% 0.31/0.49 iftrue: $o ).
% 0.31/0.49
% 0.31/0.49 thf(iftrue,definition,
% 0.31/0.49 ( iftrue
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( Xphi
% 0.31/0.49 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.31/0.49 = Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(iffalse_type,type,
% 0.31/0.49 iffalse: $o ).
% 0.31/0.49
% 0.31/0.49 thf(iffalse,definition,
% 0.31/0.49 ( iffalse
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( ~ Xphi
% 0.31/0.49 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.31/0.49 = Xy ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(iftrueorfalse_type,type,
% 0.31/0.49 iftrueorfalse: $o ).
% 0.31/0.49
% 0.31/0.49 thf(iftrueorfalse,definition,
% 0.31/0.49 ( iftrueorfalse
% 0.31/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectT_lem_type,type,
% 0.31/0.49 binintersectT_lem: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectT_lem,definition,
% 0.31/0.49 ( binintersectT_lem
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionT_lem_type,type,
% 0.31/0.49 binunionT_lem: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionT_lem,definition,
% 0.31/0.49 ( binunionT_lem
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powersetT_lem_type,type,
% 0.31/0.49 powersetT_lem: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powersetT_lem,definition,
% 0.31/0.49 ( powersetT_lem
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setminusT_lem_type,type,
% 0.31/0.49 setminusT_lem: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setminusT_lem,definition,
% 0.31/0.49 ( setminusT_lem
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementT_lem_type,type,
% 0.31/0.49 complementT_lem: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementT_lem,definition,
% 0.31/0.49 ( complementT_lem
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(setextT_type,type,
% 0.31/0.49 setextT: $o ).
% 0.31/0.49
% 0.31/0.49 thf(setextT,definition,
% 0.31/0.49 ( setextT
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ X )
% 0.31/0.49 => ( in @ Xx @ Y ) ) )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ Y )
% 0.31/0.49 => ( in @ Xx @ X ) ) )
% 0.31/0.49 => ( X = Y ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(subsetTI_type,type,
% 0.31/0.49 subsetTI: $o ).
% 0.31/0.49
% 0.31/0.49 thf(subsetTI,definition,
% 0.31/0.49 ( subsetTI
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ X )
% 0.31/0.49 => ( in @ Xx @ Y ) ) )
% 0.31/0.49 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powersetTI1_type,type,
% 0.31/0.49 powersetTI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powersetTI1,definition,
% 0.31/0.49 ( powersetTI1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ X )
% 0.31/0.49 => ( in @ Xx @ Y ) ) )
% 0.31/0.49 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(powersetTE1_type,type,
% 0.31/0.49 powersetTE1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(powersetTE1,definition,
% 0.31/0.49 ( powersetTE1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ X @ ( powerset @ Y ) )
% 0.31/0.49 => ( ( in @ Xx @ X )
% 0.31/0.49 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementTI1_type,type,
% 0.31/0.49 complementTI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementTI1,definition,
% 0.31/0.49 ( complementTI1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ X )
% 0.31/0.49 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementTE1_type,type,
% 0.31/0.49 complementTE1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementTE1,definition,
% 0.31/0.49 ( complementTE1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.49 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectTELcontra_type,type,
% 0.31/0.49 binintersectTELcontra: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectTELcontra,definition,
% 0.31/0.49 ( binintersectTELcontra
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ X )
% 0.31/0.49 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectTERcontra_type,type,
% 0.31/0.49 binintersectTERcontra: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binintersectTERcontra,definition,
% 0.31/0.49 ( binintersectTERcontra
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ Y )
% 0.31/0.49 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(contrasubsetT_type,type,
% 0.31/0.49 contrasubsetT: $o ).
% 0.31/0.49
% 0.31/0.49 thf(contrasubsetT,definition,
% 0.31/0.49 ( contrasubsetT
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.49 => ( ( in @ Xx @ Y )
% 0.31/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(contrasubsetT1_type,type,
% 0.31/0.49 contrasubsetT1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(contrasubsetT1,definition,
% 0.31/0.49 ( contrasubsetT1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( subset @ X @ Y )
% 0.31/0.49 => ( ~ ( in @ Xx @ Y )
% 0.31/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(contrasubsetT2_type,type,
% 0.31/0.49 contrasubsetT2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(contrasubsetT2,definition,
% 0.31/0.49 ( contrasubsetT2
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ Y )
% 0.31/0.49 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(contrasubsetT3_type,type,
% 0.31/0.49 contrasubsetT3: $o ).
% 0.31/0.49
% 0.31/0.49 thf(contrasubsetT3,definition,
% 0.31/0.49 ( contrasubsetT3
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.31/0.49 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementI1_type,type,
% 0.31/0.49 doubleComplementI1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementI1,definition,
% 0.31/0.49 ( doubleComplementI1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ X )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementE1_type,type,
% 0.31/0.49 doubleComplementE1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementE1,definition,
% 0.31/0.49 ( doubleComplementE1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.31/0.49 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementSub1_type,type,
% 0.31/0.49 doubleComplementSub1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementSub1,definition,
% 0.31/0.49 ( doubleComplementSub1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementSub2_type,type,
% 0.31/0.49 doubleComplementSub2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementSub2,definition,
% 0.31/0.49 ( doubleComplementSub2
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementEq_type,type,
% 0.31/0.49 doubleComplementEq: $o ).
% 0.31/0.49
% 0.31/0.49 thf(doubleComplementEq,definition,
% 0.31/0.49 ( doubleComplementEq
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ( X
% 0.31/0.49 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementTnotintersectT_type,type,
% 0.31/0.49 complementTnotintersectT: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementTnotintersectT,definition,
% 0.31/0.49 ( complementTnotintersectT
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.49 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementImpComplementIntersect_type,type,
% 0.31/0.49 complementImpComplementIntersect: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementImpComplementIntersect,definition,
% 0.31/0.49 ( complementImpComplementIntersect
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementSubsetComplementIntersect_type,type,
% 0.31/0.49 complementSubsetComplementIntersect: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementSubsetComplementIntersect,definition,
% 0.31/0.49 ( complementSubsetComplementIntersect
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementInPowersetComplementIntersect_type,type,
% 0.31/0.49 complementInPowersetComplementIntersect: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementInPowersetComplementIntersect,definition,
% 0.31/0.49 ( complementInPowersetComplementIntersect
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(contraSubsetComplement_type,type,
% 0.31/0.49 contraSubsetComplement: $o ).
% 0.31/0.49
% 0.31/0.49 thf(contraSubsetComplement,definition,
% 0.31/0.49 ( contraSubsetComplement
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ Y )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementTcontraSubset_type,type,
% 0.31/0.49 complementTcontraSubset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementTcontraSubset,definition,
% 0.31/0.49 ( complementTcontraSubset
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.49 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTILcontra_type,type,
% 0.31/0.49 binunionTILcontra: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTILcontra,definition,
% 0.31/0.49 ( binunionTILcontra
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTIRcontra_type,type,
% 0.31/0.49 binunionTIRcontra: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTIRcontra,definition,
% 0.31/0.49 ( binunionTIRcontra
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/0.49 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(inIntersectImpInUnion_type,type,
% 0.31/0.49 inIntersectImpInUnion: $o ).
% 0.31/0.49
% 0.31/0.49 thf(inIntersectImpInUnion,definition,
% 0.31/0.49 ( inIntersectImpInUnion
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.49 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(inIntersectImpInUnion2_type,type,
% 0.31/0.49 inIntersectImpInUnion2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(inIntersectImpInUnion2,definition,
% 0.31/0.49 ( inIntersectImpInUnion2
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.49 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(inIntersectImpInIntersectUnions_type,type,
% 0.31/0.49 inIntersectImpInIntersectUnions: $o ).
% 0.31/0.49
% 0.31/0.49 thf(inIntersectImpInIntersectUnions,definition,
% 0.31/0.49 ( inIntersectImpInIntersectUnions
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.49 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(intersectInPowersetIntersectUnions_type,type,
% 0.31/0.49 intersectInPowersetIntersectUnions: $o ).
% 0.31/0.49
% 0.31/0.49 thf(intersectInPowersetIntersectUnions,definition,
% 0.31/0.49 ( intersectInPowersetIntersectUnions
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(inComplementUnionImpNotIn1_type,type,
% 0.31/0.49 inComplementUnionImpNotIn1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(inComplementUnionImpNotIn1,definition,
% 0.31/0.49 ( inComplementUnionImpNotIn1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(inComplementUnionImpInComplement1_type,type,
% 0.31/0.49 inComplementUnionImpInComplement1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(inComplementUnionImpInComplement1,definition,
% 0.31/0.49 ( inComplementUnionImpInComplement1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTE_type,type,
% 0.31/0.49 binunionTE: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTE,definition,
% 0.31/0.49 ( binunionTE
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xphi: $o,Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/0.49 => ( ( ( in @ Xx @ X )
% 0.31/0.49 => Xphi )
% 0.31/0.49 => ( ( ( in @ Xx @ Y )
% 0.31/0.49 => Xphi )
% 0.31/0.49 => Xphi ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTEcontra_type,type,
% 0.31/0.49 binunionTEcontra: $o ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTEcontra,definition,
% 0.31/0.49 ( binunionTEcontra
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ X )
% 0.31/0.49 => ( ~ ( in @ Xx @ Y )
% 0.31/0.49 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2a1_type,type,
% 0.31/0.49 demorgan2a1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2a1,definition,
% 0.31/0.49 ( demorgan2a1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementUnionInPowersetComplement_type,type,
% 0.31/0.49 complementUnionInPowersetComplement: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementUnionInPowersetComplement,definition,
% 0.31/0.49 ( complementUnionInPowersetComplement
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2a2_type,type,
% 0.31/0.49 demorgan2a2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2a2,definition,
% 0.31/0.49 ( demorgan2a2
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan1a_type,type,
% 0.31/0.49 demorgan1a: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan1a,definition,
% 0.31/0.49 ( demorgan1a
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.31/0.49 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan1b_type,type,
% 0.31/0.49 demorgan1b: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan1b,definition,
% 0.31/0.49 ( demorgan1b
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan1_type,type,
% 0.31/0.49 demorgan1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan1,definition,
% 0.31/0.49 ( demorgan1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.31/0.49 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2a_type,type,
% 0.31/0.49 demorgan2a: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2a,definition,
% 0.31/0.49 ( demorgan2a
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.49 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2b2_type,type,
% 0.31/0.49 demorgan2b2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2b2,definition,
% 0.31/0.49 ( demorgan2b2
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2b_type,type,
% 0.31/0.49 demorgan2b: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2b,definition,
% 0.31/0.49 ( demorgan2b
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2_type,type,
% 0.31/0.49 demorgan2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(demorgan2,definition,
% 0.31/0.49 ( demorgan2
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.31/0.49 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule0_type,type,
% 0.31/0.49 woz13rule0: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule0,definition,
% 0.31/0.49 ( woz13rule0
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.49 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule1_type,type,
% 0.31/0.49 woz13rule1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule1,definition,
% 0.31/0.49 ( woz13rule1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ Z )
% 0.31/0.49 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule2_type,type,
% 0.31/0.49 woz13rule2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule2,definition,
% 0.31/0.49 ( woz13rule2
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ Y @ Z )
% 0.31/0.49 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule3_type,type,
% 0.31/0.49 woz13rule3: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule3,definition,
% 0.31/0.49 ( woz13rule3
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ Y )
% 0.31/0.49 => ( ( subset @ X @ Z )
% 0.31/0.49 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule4_type,type,
% 0.31/0.49 woz13rule4: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz13rule4,definition,
% 0.31/0.49 ( woz13rule4
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ! [W: $i] :
% 0.31/0.49 ( ( in @ W @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ Z )
% 0.31/0.49 => ( ( subset @ Y @ W )
% 0.31/0.49 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_1_type,type,
% 0.31/0.49 woz1_1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_1,definition,
% 0.31/0.49 ( woz1_1
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_2_type,type,
% 0.31/0.49 woz1_2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_2,definition,
% 0.31/0.49 ( woz1_2
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ! [W: $i] :
% 0.31/0.49 ( ( in @ W @ ( powerset @ A ) )
% 0.31/0.49 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.31/0.49 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_3_type,type,
% 0.31/0.49 woz1_3: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_3,definition,
% 0.31/0.49 ( woz1_3
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Z: $i] :
% 0.31/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_4_type,type,
% 0.31/0.49 woz1_4: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_4,definition,
% 0.31/0.49 ( woz1_4
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.49 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_5_type,type,
% 0.31/0.49 woz1_5: $o ).
% 0.31/0.49
% 0.31/0.49 thf(woz1_5,definition,
% 0.31/0.49 ( woz1_5
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(breln1_type,type,
% 0.31/0.49 breln1: $i > $i > $o ).
% 0.31/0.49
% 0.31/0.49 thf(breln1all2_type,type,
% 0.31/0.49 breln1all2: $o ).
% 0.31/0.49
% 0.31/0.49 thf(breln1all2,definition,
% 0.31/0.49 ( breln1all2
% 0.31/0.49 = ( ! [A: $i,R: $i] :
% 0.31/0.49 ( ( breln1 @ A @ R )
% 0.31/0.49 => ! [Xphi: $i > $o] :
% 0.31/0.49 ( ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ! [Xy: $i] :
% 0.31/0.49 ( ( in @ Xy @ A )
% 0.31/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.49 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ R )
% 0.31/0.49 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(breln1Set_type,type,
% 0.31/0.49 breln1Set: $i > $i ).
% 0.31/0.49
% 0.31/0.49 thf(breln1SetBreln1_type,type,
% 0.31/0.49 breln1SetBreln1: $o ).
% 0.31/0.49
% 0.31/0.49 thf(breln1SetBreln1,definition,
% 0.31/0.49 ( breln1SetBreln1
% 0.31/0.49 = ( ! [A: $i,R: $i] :
% 0.31/0.49 ( ( in @ R @ ( breln1Set @ A ) )
% 0.31/0.50 => ( breln1 @ A @ R ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(transitive_type,type,
% 0.31/0.50 transitive: $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(antisymmetric_type,type,
% 0.31/0.50 antisymmetric: $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(reflexive_type,type,
% 0.31/0.50 reflexive: $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(refltransitive_type,type,
% 0.31/0.50 refltransitive: $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(refllinearorder_type,type,
% 0.31/0.50 refllinearorder: $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(reflwellordering_type,type,
% 0.31/0.50 reflwellordering: $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(choice2fnsingleton_type,type,
% 0.31/0.50 choice2fnsingleton: $o ).
% 0.31/0.50
% 0.31/0.50 thf(choice2fnsingleton,definition,
% 0.31/0.50 ( choice2fnsingleton
% 0.31/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.50 ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ? [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ B )
% 0.31/0.50 & ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.50 => ! [R: $i] :
% 0.31/0.50 ( ( in @ R @ ( breln1Set @ B ) )
% 0.31/0.50 => ( ( reflwellordering @ B @ R )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ( singleton
% 0.31/0.50 @ ( dsetconstr @ B
% 0.31/0.50 @ ^ [Xy: $i] :
% 0.31/0.50 ( ( Xphi @ Xx @ Xy )
% 0.31/0.50 & ! [Xz: $i] :
% 0.31/0.50 ( ( in @ Xz @ B )
% 0.31/0.50 => ( ( Xphi @ Xx @ Xz )
% 0.31/0.50 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(setOfPairsIsBReln1_type,type,
% 0.31/0.50 setOfPairsIsBReln1: $o ).
% 0.31/0.50
% 0.31/0.50 thf(setOfPairsIsBReln1,definition,
% 0.31/0.50 ( setOfPairsIsBReln1
% 0.31/0.50 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.31/0.50 ( breln1 @ A
% 0.31/0.50 @ ( dpsetconstr @ A @ A
% 0.31/0.50 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1all1_type,type,
% 0.31/0.50 breln1all1: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1all1,definition,
% 0.31/0.50 ( breln1all1
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [Xphi: $i > $o] :
% 0.31/0.50 ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.50 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ R )
% 0.31/0.50 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(subbreln1_type,type,
% 0.31/0.50 subbreln1: $o ).
% 0.31/0.50
% 0.31/0.50 thf(subbreln1,definition,
% 0.31/0.50 ( subbreln1
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.50 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(eqbreln1_type,type,
% 0.31/0.50 eqbreln1: $o ).
% 0.31/0.50
% 0.31/0.50 thf(eqbreln1,definition,
% 0.31/0.50 ( eqbreln1
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.50 => ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.31/0.50 => ( R = S ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1invset_type,type,
% 0.31/0.50 breln1invset: $i > $i > $i ).
% 0.31/0.50
% 0.31/0.50 thf(breln1invprop_type,type,
% 0.31/0.50 breln1invprop: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1invprop,definition,
% 0.31/0.50 ( breln1invprop
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1invI_type,type,
% 0.31/0.50 breln1invI: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1invI,definition,
% 0.31/0.50 ( breln1invI
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.50 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1invE_type,type,
% 0.31/0.50 breln1invE: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1invE,definition,
% 0.31/0.50 ( breln1invE
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.31/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compset_type,type,
% 0.31/0.50 breln1compset: $i > $i > $i > $i ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compprop_type,type,
% 0.31/0.50 breln1compprop: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compprop,definition,
% 0.31/0.50 ( breln1compprop
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compI_type,type,
% 0.31/0.50 breln1compI: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compI,definition,
% 0.31/0.50 ( breln1compI
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ! [Xz: $i] :
% 0.31/0.50 ( ( in @ Xz @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.31/0.50 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.31/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compE_type,type,
% 0.31/0.50 breln1compE: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compE,definition,
% 0.31/0.50 ( breln1compE
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.31/0.50 => ? [Xz: $i] :
% 0.31/0.50 ( ( in @ Xz @ A )
% 0.31/0.50 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.31/0.50 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compEex_type,type,
% 0.31/0.50 breln1compEex: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1compEex,definition,
% 0.31/0.50 ( breln1compEex
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.31/0.50 => ! [Xphi: $o] :
% 0.31/0.50 ( ! [Xz: $i] :
% 0.31/0.50 ( ( in @ Xz @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.31/0.50 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.31/0.50 => Xphi ) ) )
% 0.31/0.50 => Xphi ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionprop_type,type,
% 0.31/0.50 breln1unionprop: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionprop,definition,
% 0.31/0.50 ( breln1unionprop
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionIL_type,type,
% 0.31/0.50 breln1unionIL: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionIL,definition,
% 0.31/0.50 ( breln1unionIL
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionIR_type,type,
% 0.31/0.50 breln1unionIR: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionIR,definition,
% 0.31/0.50 ( breln1unionIR
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionI_type,type,
% 0.31/0.50 breln1unionI: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionI,definition,
% 0.31/0.50 ( breln1unionI
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.50 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.31/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionE_type,type,
% 0.31/0.50 breln1unionE: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionE,definition,
% 0.31/0.50 ( breln1unionE
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.50 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionEcases_type,type,
% 0.31/0.50 breln1unionEcases: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionEcases,definition,
% 0.31/0.50 ( breln1unionEcases
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.31/0.50 => ! [Xphi: $o] :
% 0.31/0.50 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.50 => Xphi )
% 0.31/0.50 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/0.50 => Xphi )
% 0.31/0.50 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionCommutes_type,type,
% 0.31/0.50 breln1unionCommutes: $o ).
% 0.31/0.50
% 0.31/0.50 thf(breln1unionCommutes,definition,
% 0.31/0.50 ( breln1unionCommutes
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ( ( binunion @ R @ S )
% 0.31/0.50 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(woz2Ex_type,type,
% 0.31/0.50 woz2Ex: $o ).
% 0.31/0.50
% 0.31/0.50 thf(woz2Ex,definition,
% 0.31/0.50 ( woz2Ex
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ( R
% 0.31/0.50 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(woz2W_type,type,
% 0.31/0.50 woz2W: $o ).
% 0.31/0.50
% 0.31/0.50 thf(woz2W,definition,
% 0.31/0.50 ( woz2W
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.31/0.50 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(woz2A_type,type,
% 0.31/0.50 woz2A: $o ).
% 0.31/0.50
% 0.31/0.50 thf(woz2A,definition,
% 0.31/0.50 ( woz2A
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [T: $i] :
% 0.31/0.50 ( ( breln1 @ A @ T )
% 0.31/0.50 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.31/0.50 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(woz2B_type,type,
% 0.31/0.50 woz2B: $o ).
% 0.31/0.50
% 0.31/0.50 thf(woz2B,definition,
% 0.31/0.50 ( woz2B
% 0.31/0.50 = ( ! [A: $i,R: $i] :
% 0.31/0.50 ( ( breln1 @ A @ R )
% 0.31/0.50 => ! [S: $i] :
% 0.31/0.50 ( ( breln1 @ A @ S )
% 0.31/0.50 => ! [T: $i] :
% 0.31/0.50 ( ( breln1 @ A @ T )
% 0.31/0.50 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.31/0.50 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(image1Ex_type,type,
% 0.31/0.50 image1Ex: $o ).
% 0.31/0.50
% 0.31/0.50 thf(image1Ex,definition,
% 0.31/0.50 ( image1Ex
% 0.31/0.50 = ( ! [A: $i,Xf: $i > $i] :
% 0.31/0.50 ? [B: $i] :
% 0.31/0.50 ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ B )
% 0.31/0.50 <=> ? [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 & ( Xx
% 0.31/0.50 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(image1Ex1_type,type,
% 0.31/0.50 image1Ex1: $o ).
% 0.31/0.50
% 0.31/0.50 thf(image1Ex1,definition,
% 0.31/0.50 ( image1Ex1
% 0.31/0.50 = ( ! [A: $i,Xf: $i > $i] :
% 0.31/0.50 ( exu
% 0.31/0.50 @ ^ [B: $i] :
% 0.31/0.50 ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ B )
% 0.31/0.50 <=> ? [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 & ( Xx
% 0.31/0.50 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(image1_type,type,
% 0.31/0.50 image1: $i > ( $i > $i ) > $i ).
% 0.31/0.50
% 0.31/0.50 thf(image1Equiv_type,type,
% 0.31/0.50 image1Equiv: $o ).
% 0.31/0.50
% 0.31/0.50 thf(image1Equiv,definition,
% 0.31/0.50 ( image1Equiv
% 0.31/0.50 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.31/0.50 ( ( in @ Xx
% 0.31/0.50 @ ( image1 @ A
% 0.31/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.31/0.50 <=> ? [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 & ( Xx
% 0.31/0.50 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(image1E_type,type,
% 0.31/0.50 image1E: $o ).
% 0.31/0.50
% 0.31/0.50 thf(image1E,definition,
% 0.31/0.50 ( image1E
% 0.31/0.50 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.31/0.50 ( ( in @ Xx
% 0.31/0.50 @ ( image1 @ A
% 0.31/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.31/0.50 => ? [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 & ( Xx
% 0.31/0.50 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(image1I_type,type,
% 0.31/0.50 image1I: $o ).
% 0.31/0.50
% 0.31/0.50 thf(image1I,definition,
% 0.31/0.50 ( image1I
% 0.31/0.50 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.31/0.50 ( ? [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ A )
% 0.31/0.50 & ( Xx
% 0.31/0.50 = ( Xf @ Xy ) ) )
% 0.31/0.50 => ( in @ Xx
% 0.31/0.50 @ ( image1 @ A
% 0.31/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(injective_type,type,
% 0.31/0.50 injective: $i > $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(injFuncSet_type,type,
% 0.31/0.50 injFuncSet: $i > $i > $i ).
% 0.31/0.50
% 0.31/0.50 thf(injFuncInInjFuncSet_type,type,
% 0.31/0.50 injFuncInInjFuncSet: $o ).
% 0.31/0.50
% 0.31/0.50 thf(injFuncInInjFuncSet,definition,
% 0.31/0.50 ( injFuncInInjFuncSet
% 0.31/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.50 => ( ( injective @ A @ B @ Xf )
% 0.31/0.50 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(injFuncSetFuncIn_type,type,
% 0.31/0.50 injFuncSetFuncIn: $o ).
% 0.31/0.50
% 0.31/0.50 thf(injFuncSetFuncIn,definition,
% 0.31/0.50 ( injFuncSetFuncIn
% 0.31/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.50 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.31/0.50 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(injFuncSetFuncInj_type,type,
% 0.31/0.50 injFuncSetFuncInj: $o ).
% 0.31/0.50
% 0.31/0.50 thf(injFuncSetFuncInj,definition,
% 0.31/0.50 ( injFuncSetFuncInj
% 0.31/0.50 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.31/0.50 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.31/0.50 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(surjective_type,type,
% 0.31/0.50 surjective: $i > $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(surjFuncSet_type,type,
% 0.31/0.50 surjFuncSet: $i > $i > $i ).
% 0.31/0.50
% 0.31/0.50 thf(surjFuncSetFuncIn_type,type,
% 0.31/0.50 surjFuncSetFuncIn: $o ).
% 0.31/0.50
% 0.31/0.50 thf(surjFuncSetFuncIn,definition,
% 0.31/0.50 ( surjFuncSetFuncIn
% 0.31/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.50 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.31/0.50 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(surjFuncSetFuncSurj_type,type,
% 0.31/0.50 surjFuncSetFuncSurj: $o ).
% 0.31/0.50
% 0.31/0.50 thf(surjFuncSetFuncSurj,definition,
% 0.31/0.50 ( surjFuncSetFuncSurj
% 0.31/0.50 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.31/0.50 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.31/0.50 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(leftInvIsSurj_type,type,
% 0.31/0.50 leftInvIsSurj: $o ).
% 0.31/0.50
% 0.31/0.50 thf(leftInvIsSurj,definition,
% 0.31/0.50 ( leftInvIsSurj
% 0.31/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.50 ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.50 => ! [Xg: $i] :
% 0.31/0.50 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.31/0.50 => ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.31/0.50 = Xx ) )
% 0.31/0.50 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(surjCantorThm_type,type,
% 0.31/0.50 surjCantorThm: $o ).
% 0.31/0.50
% 0.31/0.50 thf(surjCantorThm,definition,
% 0.31/0.50 ( surjCantorThm
% 0.31/0.50 = ( ! [A: $i,Xf: $i] :
% 0.31/0.50 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.31/0.50 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(foundation2_type,type,
% 0.31/0.50 foundation2: $o ).
% 0.31/0.50
% 0.31/0.50 thf(foundation2,definition,
% 0.31/0.50 ( foundation2
% 0.31/0.50 = ( ! [A: $i] :
% 0.31/0.50 ( ( nonempty @ A )
% 0.31/0.50 => ? [X: $i] :
% 0.31/0.50 ( ( in @ X @ A )
% 0.31/0.50 & ( ( binintersect @ X @ A )
% 0.31/0.50 = emptyset ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(notinself_type,type,
% 0.31/0.50 notinself: $o ).
% 0.31/0.50
% 0.31/0.50 thf(notinself,definition,
% 0.31/0.50 ( notinself
% 0.31/0.50 = ( ! [A: $i] :
% 0.31/0.50 ~ ( in @ A @ A ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(notinself2_type,type,
% 0.31/0.50 notinself2: $o ).
% 0.31/0.50
% 0.31/0.50 thf(notinself2,definition,
% 0.31/0.50 ( notinself2
% 0.31/0.50 = ( ! [A: $i,B: $i] :
% 0.31/0.50 ( ( in @ A @ B )
% 0.31/0.50 => ~ ( in @ B @ A ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(omegaS_type,type,
% 0.31/0.50 omegaS: $i > $i ).
% 0.31/0.50
% 0.31/0.50 thf(omegaSp_type,type,
% 0.31/0.50 omegaSp: $o ).
% 0.31/0.50
% 0.31/0.50 thf(omegaSp,definition,
% 0.31/0.50 ( omegaSp
% 0.31/0.50 = ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ omega )
% 0.31/0.50 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(omegaSclos_type,type,
% 0.31/0.50 omegaSclos: $o ).
% 0.31/0.50
% 0.31/0.50 thf(omegaSclos,definition,
% 0.31/0.50 ( omegaSclos
% 0.31/0.50 = ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ omega )
% 0.31/0.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(peano0notS_type,type,
% 0.31/0.50 peano0notS: $o ).
% 0.31/0.50
% 0.31/0.50 thf(peano0notS,definition,
% 0.31/0.50 ( peano0notS
% 0.31/0.50 = ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ omega )
% 0.31/0.50 => ( ( omegaS @ Xx )
% 0.31/0.50 != emptyset ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(peano3_type,type,
% 0.31/0.50 peano3: $i > $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(peano4_type,type,
% 0.31/0.50 peano4: $i > $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(peano5_type,type,
% 0.31/0.50 peano5: $i > $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(peanoSinj_type,type,
% 0.31/0.50 peanoSinj: $o ).
% 0.31/0.50
% 0.31/0.50 thf(peanoSinj,definition,
% 0.31/0.50 ( peanoSinj
% 0.31/0.50 = ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ omega )
% 0.31/0.50 => ! [Xy: $i] :
% 0.31/0.50 ( ( in @ Xy @ omega )
% 0.31/0.50 => ( ( ( omegaS @ Xx )
% 0.31/0.50 = ( omegaS @ Xy ) )
% 0.31/0.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(peano_type,type,
% 0.31/0.50 peano: $i > $i > $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(transitiveset_type,type,
% 0.31/0.50 transitiveset: $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(transitiveset,definition,
% 0.31/0.50 ( transitiveset
% 0.31/0.50 = ( ^ [A: $i] :
% 0.31/0.50 ! [X: $i] :
% 0.31/0.50 ( ( in @ X @ A )
% 0.31/0.50 => ( subset @ X @ A ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(transitivesetOp1_type,type,
% 0.31/0.50 transitivesetOp1: $o ).
% 0.31/0.50
% 0.31/0.50 thf(transitivesetOp1,definition,
% 0.31/0.50 ( transitivesetOp1
% 0.31/0.50 = ( ! [X: $i] :
% 0.31/0.50 ( ( transitiveset @ X )
% 0.31/0.50 => ! [A: $i] :
% 0.31/0.50 ( ( in @ A @ X )
% 0.31/0.50 => ( subset @ A @ X ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(binintTransitive_type,type,
% 0.31/0.50 binintTransitive: $o ).
% 0.31/0.50
% 0.31/0.50 thf(binintTransitive,definition,
% 0.31/0.50 ( binintTransitive
% 0.31/0.50 = ( ! [X: $i] :
% 0.31/0.50 ( ( transitiveset @ X )
% 0.31/0.50 => ! [Y: $i] :
% 0.31/0.50 ( ( transitiveset @ Y )
% 0.31/0.50 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(transitivesetOp2_type,type,
% 0.31/0.50 transitivesetOp2: $o ).
% 0.31/0.50
% 0.31/0.50 thf(transitivesetOp2,definition,
% 0.31/0.50 ( transitivesetOp2
% 0.31/0.50 = ( ! [X: $i] :
% 0.31/0.50 ( ( transitiveset @ X )
% 0.31/0.50 => ! [A: $i,Xx: $i] :
% 0.31/0.50 ( ( in @ A @ X )
% 0.31/0.50 => ( ( in @ Xx @ A )
% 0.31/0.50 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(setunionTransitive_type,type,
% 0.31/0.50 setunionTransitive: $o ).
% 0.31/0.50
% 0.31/0.50 thf(setunionTransitive,definition,
% 0.31/0.50 ( setunionTransitive
% 0.31/0.50 = ( ! [X: $i] :
% 0.31/0.50 ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ X )
% 0.31/0.50 => ( transitiveset @ Xx ) )
% 0.31/0.50 => ( transitiveset @ ( setunion @ X ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(stricttotalorderedByIn_type,type,
% 0.31/0.50 stricttotalorderedByIn: $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(stricttotalorderedByIn,definition,
% 0.31/0.50 ( stricttotalorderedByIn
% 0.31/0.50 = ( ^ [A: $i] :
% 0.31/0.50 ( ! [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ A )
% 0.31/0.50 => ! [X: $i] :
% 0.31/0.50 ( ( in @ X @ A )
% 0.31/0.50 => ! [Y: $i] :
% 0.31/0.50 ( ( in @ Y @ A )
% 0.31/0.50 => ( ( ( in @ Xx @ X )
% 0.31/0.50 & ( in @ X @ Y ) )
% 0.31/0.50 => ( in @ Xx @ Y ) ) ) ) )
% 0.31/0.50 & ! [X: $i] :
% 0.31/0.50 ( ( in @ X @ A )
% 0.31/0.50 => ! [Y: $i] :
% 0.31/0.50 ( ( in @ Y @ A )
% 0.31/0.50 => ( ( X = Y )
% 0.31/0.50 | ( in @ X @ Y )
% 0.31/0.50 | ( in @ Y @ X ) ) ) )
% 0.31/0.50 & ! [X: $i] :
% 0.31/0.50 ( ( in @ X @ A )
% 0.31/0.50 => ~ ( in @ X @ X ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(wellorderedByIn_type,type,
% 0.31/0.50 wellorderedByIn: $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(wellorderedByIn,definition,
% 0.31/0.50 ( wellorderedByIn
% 0.31/0.50 = ( ^ [A: $i] :
% 0.31/0.50 ( ( stricttotalorderedByIn @ A )
% 0.31/0.50 & ! [X: $i] :
% 0.31/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.50 => ( ( nonempty @ X )
% 0.31/0.50 => ? [Xx: $i] :
% 0.31/0.50 ( ( in @ Xx @ X )
% 0.31/0.50 & ! [Y: $i] :
% 0.31/0.50 ( ( in @ Y @ X )
% 0.31/0.50 => ( ( Xx = Y )
% 0.31/0.50 | ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(ordinal_type,type,
% 0.31/0.50 ordinal: $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(ordinal,definition,
% 0.31/0.50 ( ordinal
% 0.31/0.50 = ( ^ [Xx: $i] :
% 0.31/0.50 ( ( transitiveset @ Xx )
% 0.31/0.50 & ( wellorderedByIn @ Xx ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(limitOrdinal_type,type,
% 0.31/0.50 limitOrdinal: $i > $o ).
% 0.31/0.50
% 0.31/0.50 thf(ordinalMinLem1_type,type,
% 0.31/0.50 ordinalMinLem1: $o ).
% 0.31/0.50
% 0.31/0.50 thf(ordinalMinLem1,definition,
% 0.31/0.50 ( ordinalMinLem1
% 0.31/0.50 = ( ! [X: $i] :
% 0.31/0.50 ( ( ordinal @ X )
% 0.31/0.50 => ! [Y: $i] :
% 0.31/0.50 ( ( ordinal @ Y )
% 0.31/0.50 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 thf(ordinalTransSet,conjecture,
% 0.31/0.50 ( setextAx
% 0.31/0.50 => ( emptysetAx
% 0.31/0.50 => ( setadjoinAx
% 0.31/0.50 => ( powersetAx
% 0.31/0.50 => ( setunionAx
% 0.31/0.50 => ( omega0Ax
% 0.31/0.50 => ( omegaSAx
% 0.31/0.50 => ( omegaIndAx
% 0.31/0.50 => ( replAx
% 0.31/0.50 => ( foundationAx
% 0.31/0.50 => ( wellorderingAx
% 0.31/0.50 => ( descrp
% 0.31/0.50 => ( dsetconstrI
% 0.31/0.50 => ( dsetconstrEL
% 0.31/0.50 => ( dsetconstrER
% 0.31/0.50 => ( exuE1
% 0.31/0.50 => ( prop2setE
% 0.31/0.50 => ( emptysetE
% 0.31/0.50 => ( emptysetimpfalse
% 0.31/0.50 => ( notinemptyset
% 0.31/0.50 => ( exuE3e
% 0.31/0.50 => ( setext
% 0.31/0.50 => ( emptyI
% 0.31/0.50 => ( noeltsimpempty
% 0.31/0.50 => ( setbeta
% 0.31/0.50 => ( nonemptyE1
% 0.31/0.50 => ( nonemptyI
% 0.31/0.50 => ( nonemptyI1
% 0.31/0.50 => ( setadjoinIL
% 0.31/0.50 => ( emptyinunitempty
% 0.31/0.50 => ( setadjoinIR
% 0.31/0.50 => ( setadjoinE
% 0.31/0.50 => ( setadjoinOr
% 0.31/0.50 => ( setoftrueEq
% 0.31/0.50 => ( powersetI
% 0.31/0.50 => ( emptyinPowerset
% 0.31/0.50 => ( emptyInPowerset
% 0.31/0.50 => ( powersetE
% 0.31/0.50 => ( setunionI
% 0.31/0.50 => ( setunionE
% 0.31/0.50 => ( subPowSU
% 0.31/0.50 => ( exuE2
% 0.31/0.50 => ( nonemptyImpWitness
% 0.31/0.50 => ( uniqinunit
% 0.31/0.50 => ( notinsingleton
% 0.31/0.50 => ( eqinunit
% 0.31/0.50 => ( singletonsswitch
% 0.31/0.50 => ( upairsetE
% 0.31/0.50 => ( upairsetIL
% 0.31/0.50 => ( upairsetIR
% 0.31/0.50 => ( emptyE1
% 0.31/0.50 => ( vacuousDall
% 0.31/0.50 => ( quantDeMorgan1
% 0.31/0.50 => ( quantDeMorgan2
% 0.31/0.50 => ( quantDeMorgan3
% 0.31/0.50 => ( quantDeMorgan4
% 0.31/0.50 => ( prop2setI
% 0.31/0.50 => ( prop2set2propI
% 0.31/0.50 => ( notdexE
% 0.31/0.50 => ( notdallE
% 0.31/0.50 => ( exuI1
% 0.31/0.50 => ( exuI3
% 0.31/0.50 => ( exuI2
% 0.31/0.50 => ( inCongP
% 0.31/0.50 => ( in__Cong
% 0.31/0.50 => ( exuE3u
% 0.31/0.50 => ( exu__Cong
% 0.31/0.50 => ( emptyset__Cong
% 0.31/0.50 => ( setadjoin__Cong
% 0.31/0.50 => ( powerset__Cong
% 0.31/0.50 => ( setunion__Cong
% 0.31/0.50 => ( omega__Cong
% 0.31/0.50 => ( exuEu
% 0.31/0.50 => ( descr__Cong
% 0.31/0.50 => ( dsetconstr__Cong
% 0.31/0.50 => ( subsetI1
% 0.31/0.50 => ( eqimpsubset2
% 0.31/0.50 => ( eqimpsubset1
% 0.31/0.50 => ( subsetI2
% 0.31/0.50 => ( emptysetsubset
% 0.31/0.50 => ( subsetE
% 0.31/0.50 => ( subsetE2
% 0.31/0.50 => ( notsubsetI
% 0.31/0.50 => ( notequalI1
% 0.31/0.50 => ( notequalI2
% 0.31/0.50 => ( subsetRefl
% 0.31/0.50 => ( subsetTrans
% 0.31/0.50 => ( setadjoinSub
% 0.31/0.50 => ( setadjoinSub2
% 0.31/0.50 => ( subset2powerset
% 0.31/0.50 => ( setextsub
% 0.31/0.50 => ( subsetemptysetimpeq
% 0.31/0.50 => ( powersetI1
% 0.31/0.50 => ( powersetE1
% 0.31/0.50 => ( inPowerset
% 0.31/0.50 => ( powersetsubset
% 0.31/0.50 => ( sepInPowerset
% 0.31/0.50 => ( sepSubset
% 0.31/0.50 => ( binunionIL
% 0.31/0.50 => ( upairset2IR
% 0.31/0.50 => ( binunionIR
% 0.31/0.50 => ( binunionEcases
% 0.31/0.50 => ( binunionE
% 0.31/0.50 => ( binunionLsub
% 0.31/0.50 => ( binunionRsub
% 0.31/0.50 => ( binintersectI
% 0.31/0.50 => ( binintersectSubset5
% 0.31/0.50 => ( binintersectEL
% 0.31/0.50 => ( binintersectLsub
% 0.31/0.50 => ( binintersectSubset2
% 0.31/0.50 => ( binintersectSubset3
% 0.31/0.50 => ( binintersectER
% 0.31/0.50 => ( disjointsetsI1
% 0.31/0.50 => ( binintersectRsub
% 0.31/0.50 => ( binintersectSubset4
% 0.31/0.50 => ( binintersectSubset1
% 0.31/0.50 => ( bs114d
% 0.31/0.50 => ( setminusI
% 0.31/0.50 => ( setminusEL
% 0.31/0.50 => ( setminusER
% 0.31/0.50 => ( setminusSubset2
% 0.31/0.50 => ( setminusERneg
% 0.31/0.50 => ( setminusELneg
% 0.31/0.50 => ( setminusILneg
% 0.31/0.50 => ( setminusIRneg
% 0.31/0.50 => ( setminusLsub
% 0.31/0.50 => ( setminusSubset1
% 0.31/0.50 => ( symdiffE
% 0.31/0.50 => ( symdiffI1
% 0.31/0.50 => ( symdiffI2
% 0.31/0.50 => ( symdiffIneg1
% 0.31/0.50 => ( symdiffIneg2
% 0.31/0.50 => ( secondinupair
% 0.31/0.50 => ( setukpairIL
% 0.31/0.50 => ( setukpairIR
% 0.31/0.50 => ( kpairiskpair
% 0.31/0.50 => ( kpairp
% 0.31/0.50 => ( singletonsubset
% 0.31/0.50 => ( singletoninpowerset
% 0.31/0.50 => ( singletoninpowunion
% 0.31/0.50 => ( upairset2E
% 0.31/0.50 => ( upairsubunion
% 0.31/0.50 => ( upairinpowunion
% 0.31/0.50 => ( ubforcartprodlem1
% 0.31/0.50 => ( ubforcartprodlem2
% 0.31/0.50 => ( ubforcartprodlem3
% 0.31/0.50 => ( cartprodpairin
% 0.31/0.50 => ( cartprodmempair1
% 0.31/0.50 => ( cartprodmempair
% 0.31/0.50 => ( setunionE2
% 0.31/0.50 => ( setunionsingleton1
% 0.31/0.50 => ( setunionsingleton2
% 0.31/0.50 => ( setunionsingleton
% 0.31/0.50 => ( singletonprop
% 0.31/0.50 => ( ex1E1
% 0.31/0.50 => ( ex1I
% 0.31/0.50 => ( ex1I2
% 0.31/0.50 => ( singletonsuniq
% 0.31/0.50 => ( setukpairinjL1
% 0.31/0.50 => ( kfstsingleton
% 0.31/0.50 => ( theprop
% 0.31/0.50 => ( kfstpairEq
% 0.31/0.50 => ( cartprodfstin
% 0.31/0.50 => ( setukpairinjL2
% 0.31/0.50 => ( setukpairinjL
% 0.31/0.50 => ( setukpairinjR11
% 0.31/0.50 => ( setukpairinjR12
% 0.31/0.50 => ( setukpairinjR1
% 0.31/0.50 => ( upairequniteq
% 0.31/0.50 => ( setukpairinjR2
% 0.31/0.50 => ( setukpairinjR
% 0.31/0.50 => ( ksndsingleton
% 0.31/0.50 => ( ksndpairEq
% 0.31/0.50 => ( kpairsurjEq
% 0.31/0.50 => ( cartprodsndin
% 0.31/0.50 => ( cartprodpairmemEL
% 0.31/0.50 => ( cartprodpairmemER
% 0.31/0.50 => ( cartprodmempaircEq
% 0.31/0.50 => ( cartprodfstpairEq
% 0.31/0.50 => ( cartprodsndpairEq
% 0.31/0.50 => ( cartprodpairsurjEq
% 0.31/0.50 => ( dpsetconstrI
% 0.31/0.50 => ( dpsetconstrSub
% 0.31/0.50 => ( setOfPairsIsBReln
% 0.31/0.50 => ( dpsetconstrERa
% 0.31/0.50 => ( dpsetconstrEL1
% 0.31/0.50 => ( dpsetconstrEL2
% 0.31/0.50 => ( dpsetconstrER
% 0.31/0.50 => ( funcImageSingleton
% 0.31/0.50 => ( apProp
% 0.31/0.50 => ( app
% 0.31/0.50 => ( infuncsetfunc
% 0.31/0.50 => ( ap2p
% 0.31/0.50 => ( funcinfuncset
% 0.31/0.50 => ( lamProp
% 0.31/0.50 => ( lamp
% 0.31/0.50 => ( lam2p
% 0.31/0.50 => ( brelnall1
% 0.31/0.50 => ( brelnall2
% 0.31/0.50 => ( ex1E2
% 0.31/0.50 => ( funcGraphProp1
% 0.31/0.50 => ( funcGraphProp3
% 0.31/0.50 => ( funcGraphProp2
% 0.31/0.50 => ( funcextLem
% 0.31/0.50 => ( funcGraphProp4
% 0.31/0.50 => ( subbreln
% 0.31/0.50 => ( eqbreln
% 0.31/0.50 => ( funcext
% 0.31/0.50 => ( funcext2
% 0.31/0.50 => ( ap2apEq1
% 0.31/0.50 => ( ap2apEq2
% 0.31/0.50 => ( beta1
% 0.31/0.50 => ( eta1
% 0.31/0.50 => ( lam2lamEq
% 0.31/0.50 => ( beta2
% 0.31/0.50 => ( eta2
% 0.31/0.50 => ( iffalseProp1
% 0.31/0.50 => ( iffalseProp2
% 0.31/0.50 => ( iftrueProp1
% 0.31/0.50 => ( iftrueProp2
% 0.31/0.50 => ( ifSingleton
% 0.31/0.50 => ( ifp
% 0.31/0.50 => ( theeq
% 0.31/0.50 => ( iftrue
% 0.31/0.50 => ( iffalse
% 0.31/0.50 => ( iftrueorfalse
% 0.31/0.50 => ( binintersectT_lem
% 0.31/0.50 => ( binunionT_lem
% 0.31/0.50 => ( powersetT_lem
% 0.31/0.50 => ( setminusT_lem
% 0.31/0.50 => ( complementT_lem
% 0.31/0.50 => ( setextT
% 0.31/0.50 => ( subsetTI
% 0.31/0.50 => ( powersetTI1
% 0.31/0.50 => ( powersetTE1
% 0.31/0.50 => ( complementTI1
% 0.31/0.50 => ( complementTE1
% 0.31/0.50 => ( binintersectTELcontra
% 0.31/0.50 => ( binintersectTERcontra
% 0.31/0.50 => ( contrasubsetT
% 0.31/0.50 => ( contrasubsetT1
% 0.31/0.50 => ( contrasubsetT2
% 0.31/0.50 => ( contrasubsetT3
% 0.31/0.50 => ( doubleComplementI1
% 0.31/0.50 => ( doubleComplementE1
% 0.31/0.50 => ( doubleComplementSub1
% 0.31/0.50 => ( doubleComplementSub2
% 0.31/0.50 => ( doubleComplementEq
% 0.31/0.50 => ( complementTnotintersectT
% 0.31/0.50 => ( complementImpComplementIntersect
% 0.31/0.50 => ( complementSubsetComplementIntersect
% 0.31/0.50 => ( complementInPowersetComplementIntersect
% 0.31/0.50 => ( contraSubsetComplement
% 0.31/0.50 => ( complementTcontraSubset
% 0.31/0.50 => ( binunionTILcontra
% 0.31/0.50 => ( binunionTIRcontra
% 0.31/0.50 => ( inIntersectImpInUnion
% 0.31/0.50 => ( inIntersectImpInUnion2
% 0.31/0.50 => ( inIntersectImpInIntersectUnions
% 0.31/0.50 => ( intersectInPowersetIntersectUnions
% 0.31/0.50 => ( inComplementUnionImpNotIn1
% 0.31/0.50 => ( inComplementUnionImpInComplement1
% 0.31/0.50 => ( binunionTE
% 0.31/0.50 => ( binunionTEcontra
% 0.31/0.50 => ( demorgan2a1
% 0.31/0.50 => ( complementUnionInPowersetComplement
% 0.31/0.50 => ( demorgan2a2
% 0.31/0.50 => ( demorgan1a
% 0.31/0.50 => ( demorgan1b
% 0.31/0.50 => ( demorgan1
% 0.31/0.50 => ( demorgan2a
% 0.31/0.50 => ( demorgan2b2
% 0.31/0.50 => ( demorgan2b
% 0.31/0.50 => ( demorgan2
% 0.31/0.50 => ( woz13rule0
% 0.31/0.50 => ( woz13rule1
% 0.31/0.50 => ( woz13rule2
% 0.31/0.50 => ( woz13rule3
% 0.31/0.50 => ( woz13rule4
% 0.31/0.50 => ( woz1_1
% 0.31/0.50 => ( woz1_2
% 0.31/0.50 => ( woz1_3
% 0.31/0.50 => ( woz1_4
% 0.31/0.50 => ( woz1_5
% 0.31/0.50 => ( breln1all2
% 0.31/0.50 => ( breln1SetBreln1
% 0.31/0.50 => ( choice2fnsingleton
% 0.31/0.50 => ( setOfPairsIsBReln1
% 0.31/0.50 => ( breln1all1
% 0.31/0.50 => ( subbreln1
% 0.31/0.50 => ( eqbreln1
% 0.31/0.50 => ( breln1invprop
% 0.31/0.50 => ( breln1invI
% 0.31/0.50 => ( breln1invE
% 0.31/0.50 => ( breln1compprop
% 0.31/0.50 => ( breln1compI
% 0.31/0.50 => ( breln1compE
% 0.31/0.50 => ( breln1compEex
% 0.31/0.50 => ( breln1unionprop
% 0.31/0.50 => ( breln1unionIL
% 0.31/0.50 => ( breln1unionIR
% 0.31/0.50 => ( breln1unionI
% 0.31/0.50 => ( breln1unionE
% 0.31/0.50 => ( breln1unionEcases
% 0.31/0.50 => ( breln1unionCommutes
% 0.31/0.50 => ( woz2Ex
% 0.31/0.50 => ( woz2W
% 0.31/0.50 => ( woz2A
% 0.31/0.50 => ( woz2B
% 0.31/0.50 => ( image1Ex
% 0.31/0.50 => ( image1Ex1
% 0.31/0.50 => ( image1Equiv
% 0.31/0.50 => ( image1E
% 0.31/0.50 => ( image1I
% 0.31/0.50 => ( injFuncInInjFuncSet
% 0.31/0.50 => ( injFuncSetFuncIn
% 0.31/0.50 => ( injFuncSetFuncInj
% 0.31/0.50 => ( surjFuncSetFuncIn
% 0.31/0.50 => ( surjFuncSetFuncSurj
% 0.31/0.50 => ( leftInvIsSurj
% 0.31/0.50 => ( surjCantorThm
% 0.31/0.50 => ( foundation2
% 0.31/0.50 => ( notinself
% 0.31/0.50 => ( notinself2
% 0.31/0.50 => ( omegaSp
% 0.31/0.50 => ( omegaSclos
% 0.31/0.50 => ( peano0notS
% 0.31/0.50 => ( peanoSinj
% 0.31/0.50 => ( transitivesetOp1
% 0.31/0.50 => ( binintTransitive
% 0.31/0.50 => ( transitivesetOp2
% 0.31/0.50 => ( setunionTransitive
% 0.31/0.50 => ( ordinalMinLem1
% 0.31/0.50 => ! [X: $i] :
% 0.31/0.50 ( ( ordinal @ X )
% 0.31/0.50 => ! [Xx: $i,A: $i] :
% 0.31/0.50 ( ( in @ A @ X )
% 0.31/0.50 => ( ( in @ Xx @ A )
% 0.31/0.50 => ( in @ Xx @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.50
% 0.31/0.50 %------------------------------------------------------------------------------
% 0.31/0.50 ------------------------
% 3.03/1.66 ---- Embedded in THF ---
% 3.03/1.66 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.03/1.66 %%% Generated on Tue Feb 24 12:34:47 EST 2026
% 3.03/1.66 %%% using '$$dhol' embedding, version 1.3.0.
% 3.03/1.66 %%% Logic specification used:
% 3.03/1.66 %%% thf(spec, logic, $$dhol).
% 3.03/1.66
% 3.03/1.66 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2DTF13596.p
% See solution above
% 3.03/1.66 ------------------------
% 3.21/1.70 ---- Cleaned THF ---
% 3.21/1.70 thf(in_type,type,
% 3.21/1.70 in: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(exu_type,type,
% 3.21/1.70 exu: ( $i > $o ) > $o ).
% 3.21/1.70
% 3.21/1.70 thf(exu,definition,
% 3.21/1.70 ( exu
% 3.21/1.70 = ( ^ [Xphi: $i > $o] :
% 3.21/1.70 ? [Xx: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 & ! [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setextAx_type,type,
% 3.21/1.70 setextAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setextAx,definition,
% 3.21/1.70 ( setextAx
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 <=> ( in @ Xx @ B ) )
% 3.21/1.70 => ( A = B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyset_type,type,
% 3.21/1.70 emptyset: $i ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetAx_type,type,
% 3.21/1.70 emptysetAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetAx,definition,
% 3.21/1.70 ( emptysetAx
% 3.21/1.70 = ( ! [Xx: $i] :
% 3.21/1.70 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoin_type,type,
% 3.21/1.70 setadjoin: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinAx_type,type,
% 3.21/1.70 setadjoinAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinAx,definition,
% 3.21/1.70 ( setadjoinAx
% 3.21/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.21/1.70 <=> ( ( Xy = Xx )
% 3.21/1.70 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powerset_type,type,
% 3.21/1.70 powerset: $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(powersetAx_type,type,
% 3.21/1.70 powersetAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetAx,definition,
% 3.21/1.70 ( powersetAx
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( in @ B @ ( powerset @ A ) )
% 3.21/1.70 <=> ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunion_type,type,
% 3.21/1.70 setunion: $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(setunionAx_type,type,
% 3.21/1.70 setunionAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionAx,definition,
% 3.21/1.70 ( setunionAx
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 3.21/1.70 <=> ? [B: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 & ( in @ B @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(omega_type,type,
% 3.21/1.70 omega: $i ).
% 3.21/1.70
% 3.21/1.70 thf(omega0Ax_type,type,
% 3.21/1.70 omega0Ax: $o ).
% 3.21/1.70
% 3.21/1.70 thf(omega0Ax,definition,
% 3.21/1.70 ( omega0Ax
% 3.21/1.70 = ( in @ emptyset @ omega ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(omegaSAx_type,type,
% 3.21/1.70 omegaSAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(omegaSAx,definition,
% 3.21/1.70 ( omegaSAx
% 3.21/1.70 = ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ omega )
% 3.21/1.70 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(omegaIndAx_type,type,
% 3.21/1.70 omegaIndAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(omegaIndAx,definition,
% 3.21/1.70 ( omegaIndAx
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( ( in @ emptyset @ A )
% 3.21/1.70 & ! [Xx: $i] :
% 3.21/1.70 ( ( ( in @ Xx @ omega )
% 3.21/1.70 & ( in @ Xx @ A ) )
% 3.21/1.70 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ omega )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(replAx_type,type,
% 3.21/1.70 replAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(replAx,definition,
% 3.21/1.70 ( replAx
% 3.21/1.70 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( exu
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.21/1.70 => ? [B: $i] :
% 3.21/1.70 ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 <=> ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(foundationAx_type,type,
% 3.21/1.70 foundationAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(foundationAx,definition,
% 3.21/1.70 ( foundationAx
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.21/1.70 => ? [B: $i] :
% 3.21/1.70 ( ( in @ B @ A )
% 3.21/1.70 & ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(wellorderingAx_type,type,
% 3.21/1.70 wellorderingAx: $o ).
% 3.21/1.70
% 3.21/1.70 thf(wellorderingAx,definition,
% 3.21/1.70 ( wellorderingAx
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ? [B: $i] :
% 3.21/1.70 ( ! [C: $i] :
% 3.21/1.70 ( ( in @ C @ B )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ C )
% 3.21/1.70 => ( in @ Xx @ A ) ) )
% 3.21/1.70 & ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( ( in @ Xx @ A )
% 3.21/1.70 & ( in @ Xy @ A ) )
% 3.21/1.70 => ( ! [C: $i] :
% 3.21/1.70 ( ( in @ C @ B )
% 3.21/1.70 => ( ( in @ Xx @ C )
% 3.21/1.70 <=> ( in @ Xy @ C ) ) )
% 3.21/1.70 => ( Xx = Xy ) ) )
% 3.21/1.70 & ! [C: $i,D: $i] :
% 3.21/1.70 ( ( ( in @ C @ B )
% 3.21/1.70 & ( in @ D @ B ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ C )
% 3.21/1.70 => ( in @ Xx @ D ) )
% 3.21/1.70 | ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ D )
% 3.21/1.70 => ( in @ Xx @ C ) ) ) )
% 3.21/1.70 & ! [C: $i] :
% 3.21/1.70 ( ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ C )
% 3.21/1.70 => ( in @ Xx @ A ) )
% 3.21/1.70 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.21/1.70 => ? [D: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ D @ B )
% 3.21/1.70 & ( in @ Xx @ C )
% 3.21/1.70 & ~ ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ D )
% 3.21/1.70 & ( in @ Xy @ C ) )
% 3.21/1.70 & ! [E: $i] :
% 3.21/1.70 ( ( in @ E @ B )
% 3.21/1.70 => ( ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ E )
% 3.21/1.70 => ( in @ Xy @ D ) )
% 3.21/1.70 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(descr_type,type,
% 3.21/1.70 descr: ( $i > $o ) > $i ).
% 3.21/1.70
% 3.21/1.70 thf(descrp_type,type,
% 3.21/1.70 descrp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(descrp,definition,
% 3.21/1.70 ( descrp
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ( Xphi
% 3.21/1.70 @ ( descr
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstr_type,type,
% 3.21/1.70 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstrI_type,type,
% 3.21/1.70 dsetconstrI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstrI,definition,
% 3.21/1.70 ( dsetconstrI
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( in @ Xx
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstrEL_type,type,
% 3.21/1.70 dsetconstrEL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstrEL,definition,
% 3.21/1.70 ( dsetconstrEL
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstrER_type,type,
% 3.21/1.70 dsetconstrER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstrER,definition,
% 3.21/1.70 ( dsetconstrER
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuE1_type,type,
% 3.21/1.70 exuE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuE1,definition,
% 3.21/1.70 ( exuE1
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 & ! [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(prop2set_type,type,
% 3.21/1.70 prop2set: $o > $i ).
% 3.21/1.70
% 3.21/1.70 thf(prop2setE_type,type,
% 3.21/1.70 prop2setE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(prop2setE,definition,
% 3.21/1.70 ( prop2setE
% 3.21/1.70 = ( ! [Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.21/1.70 => Xphi ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetE_type,type,
% 3.21/1.70 emptysetE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetE,definition,
% 3.21/1.70 ( emptysetE
% 3.21/1.70 = ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ emptyset )
% 3.21/1.70 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetimpfalse_type,type,
% 3.21/1.70 emptysetimpfalse: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetimpfalse,definition,
% 3.21/1.70 ( emptysetimpfalse
% 3.21/1.70 = ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ emptyset )
% 3.21/1.70 => $false ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notinemptyset_type,type,
% 3.21/1.70 notinemptyset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notinemptyset,definition,
% 3.21/1.70 ( notinemptyset
% 3.21/1.70 = ( ! [Xx: $i] :
% 3.21/1.70 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuE3e_type,type,
% 3.21/1.70 exuE3e: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuE3e,definition,
% 3.21/1.70 ( exuE3e
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setext_type,type,
% 3.21/1.70 setext: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setext,definition,
% 3.21/1.70 ( setext
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ A ) )
% 3.21/1.70 => ( A = B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyI_type,type,
% 3.21/1.70 emptyI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyI,definition,
% 3.21/1.70 ( emptyI
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ~ ( in @ Xx @ A )
% 3.21/1.70 => ( A = emptyset ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(noeltsimpempty_type,type,
% 3.21/1.70 noeltsimpempty: $o ).
% 3.21/1.70
% 3.21/1.70 thf(noeltsimpempty,definition,
% 3.21/1.70 ( noeltsimpempty
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ~ ( in @ Xx @ A )
% 3.21/1.70 => ( A = emptyset ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setbeta_type,type,
% 3.21/1.70 setbeta: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setbeta,definition,
% 3.21/1.70 ( setbeta
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.21/1.70 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonempty_type,type,
% 3.21/1.70 nonempty: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonempty,definition,
% 3.21/1.70 ( nonempty
% 3.21/1.70 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyE1_type,type,
% 3.21/1.70 nonemptyE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyE1,definition,
% 3.21/1.70 ( nonemptyE1
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( nonempty @ A )
% 3.21/1.70 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyI_type,type,
% 3.21/1.70 nonemptyI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyI,definition,
% 3.21/1.70 ( nonemptyI
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( nonempty
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyI1_type,type,
% 3.21/1.70 nonemptyI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyI1,definition,
% 3.21/1.70 ( nonemptyI1
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.21/1.70 => ( nonempty @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinIL_type,type,
% 3.21/1.70 setadjoinIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinIL,definition,
% 3.21/1.70 ( setadjoinIL
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyinunitempty_type,type,
% 3.21/1.70 emptyinunitempty: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyinunitempty,definition,
% 3.21/1.70 ( emptyinunitempty
% 3.21/1.70 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinIR_type,type,
% 3.21/1.70 setadjoinIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinIR,definition,
% 3.21/1.70 ( setadjoinIR
% 3.21/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinE_type,type,
% 3.21/1.70 setadjoinE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinE,definition,
% 3.21/1.70 ( setadjoinE
% 3.21/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.21/1.70 => ! [Xphi: $o] :
% 3.21/1.70 ( ( ( Xy = Xx )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => ( ( ( in @ Xy @ A )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinOr_type,type,
% 3.21/1.70 setadjoinOr: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinOr,definition,
% 3.21/1.70 ( setadjoinOr
% 3.21/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.21/1.70 => ( ( Xy = Xx )
% 3.21/1.70 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setoftrueEq_type,type,
% 3.21/1.70 setoftrueEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setoftrueEq,definition,
% 3.21/1.70 ( setoftrueEq
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : $true )
% 3.21/1.70 = A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetI_type,type,
% 3.21/1.70 powersetI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetI,definition,
% 3.21/1.70 ( powersetI
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ A ) )
% 3.21/1.70 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyinPowerset_type,type,
% 3.21/1.70 emptyinPowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyinPowerset,definition,
% 3.21/1.70 ( emptyinPowerset
% 3.21/1.70 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyInPowerset_type,type,
% 3.21/1.70 emptyInPowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyInPowerset,definition,
% 3.21/1.70 ( emptyInPowerset
% 3.21/1.70 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetE_type,type,
% 3.21/1.70 powersetE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetE,definition,
% 3.21/1.70 ( powersetE
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ B @ ( powerset @ A ) )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionI_type,type,
% 3.21/1.70 setunionI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionI,definition,
% 3.21/1.70 ( setunionI
% 3.21/1.70 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( ( in @ B @ A )
% 3.21/1.70 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionE_type,type,
% 3.21/1.70 setunionE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionE,definition,
% 3.21/1.70 ( setunionE
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 3.21/1.70 => ! [Xphi: $o] :
% 3.21/1.70 ( ! [B: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( ( in @ B @ A )
% 3.21/1.70 => Xphi ) )
% 3.21/1.70 => Xphi ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subPowSU_type,type,
% 3.21/1.70 subPowSU: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subPowSU,definition,
% 3.21/1.70 ( subPowSU
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuE2_type,type,
% 3.21/1.70 exuE2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuE2,definition,
% 3.21/1.70 ( exuE2
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ! [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xy )
% 3.21/1.70 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyImpWitness_type,type,
% 3.21/1.70 nonemptyImpWitness: $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyImpWitness,definition,
% 3.21/1.70 ( nonemptyImpWitness
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( nonempty @ A )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & $true ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(uniqinunit_type,type,
% 3.21/1.70 uniqinunit: $o ).
% 3.21/1.70
% 3.21/1.70 thf(uniqinunit,definition,
% 3.21/1.70 ( uniqinunit
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notinsingleton_type,type,
% 3.21/1.70 notinsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notinsingleton,definition,
% 3.21/1.70 ( notinsingleton
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx != Xy )
% 3.21/1.70 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqinunit_type,type,
% 3.21/1.70 eqinunit: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqinunit,definition,
% 3.21/1.70 ( eqinunit
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsswitch_type,type,
% 3.21/1.70 singletonsswitch: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsswitch,definition,
% 3.21/1.70 ( singletonsswitch
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetE_type,type,
% 3.21/1.70 upairsetE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetE,definition,
% 3.21/1.70 ( upairsetE
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.21/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.21/1.70 => ( ( Xz = Xx )
% 3.21/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetIL_type,type,
% 3.21/1.70 upairsetIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetIL,definition,
% 3.21/1.70 ( upairsetIL
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetIR_type,type,
% 3.21/1.70 upairsetIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetIR,definition,
% 3.21/1.70 ( upairsetIR
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyE1_type,type,
% 3.21/1.70 emptyE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyE1,definition,
% 3.21/1.70 ( emptyE1
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ( ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 = emptyset )
% 3.21/1.70 => $false ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(vacuousDall_type,type,
% 3.21/1.70 vacuousDall: $o ).
% 3.21/1.70
% 3.21/1.70 thf(vacuousDall,definition,
% 3.21/1.70 ( vacuousDall
% 3.21/1.70 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ emptyset )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan1_type,type,
% 3.21/1.70 quantDeMorgan1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan1,definition,
% 3.21/1.70 ( quantDeMorgan1
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ~ ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan2_type,type,
% 3.21/1.70 quantDeMorgan2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan2,definition,
% 3.21/1.70 ( quantDeMorgan2
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ~ ( Xphi @ Xx ) )
% 3.21/1.70 => ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan3_type,type,
% 3.21/1.70 quantDeMorgan3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan3,definition,
% 3.21/1.70 ( quantDeMorgan3
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan4_type,type,
% 3.21/1.70 quantDeMorgan4: $o ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan4,definition,
% 3.21/1.70 ( quantDeMorgan4
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ~ ( Xphi @ Xx ) )
% 3.21/1.70 => ~ ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(prop2setI_type,type,
% 3.21/1.70 prop2setI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(prop2setI,definition,
% 3.21/1.70 ( prop2setI
% 3.21/1.70 = ( ! [Xphi: $o] :
% 3.21/1.70 ( Xphi
% 3.21/1.70 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(set2prop_type,type,
% 3.21/1.70 set2prop: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(prop2set2propI_type,type,
% 3.21/1.70 prop2set2propI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(prop2set2propI,definition,
% 3.21/1.70 ( prop2set2propI
% 3.21/1.70 = ( ! [Xphi: $o] :
% 3.21/1.70 ( Xphi
% 3.21/1.70 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notdexE_type,type,
% 3.21/1.70 notdexE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notdexE,definition,
% 3.21/1.70 ( notdexE
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notdallE_type,type,
% 3.21/1.70 notdallE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notdallE,definition,
% 3.21/1.70 ( notdallE
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ~ ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuI1_type,type,
% 3.21/1.70 exuI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuI1,definition,
% 3.21/1.70 ( exuI1
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 & ! [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) )
% 3.21/1.70 => ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuI3_type,type,
% 3.21/1.70 exuI3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuI3,definition,
% 3.21/1.70 ( exuI3
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.21/1.70 => ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) )
% 3.21/1.70 => ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuI2_type,type,
% 3.21/1.70 exuI2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuI2,definition,
% 3.21/1.70 ( exuI2
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] :
% 3.21/1.70 ! [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xy )
% 3.21/1.70 <=> ( Xy = Xx ) )
% 3.21/1.70 => ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inCongP_type,type,
% 3.21/1.70 inCongP: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inCongP,definition,
% 3.21/1.70 ( inCongP
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(in__Cong_type,type,
% 3.21/1.70 in__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(in__Cong,definition,
% 3.21/1.70 ( in__Cong
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuE3u_type,type,
% 3.21/1.70 exuE3u: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuE3u,definition,
% 3.21/1.70 ( exuE3u
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exu__Cong_type,type,
% 3.21/1.70 exu__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exu__Cong,definition,
% 3.21/1.70 ( exu__Cong
% 3.21/1.70 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 <=> ( Xpsi @ Xy ) ) )
% 3.21/1.70 => ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 <=> ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyset__Cong_type,type,
% 3.21/1.70 emptyset__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyset__Cong,definition,
% 3.21/1.70 ( emptyset__Cong
% 3.21/1.70 = ( emptyset = emptyset ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoin__Cong_type,type,
% 3.21/1.70 setadjoin__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoin__Cong,definition,
% 3.21/1.70 ( setadjoin__Cong
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ! [Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( Xz = Xu )
% 3.21/1.70 => ( ( setadjoin @ Xx @ Xz )
% 3.21/1.70 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powerset__Cong_type,type,
% 3.21/1.70 powerset__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powerset__Cong,definition,
% 3.21/1.70 ( powerset__Cong
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ( ( powerset @ A )
% 3.21/1.70 = ( powerset @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunion__Cong_type,type,
% 3.21/1.70 setunion__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunion__Cong,definition,
% 3.21/1.70 ( setunion__Cong
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ( ( setunion @ A )
% 3.21/1.70 = ( setunion @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(omega__Cong_type,type,
% 3.21/1.70 omega__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(omega__Cong,definition,
% 3.21/1.70 ( omega__Cong
% 3.21/1.70 = ( omega = omega ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuEu_type,type,
% 3.21/1.70 exuEu: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuEu,definition,
% 3.21/1.70 ( exuEu
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(descr__Cong_type,type,
% 3.21/1.70 descr__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(descr__Cong,definition,
% 3.21/1.70 ( descr__Cong
% 3.21/1.70 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 <=> ( Xpsi @ Xy ) ) )
% 3.21/1.70 => ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.21/1.70 => ( ( descr
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 = ( descr
% 3.21/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstr__Cong_type,type,
% 3.21/1.70 dsetconstr__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstr__Cong,definition,
% 3.21/1.70 ( dsetconstr__Cong
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( Xx = Xy )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.21/1.70 => ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 = ( dsetconstr @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subset_type,type,
% 3.21/1.70 subset: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(disjoint_type,type,
% 3.21/1.70 disjoint: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(setsmeet_type,type,
% 3.21/1.70 setsmeet: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetI1_type,type,
% 3.21/1.70 subsetI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetI1,definition,
% 3.21/1.70 ( subsetI1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqimpsubset2_type,type,
% 3.21/1.70 eqimpsubset2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqimpsubset2,definition,
% 3.21/1.70 ( eqimpsubset2
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ( subset @ B @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqimpsubset1_type,type,
% 3.21/1.70 eqimpsubset1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqimpsubset1,definition,
% 3.21/1.70 ( eqimpsubset1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetI2_type,type,
% 3.21/1.70 subsetI2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetI2,definition,
% 3.21/1.70 ( subsetI2
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetsubset_type,type,
% 3.21/1.70 emptysetsubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetsubset,definition,
% 3.21/1.70 ( emptysetsubset
% 3.21/1.70 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetE_type,type,
% 3.21/1.70 subsetE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetE,definition,
% 3.21/1.70 ( subsetE
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetE2_type,type,
% 3.21/1.70 subsetE2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetE2,definition,
% 3.21/1.70 ( subsetE2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notsubsetI_type,type,
% 3.21/1.70 notsubsetI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notsubsetI,definition,
% 3.21/1.70 ( notsubsetI
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notequalI1_type,type,
% 3.21/1.70 notequalI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notequalI1,definition,
% 3.21/1.70 ( notequalI1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ~ ( subset @ A @ B )
% 3.21/1.70 => ( A != B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notequalI2_type,type,
% 3.21/1.70 notequalI2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notequalI2,definition,
% 3.21/1.70 ( notequalI2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ( A != B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetRefl_type,type,
% 3.21/1.70 subsetRefl: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetRefl,definition,
% 3.21/1.70 ( subsetRefl
% 3.21/1.70 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetTrans_type,type,
% 3.21/1.70 subsetTrans: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetTrans,definition,
% 3.21/1.70 ( subsetTrans
% 3.21/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( subset @ B @ C )
% 3.21/1.70 => ( subset @ A @ C ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinSub_type,type,
% 3.21/1.70 setadjoinSub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinSub,definition,
% 3.21/1.70 ( setadjoinSub
% 3.21/1.70 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinSub2_type,type,
% 3.21/1.70 setadjoinSub2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinSub2,definition,
% 3.21/1.70 ( setadjoinSub2
% 3.21/1.70 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subset2powerset_type,type,
% 3.21/1.70 subset2powerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subset2powerset,definition,
% 3.21/1.70 ( subset2powerset
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setextsub_type,type,
% 3.21/1.70 setextsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setextsub,definition,
% 3.21/1.70 ( setextsub
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( subset @ B @ A )
% 3.21/1.70 => ( A = B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetemptysetimpeq_type,type,
% 3.21/1.70 subsetemptysetimpeq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetemptysetimpeq,definition,
% 3.21/1.70 ( subsetemptysetimpeq
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( subset @ A @ emptyset )
% 3.21/1.70 => ( A = emptyset ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetI1_type,type,
% 3.21/1.70 powersetI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetI1,definition,
% 3.21/1.70 ( powersetI1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ B @ A )
% 3.21/1.70 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetE1_type,type,
% 3.21/1.70 powersetE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetE1,definition,
% 3.21/1.70 ( powersetE1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( in @ B @ ( powerset @ A ) )
% 3.21/1.70 => ( subset @ B @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inPowerset_type,type,
% 3.21/1.70 inPowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inPowerset,definition,
% 3.21/1.70 ( inPowerset
% 3.21/1.70 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetsubset_type,type,
% 3.21/1.70 powersetsubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetsubset,definition,
% 3.21/1.70 ( powersetsubset
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(sepInPowerset_type,type,
% 3.21/1.70 sepInPowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(sepInPowerset,definition,
% 3.21/1.70 ( sepInPowerset
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( in
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 @ ( powerset @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(sepSubset_type,type,
% 3.21/1.70 sepSubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(sepSubset,definition,
% 3.21/1.70 ( sepSubset
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( subset
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunion_type,type,
% 3.21/1.70 binunion: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(binunionIL_type,type,
% 3.21/1.70 binunionIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionIL,definition,
% 3.21/1.70 ( binunionIL
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairset2IR_type,type,
% 3.21/1.70 upairset2IR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairset2IR,definition,
% 3.21/1.70 ( upairset2IR
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionIR_type,type,
% 3.21/1.70 binunionIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionIR,definition,
% 3.21/1.70 ( binunionIR
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionEcases_type,type,
% 3.21/1.70 binunionEcases: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionEcases,definition,
% 3.21/1.70 ( binunionEcases
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.21/1.70 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.21/1.70 => ( ( ( in @ Xx @ A )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => ( ( ( in @ Xx @ B )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionE_type,type,
% 3.21/1.70 binunionE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionE,definition,
% 3.21/1.70 ( binunionE
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionLsub_type,type,
% 3.21/1.70 binunionLsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionLsub,definition,
% 3.21/1.70 ( binunionLsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionRsub_type,type,
% 3.21/1.70 binunionRsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionRsub,definition,
% 3.21/1.70 ( binunionRsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersect_type,type,
% 3.21/1.70 binintersect: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectI_type,type,
% 3.21/1.70 binintersectI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectI,definition,
% 3.21/1.70 ( binintersectI
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset5_type,type,
% 3.21/1.70 binintersectSubset5: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset5,definition,
% 3.21/1.70 ( binintersectSubset5
% 3.21/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 3.21/1.70 ( ( subset @ C @ A )
% 3.21/1.70 => ( ( subset @ C @ B )
% 3.21/1.70 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectEL_type,type,
% 3.21/1.70 binintersectEL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectEL,definition,
% 3.21/1.70 ( binintersectEL
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectLsub_type,type,
% 3.21/1.70 binintersectLsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectLsub,definition,
% 3.21/1.70 ( binintersectLsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset2_type,type,
% 3.21/1.70 binintersectSubset2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset2,definition,
% 3.21/1.70 ( binintersectSubset2
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( binintersect @ A @ B )
% 3.21/1.70 = A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset3_type,type,
% 3.21/1.70 binintersectSubset3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset3,definition,
% 3.21/1.70 ( binintersectSubset3
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( ( binintersect @ A @ B )
% 3.21/1.70 = B )
% 3.21/1.70 => ( subset @ B @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectER_type,type,
% 3.21/1.70 binintersectER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectER,definition,
% 3.21/1.70 ( binintersectER
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.21/1.70 => ( in @ Xx @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(disjointsetsI1_type,type,
% 3.21/1.70 disjointsetsI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(disjointsetsI1,definition,
% 3.21/1.70 ( disjointsetsI1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( in @ Xx @ B ) )
% 3.21/1.70 => ( ( binintersect @ A @ B )
% 3.21/1.70 = emptyset ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectRsub_type,type,
% 3.21/1.70 binintersectRsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectRsub,definition,
% 3.21/1.70 ( binintersectRsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset4_type,type,
% 3.21/1.70 binintersectSubset4: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset4,definition,
% 3.21/1.70 ( binintersectSubset4
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ B @ A )
% 3.21/1.70 => ( ( binintersect @ A @ B )
% 3.21/1.70 = B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset1_type,type,
% 3.21/1.70 binintersectSubset1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset1,definition,
% 3.21/1.70 ( binintersectSubset1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( ( binintersect @ A @ B )
% 3.21/1.70 = A )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(bs114d_type,type,
% 3.21/1.70 bs114d: $o ).
% 3.21/1.70
% 3.21/1.70 thf(bs114d,definition,
% 3.21/1.70 ( bs114d
% 3.21/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 3.21/1.70 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.21/1.70 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(regular_type,type,
% 3.21/1.70 regular: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminus_type,type,
% 3.21/1.70 setminus: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(setminusI_type,type,
% 3.21/1.70 setminusI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusI,definition,
% 3.21/1.70 ( setminusI
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusEL_type,type,
% 3.21/1.70 setminusEL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusEL,definition,
% 3.21/1.70 ( setminusEL
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusER_type,type,
% 3.21/1.70 setminusER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusER,definition,
% 3.21/1.70 ( setminusER
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.21/1.70 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusSubset2_type,type,
% 3.21/1.70 setminusSubset2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusSubset2,definition,
% 3.21/1.70 ( setminusSubset2
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( setminus @ A @ B )
% 3.21/1.70 = emptyset ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusERneg_type,type,
% 3.21/1.70 setminusERneg: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusERneg,definition,
% 3.21/1.70 ( setminusERneg
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusELneg_type,type,
% 3.21/1.70 setminusELneg: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusELneg,definition,
% 3.21/1.70 ( setminusELneg
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusILneg_type,type,
% 3.21/1.70 setminusILneg: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusILneg,definition,
% 3.21/1.70 ( setminusILneg
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ A )
% 3.21/1.70 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusIRneg_type,type,
% 3.21/1.70 setminusIRneg: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusIRneg,definition,
% 3.21/1.70 ( setminusIRneg
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusLsub_type,type,
% 3.21/1.70 setminusLsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusLsub,definition,
% 3.21/1.70 ( setminusLsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusSubset1_type,type,
% 3.21/1.70 setminusSubset1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusSubset1,definition,
% 3.21/1.70 ( setminusSubset1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( ( setminus @ A @ B )
% 3.21/1.70 = emptyset )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiff_type,type,
% 3.21/1.70 symdiff: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffE_type,type,
% 3.21/1.70 symdiffE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffE,definition,
% 3.21/1.70 ( symdiffE
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.21/1.70 => ! [Xphi: $o] :
% 3.21/1.70 ( ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => Xphi ) )
% 3.21/1.70 => ( ( ~ ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => Xphi ) )
% 3.21/1.70 => Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffI1_type,type,
% 3.21/1.70 symdiffI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffI1,definition,
% 3.21/1.70 ( symdiffI1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffI2_type,type,
% 3.21/1.70 symdiffI2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffI2,definition,
% 3.21/1.70 ( symdiffI2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffIneg1_type,type,
% 3.21/1.70 symdiffIneg1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffIneg1,definition,
% 3.21/1.70 ( symdiffIneg1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffIneg2_type,type,
% 3.21/1.70 symdiffIneg2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffIneg2,definition,
% 3.21/1.70 ( symdiffIneg2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iskpair_type,type,
% 3.21/1.70 iskpair: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(secondinupair_type,type,
% 3.21/1.70 secondinupair: $o ).
% 3.21/1.70
% 3.21/1.70 thf(secondinupair,definition,
% 3.21/1.70 ( secondinupair
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairIL_type,type,
% 3.21/1.70 setukpairIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairIL,definition,
% 3.21/1.70 ( setukpairIL
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairIR_type,type,
% 3.21/1.70 setukpairIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairIR,definition,
% 3.21/1.70 ( setukpairIR
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kpairiskpair_type,type,
% 3.21/1.70 kpairiskpair: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kpairiskpair,definition,
% 3.21/1.70 ( kpairiskpair
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kpair_type,type,
% 3.21/1.70 kpair: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(kpairp_type,type,
% 3.21/1.70 kpairp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kpairp,definition,
% 3.21/1.70 ( kpairp
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprod_type,type,
% 3.21/1.70 cartprod: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsubset_type,type,
% 3.21/1.70 singletonsubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsubset,definition,
% 3.21/1.70 ( singletonsubset
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singletoninpowerset_type,type,
% 3.21/1.70 singletoninpowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletoninpowerset,definition,
% 3.21/1.70 ( singletoninpowerset
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singletoninpowunion_type,type,
% 3.21/1.70 singletoninpowunion: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletoninpowunion,definition,
% 3.21/1.70 ( singletoninpowunion
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairset2E_type,type,
% 3.21/1.70 upairset2E: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairset2E,definition,
% 3.21/1.70 ( upairset2E
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.21/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.21/1.70 => ( ( Xz = Xx )
% 3.21/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairsubunion_type,type,
% 3.21/1.70 upairsubunion: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsubunion,definition,
% 3.21/1.70 ( upairsubunion
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairinpowunion_type,type,
% 3.21/1.70 upairinpowunion: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairinpowunion,definition,
% 3.21/1.70 ( upairinpowunion
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem1_type,type,
% 3.21/1.70 ubforcartprodlem1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem1,definition,
% 3.21/1.70 ( ubforcartprodlem1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem2_type,type,
% 3.21/1.70 ubforcartprodlem2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem2,definition,
% 3.21/1.70 ( ubforcartprodlem2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem3_type,type,
% 3.21/1.70 ubforcartprodlem3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem3,definition,
% 3.21/1.70 ( ubforcartprodlem3
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairin_type,type,
% 3.21/1.70 cartprodpairin: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairin,definition,
% 3.21/1.70 ( cartprodpairin
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempair1_type,type,
% 3.21/1.70 cartprodmempair1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempair1,definition,
% 3.21/1.70 ( cartprodmempair1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 & ( Xu
% 3.21/1.70 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempair_type,type,
% 3.21/1.70 cartprodmempair: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempair,definition,
% 3.21/1.70 ( cartprodmempair
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( iskpair @ Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionE2_type,type,
% 3.21/1.70 setunionE2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionE2,definition,
% 3.21/1.70 ( setunionE2
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 3.21/1.70 => ? [X: $i] :
% 3.21/1.70 ( ( in @ X @ A )
% 3.21/1.70 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton1_type,type,
% 3.21/1.70 setunionsingleton1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton1,definition,
% 3.21/1.70 ( setunionsingleton1
% 3.21/1.70 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton2_type,type,
% 3.21/1.70 setunionsingleton2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton2,definition,
% 3.21/1.70 ( setunionsingleton2
% 3.21/1.70 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton_type,type,
% 3.21/1.70 setunionsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton,definition,
% 3.21/1.70 ( setunionsingleton
% 3.21/1.70 = ( ! [Xx: $i] :
% 3.21/1.70 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.21/1.70 = Xx ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singleton_type,type,
% 3.21/1.70 singleton: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonprop_type,type,
% 3.21/1.70 singletonprop: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonprop,definition,
% 3.21/1.70 ( singletonprop
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) )
% 3.21/1.70 => ( ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ex1_type,type,
% 3.21/1.70 ex1: $i > ( $i > $o ) > $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1E1_type,type,
% 3.21/1.70 ex1E1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1E1,definition,
% 3.21/1.70 ( ex1E1
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ( ex1 @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ex1I_type,type,
% 3.21/1.70 ex1I: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1I,definition,
% 3.21/1.70 ( ex1I
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xy = Xx ) ) )
% 3.21/1.70 => ( ex1 @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ex1I2_type,type,
% 3.21/1.70 ex1I2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1I2,definition,
% 3.21/1.70 ( ex1I2
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) )
% 3.21/1.70 => ( ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ( ex1 @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsuniq_type,type,
% 3.21/1.70 singletonsuniq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsuniq,definition,
% 3.21/1.70 ( singletonsuniq
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( ( setadjoin @ Xx @ emptyset )
% 3.21/1.70 = ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(atmost1p_type,type,
% 3.21/1.70 atmost1p: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(atleast2p_type,type,
% 3.21/1.70 atleast2p: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(atmost2p_type,type,
% 3.21/1.70 atmost2p: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetp_type,type,
% 3.21/1.70 upairsetp: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL1_type,type,
% 3.21/1.70 setukpairinjL1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL1,definition,
% 3.21/1.70 ( setukpairinjL1
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.21/1.70 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.21/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kfstsingleton_type,type,
% 3.21/1.70 kfstsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kfstsingleton,definition,
% 3.21/1.70 ( kfstsingleton
% 3.21/1.70 = ( ! [Xu: $i] :
% 3.21/1.70 ( ( iskpair @ Xu )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ ( setunion @ Xu )
% 3.21/1.70 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(theprop_type,type,
% 3.21/1.70 theprop: $o ).
% 3.21/1.70
% 3.21/1.70 thf(theprop,definition,
% 3.21/1.70 ( theprop
% 3.21/1.70 = ( ! [X: $i] :
% 3.21/1.70 ( ( singleton @ X )
% 3.21/1.70 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kfst_type,type,
% 3.21/1.70 kfst: $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(kfstpairEq_type,type,
% 3.21/1.70 kfstpairEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kfstpairEq,definition,
% 3.21/1.70 ( kfstpairEq
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.21/1.70 = Xx ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodfstin_type,type,
% 3.21/1.70 cartprodfstin: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodfstin,definition,
% 3.21/1.70 ( cartprodfstin
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL2_type,type,
% 3.21/1.70 setukpairinjL2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL2,definition,
% 3.21/1.70 ( setukpairinjL2
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.21/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL_type,type,
% 3.21/1.70 setukpairinjL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL,definition,
% 3.21/1.70 ( setukpairinjL
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( kpair @ Xx @ Xy )
% 3.21/1.70 = ( kpair @ Xz @ Xu ) )
% 3.21/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR11_type,type,
% 3.21/1.70 setukpairinjR11: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR11,definition,
% 3.21/1.70 ( setukpairinjR11
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR12_type,type,
% 3.21/1.70 setukpairinjR12: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR12,definition,
% 3.21/1.70 ( setukpairinjR12
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR1_type,type,
% 3.21/1.70 setukpairinjR1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR1,definition,
% 3.21/1.70 ( setukpairinjR1
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.21/1.70 => ( ( Xz = Xu )
% 3.21/1.70 => ( Xy = Xu ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairequniteq_type,type,
% 3.21/1.70 upairequniteq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairequniteq,definition,
% 3.21/1.70 ( upairequniteq
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.21/1.70 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ Xz @ emptyset ) )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR2_type,type,
% 3.21/1.70 setukpairinjR2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR2,definition,
% 3.21/1.70 ( setukpairinjR2
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.21/1.70 => ( Xy = Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR_type,type,
% 3.21/1.70 setukpairinjR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR,definition,
% 3.21/1.70 ( setukpairinjR
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( kpair @ Xx @ Xy )
% 3.21/1.70 = ( kpair @ Xz @ Xu ) )
% 3.21/1.70 => ( Xy = Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ksndsingleton_type,type,
% 3.21/1.70 ksndsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ksndsingleton,definition,
% 3.21/1.70 ( ksndsingleton
% 3.21/1.70 = ( ! [Xu: $i] :
% 3.21/1.70 ( ( iskpair @ Xu )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ ( setunion @ Xu )
% 3.21/1.70 @ ^ [Xx: $i] :
% 3.21/1.70 ( Xu
% 3.21/1.70 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ksnd_type,type,
% 3.21/1.70 ksnd: $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(ksndpairEq_type,type,
% 3.21/1.70 ksndpairEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ksndpairEq,definition,
% 3.21/1.70 ( ksndpairEq
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.21/1.70 = Xy ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kpairsurjEq_type,type,
% 3.21/1.70 kpairsurjEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kpairsurjEq,definition,
% 3.21/1.70 ( kpairsurjEq
% 3.21/1.70 = ( ! [Xu: $i] :
% 3.21/1.70 ( ( iskpair @ Xu )
% 3.21/1.70 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.21/1.70 = Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodsndin_type,type,
% 3.21/1.70 cartprodsndin: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodsndin,definition,
% 3.21/1.70 ( cartprodsndin
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairmemEL_type,type,
% 3.21/1.70 cartprodpairmemEL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairmemEL,definition,
% 3.21/1.70 ( cartprodpairmemEL
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairmemER_type,type,
% 3.21/1.70 cartprodpairmemER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairmemER,definition,
% 3.21/1.70 ( cartprodpairmemER
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( in @ Xy @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempaircEq_type,type,
% 3.21/1.70 cartprodmempaircEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempaircEq,definition,
% 3.21/1.70 ( cartprodmempaircEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( kpair @ Xx @ Xy )
% 3.21/1.70 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodfstpairEq_type,type,
% 3.21/1.70 cartprodfstpairEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodfstpairEq,definition,
% 3.21/1.70 ( cartprodfstpairEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.21/1.70 = Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodsndpairEq_type,type,
% 3.21/1.70 cartprodsndpairEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodsndpairEq,definition,
% 3.21/1.70 ( cartprodsndpairEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.21/1.70 = Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairsurjEq_type,type,
% 3.21/1.70 cartprodpairsurjEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairsurjEq,definition,
% 3.21/1.70 ( cartprodpairsurjEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.21/1.70 = Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln_type,type,
% 3.21/1.70 breln: $i > $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstr_type,type,
% 3.21/1.70 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrI_type,type,
% 3.21/1.70 dpsetconstrI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrI,definition,
% 3.21/1.70 ( dpsetconstrI
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( Xphi @ Xx @ Xy )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrSub_type,type,
% 3.21/1.70 dpsetconstrSub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrSub,definition,
% 3.21/1.70 ( dpsetconstrSub
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.21/1.70 ( subset
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.21/1.70 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setOfPairsIsBReln_type,type,
% 3.21/1.70 setOfPairsIsBReln: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setOfPairsIsBReln,definition,
% 3.21/1.70 ( setOfPairsIsBReln
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.21/1.70 ( breln @ A @ B
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrERa_type,type,
% 3.21/1.70 dpsetconstrERa: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrERa,definition,
% 3.21/1.70 ( dpsetconstrERa
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.21/1.70 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrEL1_type,type,
% 3.21/1.70 dpsetconstrEL1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrEL1,definition,
% 3.21/1.70 ( dpsetconstrEL1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrEL2_type,type,
% 3.21/1.70 dpsetconstrEL2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrEL2,definition,
% 3.21/1.70 ( dpsetconstrEL2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.21/1.70 => ( in @ Xy @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrER_type,type,
% 3.21/1.70 dpsetconstrER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrER,definition,
% 3.21/1.70 ( dpsetconstrER
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.21/1.70 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(func_type,type,
% 3.21/1.70 func: $i > $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcSet_type,type,
% 3.21/1.70 funcSet: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(funcImageSingleton_type,type,
% 3.21/1.70 funcImageSingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcImageSingleton,definition,
% 3.21/1.70 ( funcImageSingleton
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ B
% 3.21/1.70 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(apProp_type,type,
% 3.21/1.70 apProp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(apProp,definition,
% 3.21/1.70 ( apProp
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in
% 3.21/1.70 @ ( setunion
% 3.21/1.70 @ ( dsetconstr @ B
% 3.21/1.70 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.21/1.70 @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ap_type,type,
% 3.21/1.70 ap: $i > $i > $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(app_type,type,
% 3.21/1.70 app: $o ).
% 3.21/1.70
% 3.21/1.70 thf(app,definition,
% 3.21/1.70 ( app
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(infuncsetfunc_type,type,
% 3.21/1.70 infuncsetfunc: $o ).
% 3.21/1.70
% 3.21/1.70 thf(infuncsetfunc,definition,
% 3.21/1.70 ( infuncsetfunc
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ap2p_type,type,
% 3.21/1.70 ap2p: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ap2p,definition,
% 3.21/1.70 ( ap2p
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcinfuncset_type,type,
% 3.21/1.70 funcinfuncset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcinfuncset,definition,
% 3.21/1.70 ( funcinfuncset
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(lamProp_type,type,
% 3.21/1.70 lamProp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(lamProp,definition,
% 3.21/1.70 ( lamProp
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ( func @ A @ B
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xf @ Xx )
% 3.21/1.70 = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(lam_type,type,
% 3.21/1.70 lam: $i > $i > ( $i > $i ) > $i ).
% 3.21/1.70
% 3.21/1.70 thf(lamp_type,type,
% 3.21/1.70 lamp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(lamp,definition,
% 3.21/1.70 ( lamp
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ( func @ A @ B
% 3.21/1.70 @ ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(lam2p_type,type,
% 3.21/1.70 lam2p: $o ).
% 3.21/1.70
% 3.21/1.70 thf(lam2p,definition,
% 3.21/1.70 ( lam2p
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ( in
% 3.21/1.70 @ ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.21/1.70 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(brelnall1_type,type,
% 3.21/1.70 brelnall1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(brelnall1,definition,
% 3.21/1.70 ( brelnall1
% 3.21/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ R )
% 3.21/1.70 => ! [Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ R )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(brelnall2_type,type,
% 3.21/1.70 brelnall2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(brelnall2,definition,
% 3.21/1.70 ( brelnall2
% 3.21/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ R )
% 3.21/1.70 => ! [Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ R )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ex1E2_type,type,
% 3.21/1.70 ex1E2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1E2,definition,
% 3.21/1.70 ( ex1E2
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ( ex1 @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp1_type,type,
% 3.21/1.70 funcGraphProp1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp1,definition,
% 3.21/1.70 ( funcGraphProp1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp3_type,type,
% 3.21/1.70 funcGraphProp3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp3,definition,
% 3.21/1.70 ( funcGraphProp3
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp2_type,type,
% 3.21/1.70 funcGraphProp2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp2,definition,
% 3.21/1.70 ( funcGraphProp2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcextLem_type,type,
% 3.21/1.70 funcextLem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcextLem,definition,
% 3.21/1.70 ( funcextLem
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xg: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xg )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp4_type,type,
% 3.21/1.70 funcGraphProp4: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp4,definition,
% 3.21/1.70 ( funcGraphProp4
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subbreln_type,type,
% 3.21/1.70 subbreln: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subbreln,definition,
% 3.21/1.70 ( subbreln
% 3.21/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ S )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.21/1.70 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqbreln_type,type,
% 3.21/1.70 eqbreln: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqbreln,definition,
% 3.21/1.70 ( eqbreln
% 3.21/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ S )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.21/1.70 => ( R = S ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcext_type,type,
% 3.21/1.70 funcext: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcext,definition,
% 3.21/1.70 ( funcext
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xg: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xg )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.21/1.70 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcext2_type,type,
% 3.21/1.70 funcext2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcext2,definition,
% 3.21/1.70 ( funcext2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xg: $i] :
% 3.21/1.70 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.21/1.70 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ap2apEq1_type,type,
% 3.21/1.70 ap2apEq1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ap2apEq1,definition,
% 3.21/1.70 ( ap2apEq1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ap2apEq2_type,type,
% 3.21/1.70 ap2apEq2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ap2apEq2,definition,
% 3.21/1.70 ( ap2apEq2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(beta1_type,type,
% 3.21/1.70 beta1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(beta1,definition,
% 3.21/1.70 ( beta1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B
% 3.21/1.70 @ ( lam @ A @ B
% 3.21/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.21/1.70 @ Xx )
% 3.21/1.70 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eta1_type,type,
% 3.21/1.70 eta1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eta1,definition,
% 3.21/1.70 ( eta1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ( ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.21/1.70 = Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(lam2lamEq_type,type,
% 3.21/1.70 lam2lamEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(lam2lamEq,definition,
% 3.21/1.70 ( lam2lamEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ( ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.21/1.70 = ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(beta2_type,type,
% 3.21/1.70 beta2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(beta2,definition,
% 3.21/1.70 ( beta2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B
% 3.21/1.70 @ ( lam @ A @ B
% 3.21/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.21/1.70 @ Xx )
% 3.21/1.70 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eta2_type,type,
% 3.21/1.70 eta2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eta2,definition,
% 3.21/1.70 ( eta2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ( ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.21/1.70 = Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iffalseProp1_type,type,
% 3.21/1.70 iffalseProp1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iffalseProp1,definition,
% 3.21/1.70 ( iffalseProp1
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ~ Xphi
% 3.21/1.70 => ( in @ Xy
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iffalseProp2_type,type,
% 3.21/1.70 iffalseProp2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iffalseProp2,definition,
% 3.21/1.70 ( iffalseProp2
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ~ Xphi
% 3.21/1.70 => ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) )
% 3.21/1.70 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueProp1_type,type,
% 3.21/1.70 iftrueProp1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueProp1,definition,
% 3.21/1.70 ( iftrueProp1
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( Xphi
% 3.21/1.70 => ( in @ Xx
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueProp2_type,type,
% 3.21/1.70 iftrueProp2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueProp2,definition,
% 3.21/1.70 ( iftrueProp2
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( Xphi
% 3.21/1.70 => ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) )
% 3.21/1.70 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ifSingleton_type,type,
% 3.21/1.70 ifSingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ifSingleton,definition,
% 3.21/1.70 ( ifSingleton
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(if_type,type,
% 3.21/1.70 if: $i > $o > $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(ifp_type,type,
% 3.21/1.70 ifp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ifp,definition,
% 3.21/1.70 ( ifp
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(theeq_type,type,
% 3.21/1.70 theeq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(theeq,definition,
% 3.21/1.70 ( theeq
% 3.21/1.70 = ( ! [X: $i] :
% 3.21/1.70 ( ( singleton @ X )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ X )
% 3.21/1.70 => ( ( setunion @ X )
% 3.21/1.70 = Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iftrue_type,type,
% 3.21/1.70 iftrue: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iftrue,definition,
% 3.21/1.70 ( iftrue
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( Xphi
% 3.21/1.70 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.21/1.70 = Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iffalse_type,type,
% 3.21/1.70 iffalse: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iffalse,definition,
% 3.21/1.70 ( iffalse
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ~ Xphi
% 3.21/1.70 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.21/1.70 = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueorfalse_type,type,
% 3.21/1.70 iftrueorfalse: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueorfalse,definition,
% 3.21/1.70 ( iftrueorfalse
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectT_lem_type,type,
% 3.21/1.70 binintersectT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectT_lem,definition,
% 3.21/1.70 ( binintersectT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionT_lem_type,type,
% 3.21/1.70 binunionT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionT_lem,definition,
% 3.21/1.70 ( binunionT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetT_lem_type,type,
% 3.21/1.70 powersetT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetT_lem,definition,
% 3.21/1.70 ( powersetT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusT_lem_type,type,
% 3.21/1.70 setminusT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusT_lem,definition,
% 3.21/1.70 ( setminusT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementT_lem_type,type,
% 3.21/1.70 complementT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementT_lem,definition,
% 3.21/1.70 ( complementT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setextT_type,type,
% 3.21/1.70 setextT: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setextT,definition,
% 3.21/1.70 ( setextT
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ Y ) ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ Y )
% 3.21/1.70 => ( in @ Xx @ X ) ) )
% 3.21/1.70 => ( X = Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetTI_type,type,
% 3.21/1.70 subsetTI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetTI,definition,
% 3.21/1.70 ( subsetTI
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ Y ) ) )
% 3.21/1.70 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetTI1_type,type,
% 3.21/1.70 powersetTI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetTI1,definition,
% 3.21/1.70 ( powersetTI1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ Y ) ) )
% 3.21/1.70 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetTE1_type,type,
% 3.21/1.70 powersetTE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetTE1,definition,
% 3.21/1.70 ( powersetTE1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ X @ ( powerset @ Y ) )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementTI1_type,type,
% 3.21/1.70 complementTI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementTI1,definition,
% 3.21/1.70 ( complementTI1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementTE1_type,type,
% 3.21/1.70 complementTE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementTE1,definition,
% 3.21/1.70 ( complementTE1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.21/1.70 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectTELcontra_type,type,
% 3.21/1.70 binintersectTELcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectTELcontra,definition,
% 3.21/1.70 ( binintersectTELcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ X )
% 3.21/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectTERcontra_type,type,
% 3.21/1.70 binintersectTERcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectTERcontra,definition,
% 3.21/1.70 ( binintersectTERcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ Y )
% 3.21/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT_type,type,
% 3.21/1.70 contrasubsetT: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT,definition,
% 3.21/1.70 ( contrasubsetT
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.21/1.70 => ( ( in @ Xx @ Y )
% 3.21/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT1_type,type,
% 3.21/1.70 contrasubsetT1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT1,definition,
% 3.21/1.70 ( contrasubsetT1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( subset @ X @ Y )
% 3.21/1.70 => ( ~ ( in @ Xx @ Y )
% 3.21/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT2_type,type,
% 3.21/1.70 contrasubsetT2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT2,definition,
% 3.21/1.70 ( contrasubsetT2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ Y )
% 3.21/1.70 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT3_type,type,
% 3.21/1.70 contrasubsetT3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT3,definition,
% 3.21/1.70 ( contrasubsetT3
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.21/1.70 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementI1_type,type,
% 3.21/1.70 doubleComplementI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementI1,definition,
% 3.21/1.70 ( doubleComplementI1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementE1_type,type,
% 3.21/1.70 doubleComplementE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementE1,definition,
% 3.21/1.70 ( doubleComplementE1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.21/1.70 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementSub1_type,type,
% 3.21/1.70 doubleComplementSub1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementSub1,definition,
% 3.21/1.70 ( doubleComplementSub1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementSub2_type,type,
% 3.21/1.70 doubleComplementSub2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementSub2,definition,
% 3.21/1.70 ( doubleComplementSub2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementEq_type,type,
% 3.21/1.70 doubleComplementEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementEq,definition,
% 3.21/1.70 ( doubleComplementEq
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( X
% 3.21/1.70 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementTnotintersectT_type,type,
% 3.21/1.70 complementTnotintersectT: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementTnotintersectT,definition,
% 3.21/1.70 ( complementTnotintersectT
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.21/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementImpComplementIntersect_type,type,
% 3.21/1.70 complementImpComplementIntersect: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementImpComplementIntersect,definition,
% 3.21/1.70 ( complementImpComplementIntersect
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementSubsetComplementIntersect_type,type,
% 3.21/1.70 complementSubsetComplementIntersect: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementSubsetComplementIntersect,definition,
% 3.21/1.70 ( complementSubsetComplementIntersect
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementInPowersetComplementIntersect_type,type,
% 3.21/1.70 complementInPowersetComplementIntersect: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementInPowersetComplementIntersect,definition,
% 3.21/1.70 ( complementInPowersetComplementIntersect
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contraSubsetComplement_type,type,
% 3.21/1.70 contraSubsetComplement: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contraSubsetComplement,definition,
% 3.21/1.70 ( contraSubsetComplement
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ Y )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementTcontraSubset_type,type,
% 3.21/1.70 complementTcontraSubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementTcontraSubset,definition,
% 3.21/1.70 ( complementTcontraSubset
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.21/1.70 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTILcontra_type,type,
% 3.21/1.70 binunionTILcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTILcontra,definition,
% 3.21/1.70 ( binunionTILcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.21/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTIRcontra_type,type,
% 3.21/1.70 binunionTIRcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTIRcontra,definition,
% 3.21/1.70 ( binunionTIRcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.21/1.70 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inIntersectImpInUnion_type,type,
% 3.21/1.70 inIntersectImpInUnion: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inIntersectImpInUnion,definition,
% 3.21/1.70 ( inIntersectImpInUnion
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.21/1.70 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inIntersectImpInUnion2_type,type,
% 3.21/1.70 inIntersectImpInUnion2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inIntersectImpInUnion2,definition,
% 3.21/1.70 ( inIntersectImpInUnion2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.21/1.70 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inIntersectImpInIntersectUnions_type,type,
% 3.21/1.70 inIntersectImpInIntersectUnions: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inIntersectImpInIntersectUnions,definition,
% 3.21/1.70 ( inIntersectImpInIntersectUnions
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.21/1.70 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(intersectInPowersetIntersectUnions_type,type,
% 3.21/1.70 intersectInPowersetIntersectUnions: $o ).
% 3.21/1.70
% 3.21/1.70 thf(intersectInPowersetIntersectUnions,definition,
% 3.21/1.70 ( intersectInPowersetIntersectUnions
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inComplementUnionImpNotIn1_type,type,
% 3.21/1.70 inComplementUnionImpNotIn1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inComplementUnionImpNotIn1,definition,
% 3.21/1.70 ( inComplementUnionImpNotIn1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inComplementUnionImpInComplement1_type,type,
% 3.21/1.70 inComplementUnionImpInComplement1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inComplementUnionImpInComplement1,definition,
% 3.21/1.70 ( inComplementUnionImpInComplement1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTE_type,type,
% 3.21/1.70 binunionTE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTE,definition,
% 3.21/1.70 ( binunionTE
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.21/1.70 => ( ( ( in @ Xx @ X )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => ( ( ( in @ Xx @ Y )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => Xphi ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTEcontra_type,type,
% 3.21/1.70 binunionTEcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTEcontra,definition,
% 3.21/1.70 ( binunionTEcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ X )
% 3.21/1.70 => ( ~ ( in @ Xx @ Y )
% 3.21/1.70 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2a1_type,type,
% 3.21/1.70 demorgan2a1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2a1,definition,
% 3.21/1.70 ( demorgan2a1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementUnionInPowersetComplement_type,type,
% 3.21/1.70 complementUnionInPowersetComplement: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementUnionInPowersetComplement,definition,
% 3.21/1.70 ( complementUnionInPowersetComplement
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2a2_type,type,
% 3.21/1.70 demorgan2a2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2a2,definition,
% 3.21/1.70 ( demorgan2a2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan1a_type,type,
% 3.21/1.70 demorgan1a: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan1a,definition,
% 3.21/1.70 ( demorgan1a
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.21/1.70 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan1b_type,type,
% 3.21/1.70 demorgan1b: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan1b,definition,
% 3.21/1.70 ( demorgan1b
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan1_type,type,
% 3.21/1.70 demorgan1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan1,definition,
% 3.21/1.70 ( demorgan1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.21/1.70 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2a_type,type,
% 3.21/1.70 demorgan2a: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2a,definition,
% 3.21/1.70 ( demorgan2a
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.70 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2b2_type,type,
% 3.21/1.70 demorgan2b2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2b2,definition,
% 3.21/1.70 ( demorgan2b2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2b_type,type,
% 3.21/1.70 demorgan2b: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2b,definition,
% 3.21/1.70 ( demorgan2b
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2_type,type,
% 3.21/1.70 demorgan2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(demorgan2,definition,
% 3.21/1.70 ( demorgan2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.21/1.70 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule0_type,type,
% 3.21/1.70 woz13rule0: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule0,definition,
% 3.21/1.70 ( woz13rule0
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule1_type,type,
% 3.21/1.70 woz13rule1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule1,definition,
% 3.21/1.70 ( woz13rule1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ Z )
% 3.21/1.70 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule2_type,type,
% 3.21/1.70 woz13rule2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule2,definition,
% 3.21/1.70 ( woz13rule2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ Y @ Z )
% 3.21/1.70 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule3_type,type,
% 3.21/1.70 woz13rule3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule3,definition,
% 3.21/1.70 ( woz13rule3
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ Y )
% 3.21/1.70 => ( ( subset @ X @ Z )
% 3.21/1.70 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule4_type,type,
% 3.21/1.70 woz13rule4: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz13rule4,definition,
% 3.21/1.70 ( woz13rule4
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ! [W: $i] :
% 3.21/1.70 ( ( in @ W @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ Z )
% 3.21/1.70 => ( ( subset @ Y @ W )
% 3.21/1.70 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_1_type,type,
% 3.21/1.70 woz1_1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_1,definition,
% 3.21/1.70 ( woz1_1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_2_type,type,
% 3.21/1.70 woz1_2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_2,definition,
% 3.21/1.70 ( woz1_2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ! [W: $i] :
% 3.21/1.70 ( ( in @ W @ ( powerset @ A ) )
% 3.21/1.70 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.21/1.70 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_3_type,type,
% 3.21/1.70 woz1_3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_3,definition,
% 3.21/1.70 ( woz1_3
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_4_type,type,
% 3.21/1.70 woz1_4: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_4,definition,
% 3.21/1.70 ( woz1_4
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.21/1.70 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_5_type,type,
% 3.21/1.70 woz1_5: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz1_5,definition,
% 3.21/1.70 ( woz1_5
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1_type,type,
% 3.21/1.70 breln1: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1all2_type,type,
% 3.21/1.70 breln1all2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1all2,definition,
% 3.21/1.70 ( breln1all2
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ R )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1Set_type,type,
% 3.21/1.70 breln1Set: $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(breln1SetBreln1_type,type,
% 3.21/1.70 breln1SetBreln1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1SetBreln1,definition,
% 3.21/1.70 ( breln1SetBreln1
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( in @ R @ ( breln1Set @ A ) )
% 3.21/1.70 => ( breln1 @ A @ R ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(transitive_type,type,
% 3.21/1.70 transitive: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(antisymmetric_type,type,
% 3.21/1.70 antisymmetric: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(reflexive_type,type,
% 3.21/1.70 reflexive: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(refltransitive_type,type,
% 3.21/1.70 refltransitive: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(refllinearorder_type,type,
% 3.21/1.70 refllinearorder: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(reflwellordering_type,type,
% 3.21/1.70 reflwellordering: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(choice2fnsingleton_type,type,
% 3.21/1.70 choice2fnsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(choice2fnsingleton,definition,
% 3.21/1.70 ( choice2fnsingleton
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 & ( Xphi @ Xx @ Xy ) ) )
% 3.21/1.70 => ! [R: $i] :
% 3.21/1.70 ( ( in @ R @ ( breln1Set @ B ) )
% 3.21/1.70 => ( ( reflwellordering @ B @ R )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ B
% 3.21/1.70 @ ^ [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xx @ Xy )
% 3.21/1.70 & ! [Xz: $i] :
% 3.21/1.70 ( ( in @ Xz @ B )
% 3.21/1.70 => ( ( Xphi @ Xx @ Xz )
% 3.21/1.70 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setOfPairsIsBReln1_type,type,
% 3.21/1.70 setOfPairsIsBReln1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setOfPairsIsBReln1,definition,
% 3.21/1.70 ( setOfPairsIsBReln1
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.21/1.70 ( breln1 @ A
% 3.21/1.70 @ ( dpsetconstr @ A @ A
% 3.21/1.70 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1all1_type,type,
% 3.21/1.70 breln1all1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1all1,definition,
% 3.21/1.70 ( breln1all1
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ R )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subbreln1_type,type,
% 3.21/1.70 subbreln1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subbreln1,definition,
% 3.21/1.70 ( subbreln1
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.21/1.70 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqbreln1_type,type,
% 3.21/1.70 eqbreln1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqbreln1,definition,
% 3.21/1.70 ( eqbreln1
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.21/1.70 => ( R = S ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1invset_type,type,
% 3.21/1.70 breln1invset: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(breln1invprop_type,type,
% 3.21/1.70 breln1invprop: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1invprop,definition,
% 3.21/1.70 ( breln1invprop
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1invI_type,type,
% 3.21/1.70 breln1invI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1invI,definition,
% 3.21/1.70 ( breln1invI
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1invE_type,type,
% 3.21/1.70 breln1invE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1invE,definition,
% 3.21/1.70 ( breln1invE
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compset_type,type,
% 3.21/1.70 breln1compset: $i > $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compprop_type,type,
% 3.21/1.70 breln1compprop: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compprop,definition,
% 3.21/1.70 ( breln1compprop
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compI_type,type,
% 3.21/1.70 breln1compI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compI,definition,
% 3.21/1.70 ( breln1compI
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ! [Xz: $i] :
% 3.21/1.70 ( ( in @ Xz @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.21/1.70 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compE_type,type,
% 3.21/1.70 breln1compE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compE,definition,
% 3.21/1.70 ( breln1compE
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.21/1.70 => ? [Xz: $i] :
% 3.21/1.70 ( ( in @ Xz @ A )
% 3.21/1.70 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.21/1.70 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compEex_type,type,
% 3.21/1.70 breln1compEex: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1compEex,definition,
% 3.21/1.70 ( breln1compEex
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.21/1.70 => ! [Xphi: $o] :
% 3.21/1.70 ( ! [Xz: $i] :
% 3.21/1.70 ( ( in @ Xz @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.21/1.70 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.21/1.70 => Xphi ) ) )
% 3.21/1.70 => Xphi ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionprop_type,type,
% 3.21/1.70 breln1unionprop: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionprop,definition,
% 3.21/1.70 ( breln1unionprop
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionIL_type,type,
% 3.21/1.70 breln1unionIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionIL,definition,
% 3.21/1.70 ( breln1unionIL
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionIR_type,type,
% 3.21/1.70 breln1unionIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionIR,definition,
% 3.21/1.70 ( breln1unionIR
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionI_type,type,
% 3.21/1.70 breln1unionI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionI,definition,
% 3.21/1.70 ( breln1unionI
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionE_type,type,
% 3.21/1.70 breln1unionE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionE,definition,
% 3.21/1.70 ( breln1unionE
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionEcases_type,type,
% 3.21/1.70 breln1unionEcases: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionEcases,definition,
% 3.21/1.70 ( breln1unionEcases
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.21/1.70 => ! [Xphi: $o] :
% 3.21/1.70 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionCommutes_type,type,
% 3.21/1.70 breln1unionCommutes: $o ).
% 3.21/1.70
% 3.21/1.70 thf(breln1unionCommutes,definition,
% 3.21/1.70 ( breln1unionCommutes
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ( ( binunion @ R @ S )
% 3.21/1.70 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz2Ex_type,type,
% 3.21/1.70 woz2Ex: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz2Ex,definition,
% 3.21/1.70 ( woz2Ex
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ( R
% 3.21/1.70 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz2W_type,type,
% 3.21/1.70 woz2W: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz2W,definition,
% 3.21/1.70 ( woz2W
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.21/1.70 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz2A_type,type,
% 3.21/1.70 woz2A: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz2A,definition,
% 3.21/1.70 ( woz2A
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [T: $i] :
% 3.21/1.70 ( ( breln1 @ A @ T )
% 3.21/1.70 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.21/1.70 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(woz2B_type,type,
% 3.21/1.70 woz2B: $o ).
% 3.21/1.70
% 3.21/1.70 thf(woz2B,definition,
% 3.21/1.70 ( woz2B
% 3.21/1.70 = ( ! [A: $i,R: $i] :
% 3.21/1.70 ( ( breln1 @ A @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln1 @ A @ S )
% 3.21/1.70 => ! [T: $i] :
% 3.21/1.70 ( ( breln1 @ A @ T )
% 3.21/1.70 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.21/1.70 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(image1Ex_type,type,
% 3.21/1.70 image1Ex: $o ).
% 3.21/1.70
% 3.21/1.70 thf(image1Ex,definition,
% 3.21/1.70 ( image1Ex
% 3.21/1.70 = ( ! [A: $i,Xf: $i > $i] :
% 3.21/1.70 ? [B: $i] :
% 3.21/1.70 ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 <=> ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 & ( Xx
% 3.21/1.70 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(image1Ex1_type,type,
% 3.21/1.70 image1Ex1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(image1Ex1,definition,
% 3.21/1.70 ( image1Ex1
% 3.21/1.70 = ( ! [A: $i,Xf: $i > $i] :
% 3.21/1.70 ( exu
% 3.21/1.70 @ ^ [B: $i] :
% 3.21/1.70 ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 <=> ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 & ( Xx
% 3.21/1.70 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(image1_type,type,
% 3.21/1.70 image1: $i > ( $i > $i ) > $i ).
% 3.21/1.70
% 3.21/1.70 thf(image1Equiv_type,type,
% 3.21/1.70 image1Equiv: $o ).
% 3.21/1.70
% 3.21/1.70 thf(image1Equiv,definition,
% 3.21/1.70 ( image1Equiv
% 3.21/1.70 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx
% 3.21/1.70 @ ( image1 @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.21/1.70 <=> ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 & ( Xx
% 3.21/1.70 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(image1E_type,type,
% 3.21/1.70 image1E: $o ).
% 3.21/1.70
% 3.21/1.70 thf(image1E,definition,
% 3.21/1.70 ( image1E
% 3.21/1.70 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx
% 3.21/1.70 @ ( image1 @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.21/1.70 => ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 & ( Xx
% 3.21/1.70 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(image1I_type,type,
% 3.21/1.70 image1I: $o ).
% 3.21/1.70
% 3.21/1.70 thf(image1I,definition,
% 3.21/1.70 ( image1I
% 3.21/1.70 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.21/1.70 ( ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 & ( Xx
% 3.21/1.70 = ( Xf @ Xy ) ) )
% 3.21/1.70 => ( in @ Xx
% 3.21/1.70 @ ( image1 @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(injective_type,type,
% 3.21/1.70 injective: $i > $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(injFuncSet_type,type,
% 3.21/1.70 injFuncSet: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(injFuncInInjFuncSet_type,type,
% 3.21/1.70 injFuncInInjFuncSet: $o ).
% 3.21/1.70
% 3.21/1.70 thf(injFuncInInjFuncSet,definition,
% 3.21/1.70 ( injFuncInInjFuncSet
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ( ( injective @ A @ B @ Xf )
% 3.21/1.70 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(injFuncSetFuncIn_type,type,
% 3.21/1.70 injFuncSetFuncIn: $o ).
% 3.21/1.70
% 3.21/1.70 thf(injFuncSetFuncIn,definition,
% 3.21/1.70 ( injFuncSetFuncIn
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.21/1.70 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(injFuncSetFuncInj_type,type,
% 3.21/1.70 injFuncSetFuncInj: $o ).
% 3.21/1.70
% 3.21/1.70 thf(injFuncSetFuncInj,definition,
% 3.21/1.70 ( injFuncSetFuncInj
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.21/1.70 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(surjective_type,type,
% 3.21/1.70 surjective: $i > $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(surjFuncSet_type,type,
% 3.21/1.70 surjFuncSet: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(surjFuncSetFuncIn_type,type,
% 3.21/1.70 surjFuncSetFuncIn: $o ).
% 3.21/1.70
% 3.21/1.70 thf(surjFuncSetFuncIn,definition,
% 3.21/1.70 ( surjFuncSetFuncIn
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 3.21/1.70 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(surjFuncSetFuncSurj_type,type,
% 3.21/1.70 surjFuncSetFuncSurj: $o ).
% 3.21/1.70
% 3.21/1.70 thf(surjFuncSetFuncSurj,definition,
% 3.21/1.70 ( surjFuncSetFuncSurj
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 3.21/1.70 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(leftInvIsSurj_type,type,
% 3.21/1.70 leftInvIsSurj: $o ).
% 3.21/1.70
% 3.21/1.70 thf(leftInvIsSurj,definition,
% 3.21/1.70 ( leftInvIsSurj
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ! [Xg: $i] :
% 3.21/1.70 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 3.21/1.70 = Xx ) )
% 3.21/1.70 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(surjCantorThm_type,type,
% 3.21/1.70 surjCantorThm: $o ).
% 3.21/1.70
% 3.21/1.70 thf(surjCantorThm,definition,
% 3.21/1.70 ( surjCantorThm
% 3.21/1.70 = ( ! [A: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 3.21/1.70 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(foundation2_type,type,
% 3.21/1.70 foundation2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(foundation2,definition,
% 3.21/1.70 ( foundation2
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( nonempty @ A )
% 3.21/1.70 => ? [X: $i] :
% 3.21/1.70 ( ( in @ X @ A )
% 3.21/1.70 & ( ( binintersect @ X @ A )
% 3.21/1.70 = emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notinself_type,type,
% 3.21/1.70 notinself: $o ).
% 3.21/1.71
% 3.21/1.71 thf(notinself,definition,
% 3.21/1.71 ( notinself
% 3.21/1.71 = ( ! [A: $i] :
% 3.21/1.71 ~ ( in @ A @ A ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(notinself2_type,type,
% 3.21/1.71 notinself2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(notinself2,definition,
% 3.21/1.71 ( notinself2
% 3.21/1.71 = ( ! [A: $i,B: $i] :
% 3.21/1.71 ( ( in @ A @ B )
% 3.21/1.71 => ~ ( in @ B @ A ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(omegaS_type,type,
% 3.21/1.71 omegaS: $i > $i ).
% 3.21/1.71
% 3.21/1.71 thf(omegaSp_type,type,
% 3.21/1.71 omegaSp: $o ).
% 3.21/1.71
% 3.21/1.71 thf(omegaSp,definition,
% 3.21/1.71 ( omegaSp
% 3.21/1.71 = ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ omega )
% 3.21/1.71 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(omegaSclos_type,type,
% 3.21/1.71 omegaSclos: $o ).
% 3.21/1.71
% 3.21/1.71 thf(omegaSclos,definition,
% 3.21/1.71 ( omegaSclos
% 3.21/1.71 = ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ omega )
% 3.21/1.71 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(peano0notS_type,type,
% 3.21/1.71 peano0notS: $o ).
% 3.21/1.71
% 3.21/1.71 thf(peano0notS,definition,
% 3.21/1.71 ( peano0notS
% 3.21/1.71 = ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ omega )
% 3.21/1.71 => ( ( omegaS @ Xx )
% 3.21/1.71 != emptyset ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(peano3_type,type,
% 3.21/1.71 peano3: $i > $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(peano4_type,type,
% 3.21/1.71 peano4: $i > $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(peano5_type,type,
% 3.21/1.71 peano5: $i > $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(peanoSinj_type,type,
% 3.21/1.71 peanoSinj: $o ).
% 3.21/1.71
% 3.21/1.71 thf(peanoSinj,definition,
% 3.21/1.71 ( peanoSinj
% 3.21/1.71 = ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ omega )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ omega )
% 3.21/1.71 => ( ( ( omegaS @ Xx )
% 3.21/1.71 = ( omegaS @ Xy ) )
% 3.21/1.71 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(peano_type,type,
% 3.21/1.71 peano: $i > $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(transitiveset_type,type,
% 3.21/1.71 transitiveset: $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(transitiveset,definition,
% 3.21/1.71 ( transitiveset
% 3.21/1.71 = ( ^ [A: $i] :
% 3.21/1.71 ! [X: $i] :
% 3.21/1.71 ( ( in @ X @ A )
% 3.21/1.71 => ( subset @ X @ A ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(transitivesetOp1_type,type,
% 3.21/1.71 transitivesetOp1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(transitivesetOp1,definition,
% 3.21/1.71 ( transitivesetOp1
% 3.21/1.71 = ( ! [X: $i] :
% 3.21/1.71 ( ( transitiveset @ X )
% 3.21/1.71 => ! [A: $i] :
% 3.21/1.71 ( ( in @ A @ X )
% 3.21/1.71 => ( subset @ A @ X ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(binintTransitive_type,type,
% 3.21/1.71 binintTransitive: $o ).
% 3.21/1.71
% 3.21/1.71 thf(binintTransitive,definition,
% 3.21/1.71 ( binintTransitive
% 3.21/1.71 = ( ! [X: $i] :
% 3.21/1.71 ( ( transitiveset @ X )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( transitiveset @ Y )
% 3.21/1.71 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(transitivesetOp2_type,type,
% 3.21/1.71 transitivesetOp2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(transitivesetOp2,definition,
% 3.21/1.71 ( transitivesetOp2
% 3.21/1.71 = ( ! [X: $i] :
% 3.21/1.71 ( ( transitiveset @ X )
% 3.21/1.71 => ! [A: $i,Xx: $i] :
% 3.21/1.71 ( ( in @ A @ X )
% 3.21/1.71 => ( ( in @ Xx @ A )
% 3.21/1.71 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(setunionTransitive_type,type,
% 3.21/1.71 setunionTransitive: $o ).
% 3.21/1.71
% 3.21/1.71 thf(setunionTransitive,definition,
% 3.21/1.71 ( setunionTransitive
% 3.21/1.71 = ( ! [X: $i] :
% 3.21/1.71 ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ X )
% 3.21/1.71 => ( transitiveset @ Xx ) )
% 3.21/1.71 => ( transitiveset @ ( setunion @ X ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(stricttotalorderedByIn_type,type,
% 3.21/1.71 stricttotalorderedByIn: $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(stricttotalorderedByIn,definition,
% 3.21/1.71 ( stricttotalorderedByIn
% 3.21/1.71 = ( ^ [A: $i] :
% 3.21/1.71 ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [X: $i] :
% 3.21/1.71 ( ( in @ X @ A )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ A )
% 3.21/1.71 => ( ( ( in @ Xx @ X )
% 3.21/1.71 & ( in @ X @ Y ) )
% 3.21/1.71 => ( in @ Xx @ Y ) ) ) ) )
% 3.21/1.71 & ! [X: $i] :
% 3.21/1.71 ( ( in @ X @ A )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ A )
% 3.21/1.71 => ( ( X = Y )
% 3.21/1.71 | ( in @ X @ Y )
% 3.21/1.71 | ( in @ Y @ X ) ) ) )
% 3.21/1.71 & ! [X: $i] :
% 3.21/1.71 ( ( in @ X @ A )
% 3.21/1.71 => ~ ( in @ X @ X ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(wellorderedByIn_type,type,
% 3.21/1.71 wellorderedByIn: $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(wellorderedByIn,definition,
% 3.21/1.71 ( wellorderedByIn
% 3.21/1.71 = ( ^ [A: $i] :
% 3.21/1.71 ( ( stricttotalorderedByIn @ A )
% 3.21/1.71 & ! [X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ( ( nonempty @ X )
% 3.21/1.71 => ? [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ X )
% 3.21/1.71 & ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ X )
% 3.21/1.71 => ( ( Xx = Y )
% 3.21/1.71 | ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(ordinal_type,type,
% 3.21/1.71 ordinal: $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(ordinal,definition,
% 3.21/1.71 ( ordinal
% 3.21/1.71 = ( ^ [Xx: $i] :
% 3.21/1.71 ( ( transitiveset @ Xx )
% 3.21/1.71 & ( wellorderedByIn @ Xx ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(limitOrdinal_type,type,
% 3.21/1.71 limitOrdinal: $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(ordinalMinLem1_type,type,
% 3.21/1.71 ordinalMinLem1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(ordinalMinLem1,definition,
% 3.21/1.71 ( ordinalMinLem1
% 3.21/1.71 = ( ! [X: $i] :
% 3.21/1.71 ( ( ordinal @ X )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( ordinal @ Y )
% 3.21/1.71 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(ordinalTransSet,conjecture,
% 3.21/1.71 ( setextAx
% 3.21/1.71 => ( emptysetAx
% 3.21/1.71 => ( setadjoinAx
% 3.21/1.71 => ( powersetAx
% 3.21/1.71 => ( setunionAx
% 3.21/1.71 => ( omega0Ax
% 3.21/1.71 => ( omegaSAx
% 3.21/1.71 => ( omegaIndAx
% 3.21/1.71 => ( replAx
% 3.21/1.71 => ( foundationAx
% 3.21/1.71 => ( wellorderingAx
% 3.21/1.71 => ( descrp
% 3.21/1.71 => ( dsetconstrI
% 3.21/1.71 => ( dsetconstrEL
% 3.21/1.71 => ( dsetconstrER
% 3.21/1.71 => ( exuE1
% 3.21/1.71 => ( prop2setE
% 3.21/1.71 => ( emptysetE
% 3.21/1.71 => ( emptysetimpfalse
% 3.21/1.71 => ( notinemptyset
% 3.21/1.71 => ( exuE3e
% 3.21/1.71 => ( setext
% 3.21/1.71 => ( emptyI
% 3.21/1.71 => ( noeltsimpempty
% 3.21/1.71 => ( setbeta
% 3.21/1.71 => ( nonemptyE1
% 3.21/1.71 => ( nonemptyI
% 3.21/1.71 => ( nonemptyI1
% 3.21/1.71 => ( setadjoinIL
% 3.21/1.71 => ( emptyinunitempty
% 3.21/1.71 => ( setadjoinIR
% 3.21/1.71 => ( setadjoinE
% 3.21/1.71 => ( setadjoinOr
% 3.21/1.71 => ( setoftrueEq
% 3.21/1.71 => ( powersetI
% 3.21/1.71 => ( emptyinPowerset
% 3.21/1.71 => ( emptyInPowerset
% 3.21/1.71 => ( powersetE
% 3.21/1.71 => ( setunionI
% 3.21/1.71 => ( setunionE
% 3.21/1.71 => ( subPowSU
% 3.21/1.71 => ( exuE2
% 3.21/1.71 => ( nonemptyImpWitness
% 3.21/1.71 => ( uniqinunit
% 3.21/1.71 => ( notinsingleton
% 3.21/1.71 => ( eqinunit
% 3.21/1.71 => ( singletonsswitch
% 3.21/1.71 => ( upairsetE
% 3.21/1.71 => ( upairsetIL
% 3.21/1.71 => ( upairsetIR
% 3.21/1.71 => ( emptyE1
% 3.21/1.71 => ( vacuousDall
% 3.21/1.71 => ( quantDeMorgan1
% 3.21/1.71 => ( quantDeMorgan2
% 3.21/1.71 => ( quantDeMorgan3
% 3.21/1.71 => ( quantDeMorgan4
% 3.21/1.71 => ( prop2setI
% 3.21/1.71 => ( prop2set2propI
% 3.21/1.71 => ( notdexE
% 3.21/1.71 => ( notdallE
% 3.21/1.71 => ( exuI1
% 3.21/1.71 => ( exuI3
% 3.21/1.71 => ( exuI2
% 3.21/1.71 => ( inCongP
% 3.21/1.71 => ( in__Cong
% 3.21/1.71 => ( exuE3u
% 3.21/1.71 => ( exu__Cong
% 3.21/1.71 => ( emptyset__Cong
% 3.21/1.71 => ( setadjoin__Cong
% 3.21/1.71 => ( powerset__Cong
% 3.21/1.71 => ( setunion__Cong
% 3.21/1.71 => ( omega__Cong
% 3.21/1.71 => ( exuEu
% 3.21/1.71 => ( descr__Cong
% 3.21/1.71 => ( dsetconstr__Cong
% 3.21/1.71 => ( subsetI1
% 3.21/1.71 => ( eqimpsubset2
% 3.21/1.71 => ( eqimpsubset1
% 3.21/1.71 => ( subsetI2
% 3.21/1.71 => ( emptysetsubset
% 3.21/1.71 => ( subsetE
% 3.21/1.71 => ( subsetE2
% 3.21/1.71 => ( notsubsetI
% 3.21/1.71 => ( notequalI1
% 3.21/1.71 => ( notequalI2
% 3.21/1.71 => ( subsetRefl
% 3.21/1.71 => ( subsetTrans
% 3.21/1.71 => ( setadjoinSub
% 3.21/1.71 => ( setadjoinSub2
% 3.21/1.71 => ( subset2powerset
% 3.21/1.71 => ( setextsub
% 3.21/1.71 => ( subsetemptysetimpeq
% 3.21/1.71 => ( powersetI1
% 3.21/1.71 => ( powersetE1
% 3.21/1.71 => ( inPowerset
% 3.21/1.71 => ( powersetsubset
% 3.21/1.71 => ( sepInPowerset
% 3.21/1.71 => ( sepSubset
% 3.21/1.71 => ( binunionIL
% 3.21/1.71 => ( upairset2IR
% 3.21/1.71 => ( binunionIR
% 3.21/1.71 => ( binunionEcases
% 3.21/1.71 => ( binunionE
% 3.21/1.71 => ( binunionLsub
% 3.21/1.71 => ( binunionRsub
% 3.21/1.71 => ( binintersectI
% 3.21/1.71 => ( binintersectSubset5
% 3.21/1.71 => ( binintersectEL
% 3.21/1.71 => ( binintersectLsub
% 3.21/1.71 => ( binintersectSubset2
% 3.21/1.71 => ( binintersectSubset3
% 3.21/1.71 => ( binintersectER
% 3.21/1.71 => ( disjointsetsI1
% 3.21/1.71 => ( binintersectRsub
% 3.21/1.71 => ( binintersectSubset4
% 3.21/1.71 => ( binintersectSubset1
% 3.21/1.71 => ( bs114d
% 3.21/1.71 => ( setminusI
% 3.21/1.71 => ( setminusEL
% 3.21/1.71 => ( setminusER
% 3.21/1.71 => ( setminusSubset2
% 3.21/1.71 => ( setminusERneg
% 3.21/1.71 => ( setminusELneg
% 3.21/1.71 => ( setminusILneg
% 3.21/1.71 => ( setminusIRneg
% 3.21/1.71 => ( setminusLsub
% 3.21/1.71 => ( setminusSubset1
% 3.21/1.71 => ( symdiffE
% 3.21/1.71 => ( symdiffI1
% 3.21/1.71 => ( symdiffI2
% 3.21/1.71 => ( symdiffIneg1
% 3.21/1.71 => ( symdiffIneg2
% 3.21/1.71 => ( secondinupair
% 3.21/1.71 => ( setukpairIL
% 3.21/1.71 => ( setukpairIR
% 3.21/1.71 => ( kpairiskpair
% 3.21/1.71 => ( kpairp
% 3.21/1.71 => ( singletonsubset
% 3.21/1.71 => ( singletoninpowerset
% 3.21/1.71 => ( singletoninpowunion
% 3.21/1.71 => ( upairset2E
% 3.21/1.71 => ( upairsubunion
% 3.21/1.71 => ( upairinpowunion
% 3.21/1.71 => ( ubforcartprodlem1
% 3.21/1.71 => ( ubforcartprodlem2
% 3.21/1.71 => ( ubforcartprodlem3
% 3.21/1.71 => ( cartprodpairin
% 3.21/1.71 => ( cartprodmempair1
% 3.21/1.71 => ( cartprodmempair
% 3.21/1.71 => ( setunionE2
% 3.21/1.71 => ( setunionsingleton1
% 3.21/1.71 => ( setunionsingleton2
% 3.21/1.71 => ( setunionsingleton
% 3.21/1.71 => ( singletonprop
% 3.21/1.71 => ( ex1E1
% 3.21/1.71 => ( ex1I
% 3.21/1.71 => ( ex1I2
% 3.21/1.71 => ( singletonsuniq
% 3.21/1.71 => ( setukpairinjL1
% 3.21/1.71 => ( kfstsingleton
% 3.21/1.71 => ( theprop
% 3.21/1.71 => ( kfstpairEq
% 3.21/1.71 => ( cartprodfstin
% 3.21/1.71 => ( setukpairinjL2
% 3.21/1.71 => ( setukpairinjL
% 3.21/1.71 => ( setukpairinjR11
% 3.21/1.71 => ( setukpairinjR12
% 3.21/1.71 => ( setukpairinjR1
% 3.21/1.71 => ( upairequniteq
% 3.21/1.71 => ( setukpairinjR2
% 3.21/1.71 => ( setukpairinjR
% 3.21/1.71 => ( ksndsingleton
% 3.21/1.71 => ( ksndpairEq
% 3.21/1.71 => ( kpairsurjEq
% 3.21/1.71 => ( cartprodsndin
% 3.21/1.71 => ( cartprodpairmemEL
% 3.21/1.71 => ( cartprodpairmemER
% 3.21/1.71 => ( cartprodmempaircEq
% 3.21/1.71 => ( cartprodfstpairEq
% 3.21/1.71 => ( cartprodsndpairEq
% 3.21/1.71 => ( cartprodpairsurjEq
% 3.21/1.71 => ( dpsetconstrI
% 3.21/1.71 => ( dpsetconstrSub
% 3.21/1.71 => ( setOfPairsIsBReln
% 3.21/1.71 => ( dpsetconstrERa
% 3.21/1.71 => ( dpsetconstrEL1
% 3.21/1.71 => ( dpsetconstrEL2
% 3.21/1.71 => ( dpsetconstrER
% 3.21/1.71 => ( funcImageSingleton
% 3.21/1.71 => ( apProp
% 3.21/1.71 => ( app
% 3.21/1.71 => ( infuncsetfunc
% 3.21/1.71 => ( ap2p
% 3.21/1.71 => ( funcinfuncset
% 3.21/1.71 => ( lamProp
% 3.21/1.71 => ( lamp
% 3.21/1.71 => ( lam2p
% 3.21/1.71 => ( brelnall1
% 3.21/1.71 => ( brelnall2
% 3.21/1.71 => ( ex1E2
% 3.21/1.71 => ( funcGraphProp1
% 3.21/1.71 => ( funcGraphProp3
% 3.21/1.71 => ( funcGraphProp2
% 3.21/1.71 => ( funcextLem
% 3.21/1.71 => ( funcGraphProp4
% 3.21/1.71 => ( subbreln
% 3.21/1.71 => ( eqbreln
% 3.21/1.71 => ( funcext
% 3.21/1.71 => ( funcext2
% 3.21/1.71 => ( ap2apEq1
% 3.21/1.71 => ( ap2apEq2
% 3.21/1.71 => ( beta1
% 3.21/1.71 => ( eta1
% 3.21/1.71 => ( lam2lamEq
% 3.21/1.71 => ( beta2
% 3.21/1.71 => ( eta2
% 3.21/1.71 => ( iffalseProp1
% 3.21/1.71 => ( iffalseProp2
% 3.21/1.71 => ( iftrueProp1
% 3.21/1.71 => ( iftrueProp2
% 3.21/1.71 => ( ifSingleton
% 3.21/1.71 => ( ifp
% 3.21/1.71 => ( theeq
% 3.21/1.71 => ( iftrue
% 3.21/1.71 => ( iffalse
% 3.21/1.71 => ( iftrueorfalse
% 3.21/1.71 => ( binintersectT_lem
% 3.21/1.71 => ( binunionT_lem
% 3.21/1.71 => ( powersetT_lem
% 3.21/1.71 => ( setminusT_lem
% 3.21/1.71 => ( complementT_lem
% 3.21/1.71 => ( setextT
% 3.21/1.71 => ( subsetTI
% 3.21/1.71 => ( powersetTI1
% 3.21/1.71 => ( powersetTE1
% 3.21/1.71 => ( complementTI1
% 3.21/1.71 => ( complementTE1
% 3.21/1.71 => ( binintersectTELcontra
% 3.21/1.71 => ( binintersectTERcontra
% 3.21/1.71 => ( contrasubsetT
% 3.21/1.71 => ( contrasubsetT1
% 3.21/1.71 => ( contrasubsetT2
% 3.21/1.71 => ( contrasubsetT3
% 3.21/1.71 => ( doubleComplementI1
% 3.21/1.71 => ( doubleComplementE1
% 3.21/1.71 => ( doubleComplementSub1
% 3.21/1.71 => ( doubleComplementSub2
% 3.21/1.71 => ( doubleComplementEq
% 3.21/1.71 => ( complementTnotintersectT
% 3.21/1.71 => ( complementImpComplementIntersect
% 3.21/1.71 => ( complementSubsetComplementIntersect
% 3.21/1.71 => ( complementInPowersetComplementIntersect
% 3.21/1.71 => ( contraSubsetComplement
% 3.21/1.71 => ( complementTcontraSubset
% 3.21/1.71 => ( binunionTILcontra
% 3.21/1.71 => ( binunionTIRcontra
% 3.21/1.71 => ( inIntersectImpInUnion
% 3.21/1.71 => ( inIntersectImpInUnion2
% 3.21/1.71 => ( inIntersectImpInIntersectUnions
% 3.21/1.71 => ( intersectInPowersetIntersectUnions
% 3.21/1.71 => ( inComplementUnionImpNotIn1
% 3.21/1.71 => ( inComplementUnionImpInComplement1
% 3.21/1.71 => ( binunionTE
% 3.21/1.71 => ( binunionTEcontra
% 3.21/1.71 => ( demorgan2a1
% 3.21/1.71 => ( complementUnionInPowersetComplement
% 3.21/1.71 => ( demorgan2a2
% 3.21/1.71 => ( demorgan1a
% 3.21/1.71 => ( demorgan1b
% 3.21/1.71 => ( demorgan1
% 3.21/1.71 => ( demorgan2a
% 3.21/1.71 => ( demorgan2b2
% 3.21/1.71 => ( demorgan2b
% 3.21/1.71 => ( demorgan2
% 3.21/1.71 => ( woz13rule0
% 3.21/1.71 => ( woz13rule1
% 3.21/1.71 => ( woz13rule2
% 3.21/1.71 => ( woz13rule3
% 3.21/1.71 => ( woz13rule4
% 3.21/1.71 => ( woz1_1
% 3.21/1.71 => ( woz1_2
% 3.21/1.71 => ( woz1_3
% 3.21/1.71 => ( woz1_4
% 3.21/1.71 => ( woz1_5
% 3.21/1.71 => ( breln1all2
% 3.21/1.71 => ( breln1SetBreln1
% 3.21/1.71 => ( choice2fnsingleton
% 3.21/1.71 => ( setOfPairsIsBReln1
% 3.21/1.71 => ( breln1all1
% 3.21/1.71 => ( subbreln1
% 3.21/1.71 => ( eqbreln1
% 3.21/1.71 => ( breln1invprop
% 3.21/1.71 => ( breln1invI
% 3.21/1.71 => ( breln1invE
% 3.21/1.71 => ( breln1compprop
% 3.21/1.71 => ( breln1compI
% 3.21/1.71 => ( breln1compE
% 3.21/1.71 => ( breln1compEex
% 3.21/1.71 => ( breln1unionprop
% 3.21/1.71 => ( breln1unionIL
% 3.21/1.71 => ( breln1unionIR
% 3.21/1.71 => ( breln1unionI
% 3.21/1.71 => ( breln1unionE
% 3.21/1.71 => ( breln1unionEcases
% 3.21/1.71 => ( breln1unionCommutes
% 3.21/1.71 => ( woz2Ex
% 3.21/1.71 => ( woz2W
% 3.21/1.71 => ( woz2A
% 3.21/1.71 => ( woz2B
% 3.21/1.71 => ( image1Ex
% 3.21/1.71 => ( image1Ex1
% 3.21/1.71 => ( image1Equiv
% 3.21/1.71 => ( image1E
% 3.21/1.71 => ( image1I
% 3.21/1.71 => ( injFuncInInjFuncSet
% 3.21/1.71 => ( injFuncSetFuncIn
% 3.21/1.71 => ( injFuncSetFuncInj
% 3.21/1.71 => ( surjFuncSetFuncIn
% 3.21/1.71 => ( surjFuncSetFuncSurj
% 3.21/1.71 => ( leftInvIsSurj
% 3.21/1.71 => ( surjCantorThm
% 3.21/1.71 => ( foundation2
% 3.21/1.71 => ( notinself
% 3.21/1.71 => ( notinself2
% 3.21/1.71 => ( omegaSp
% 3.21/1.71 => ( omegaSclos
% 3.21/1.71 => ( peano0notS
% 3.21/1.71 => ( peanoSinj
% 3.21/1.71 => ( transitivesetOp1
% 3.21/1.71 => ( binintTransitive
% 3.21/1.71 => ( transitivesetOp2
% 3.21/1.71 => ( setunionTransitive
% 3.21/1.71 => ( ordinalMinLem1
% 3.21/1.71 => ! [X: $i] :
% 3.21/1.71 ( ( ordinal @ X )
% 3.21/1.71 => ! [Xx: $i,A: $i] :
% 3.21/1.71 ( ( in @ A @ X )
% 3.21/1.71 => ( ( in @ Xx @ A )
% 3.21/1.71 => ( in @ Xx @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71 ------------------------
% 3.21/1.75 % (13914)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_13596 for (2999ds/4Mi)
% 3.21/1.75 % (13916)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/2Mi)
% 3.21/1.75 % (13916)Instruction limit reached!
% 3.21/1.75 % (13916)------------------------------
% 3.21/1.75 % (13916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.75 % (13916)Termination reason: Unknown
% 3.21/1.75 % (13916)Termination phase: shuffling
% 3.21/1.75
% 3.21/1.75 % (13916)Memory used [KB]: 1663
% 3.21/1.75 % (13916)Time elapsed: 0.002 s
% 3.21/1.75 % (13916)Instructions burned: 2 (million)
% 3.21/1.75 % (13916)------------------------------
% 3.21/1.75 % (13916)------------------------------
% 3.21/1.75 % (13919)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/18Mi)
% 3.21/1.75 % (13914)Instruction limit reached!
% 3.21/1.75 % (13914)------------------------------
% 3.21/1.75 % (13914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.75 % (13914)Termination reason: Unknown
% 3.21/1.75 % (13914)Termination phase: shuffling
% 3.21/1.75
% 3.21/1.75 % (13914)Memory used [KB]: 1663
% 3.21/1.75 % (13914)Time elapsed: 0.003 s
% 3.21/1.75 % (13914)Instructions burned: 5 (million)
% 3.21/1.75 % (13914)------------------------------
% 3.21/1.75 % (13914)------------------------------
% 3.21/1.76 % (13919)Instruction limit reached!
% 3.21/1.76 % (13919)------------------------------
% 3.21/1.76 % (13919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.76 % (13919)Termination reason: Unknown
% 3.21/1.76 % (13919)Termination phase: shuffling
% 3.21/1.76
% 3.21/1.76 % (13919)Memory used [KB]: 1918
% 3.21/1.76 % (13919)Time elapsed: 0.007 s
% 3.21/1.76 % (13919)Instructions burned: 19 (million)
% 3.21/1.76 % (13919)------------------------------
% 3.21/1.76 % (13919)------------------------------
% 3.21/1.76 % (13913)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/183Mi)
% 3.21/1.76 % (13915)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_13596 for (2999ds/27Mi)
% 3.21/1.76 % (13917)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_13596 for (2999ds/2Mi)
% 3.21/1.76 % (13918)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_13596 for (2999ds/275Mi)
% 3.21/1.76 % (13917)Instruction limit reached!
% 3.21/1.76 % (13917)------------------------------
% 3.21/1.76 % (13917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.76 % (13917)Termination reason: Unknown
% 3.21/1.76 % (13917)Termination phase: shuffling
% 3.21/1.76
% 3.21/1.76 % (13917)Memory used [KB]: 1663
% 3.21/1.76 % (13917)Time elapsed: 0.003 s
% 3.21/1.76 % (13917)Instructions burned: 2 (million)
% 3.21/1.76 % (13917)------------------------------
% 3.21/1.76 % (13917)------------------------------
% 3.21/1.76 % (13925)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/3Mi)
% 3.21/1.76 % (13927)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_13596 for (2999ds/37Mi)
% 3.21/1.76 % (13925)Instruction limit reached!
% 3.21/1.76 % (13925)------------------------------
% 3.21/1.76 % (13925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.76 % (13925)Termination reason: Unknown
% 3.21/1.76 % (13925)Termination phase: shuffling
% 3.21/1.76
% 3.21/1.76 % (13925)Memory used [KB]: 1791
% 3.21/1.76 % (13925)Time elapsed: 0.003 s
% 3.21/1.76 % (13925)Instructions burned: 6 (million)
% 3.21/1.76 % (13925)------------------------------
% 3.21/1.76 % (13925)------------------------------
% 3.21/1.77 % (13931)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_13596 for (2999ds/15Mi)
% 3.21/1.77 % (13935)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_13596 for (2999ds/1041Mi)
% 3.21/1.77 % (13927)Instruction limit reached!
% 3.21/1.77 % (13927)------------------------------
% 3.21/1.77 % (13927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.77 % (13927)Termination reason: Unknown
% 3.21/1.77 % (13927)Termination phase: shuffling
% 3.21/1.77
% 3.21/1.77 % (13927)Memory used [KB]: 2430
% 3.21/1.77 % (13927)Time elapsed: 0.012 s
% 3.21/1.77 % (13927)Instructions burned: 40 (million)
% 3.21/1.77 % (13927)------------------------------
% 3.21/1.77 % (13927)------------------------------
% 3.21/1.77 % (13915)Instruction limit reached!
% 3.21/1.77 % (13915)------------------------------
% 3.21/1.77 % (13915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.77 % (13915)Termination reason: Unknown
% 3.21/1.77 % (13915)Termination phase: shuffling
% 3.21/1.77
% 3.21/1.77 % (13915)Memory used [KB]: 2174
% 3.21/1.77 % (13915)Time elapsed: 0.016 s
% 3.21/1.77 % (13915)Instructions burned: 28 (million)
% 3.21/1.77 % (13915)------------------------------
% 3.21/1.77 % (13915)------------------------------
% 3.21/1.78 % (13931)Instruction limit reached!
% 3.21/1.78 % (13931)------------------------------
% 3.21/1.78 % (13931)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.78 % (13931)Termination reason: Unknown
% 3.21/1.78 % (13931)Termination phase: shuffling
% 3.21/1.78
% 3.21/1.78 % (13931)Memory used [KB]: 1918
% 3.21/1.78 % (13931)Time elapsed: 0.007 s
% 3.21/1.78 % (13931)Instructions burned: 17 (million)
% 3.21/1.78 % (13931)------------------------------
% 3.21/1.78 % (13931)------------------------------
% 3.21/1.78 % (13933)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_13596 for (2999ds/3Mi)
% 3.21/1.78 % (13933)Instruction limit reached!
% 3.21/1.78 % (13933)------------------------------
% 3.21/1.78 % (13933)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.78 % (13933)Termination reason: Unknown
% 3.21/1.78 % (13933)Termination phase: shuffling
% 3.21/1.78
% 3.21/1.78 % (13933)Memory used [KB]: 1663
% 3.21/1.78 % (13933)Time elapsed: 0.004 s
% 3.21/1.78 % (13933)Instructions burned: 3 (million)
% 3.21/1.78 % (13933)------------------------------
% 3.21/1.78 % (13933)------------------------------
% 3.21/1.78 % (13939)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/7Mi)
% 3.21/1.79 % (13941)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_13596 for (2999ds/3Mi)
% 3.21/1.79 % (13941)Instruction limit reached!
% 3.21/1.79 % (13941)------------------------------
% 3.21/1.79 % (13941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.79 % (13941)Termination reason: Unknown
% 3.21/1.79 % (13941)Termination phase: shuffling
% 3.21/1.79
% 3.21/1.79 % (13941)Memory used [KB]: 1663
% 3.21/1.79 % (13941)Time elapsed: 0.002 s
% 3.21/1.79 % (13941)Instructions burned: 3 (million)
% 3.21/1.79 % (13941)------------------------------
% 3.21/1.79 % (13941)------------------------------
% 3.21/1.79 % (13939)Instruction limit reached!
% 3.21/1.79 % (13939)------------------------------
% 3.21/1.79 % (13939)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.79 % (13939)Termination reason: Unknown
% 3.21/1.79 % (13939)Termination phase: shuffling
% 3.21/1.79
% 3.21/1.79 % (13939)Memory used [KB]: 1918
% 3.21/1.79 % (13939)Time elapsed: 0.004 s
% 3.21/1.79 % (13939)Instructions burned: 10 (million)
% 3.21/1.79 % (13939)------------------------------
% 3.21/1.79 % (13939)------------------------------
% 3.21/1.79 % (13940)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/16Mi)
% 3.21/1.79 % (13940)Instruction limit reached!
% 3.21/1.79 % (13940)------------------------------
% 3.21/1.79 % (13940)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.79 % (13940)Termination reason: Unknown
% 3.21/1.79 % (13940)Termination phase: shuffling
% 3.21/1.79
% 3.21/1.79 % (13940)Memory used [KB]: 1918
% 3.21/1.79 % (13940)Time elapsed: 0.007 s
% 3.21/1.79 % (13940)Instructions burned: 16 (million)
% 3.21/1.79 % (13940)------------------------------
% 3.21/1.79 % (13940)------------------------------
% 3.21/1.80 % (13946)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/7Mi)
% 3.21/1.80 % (13947)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/3Mi)
% 3.24/1.80 % (13947)Instruction limit reached!
% 3.24/1.80 % (13947)------------------------------
% 3.24/1.80 % (13947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.80 % (13947)Termination reason: Unknown
% 3.24/1.80 % (13947)Termination phase: shuffling
% 3.24/1.80
% 3.24/1.80 % (13947)Memory used [KB]: 1663
% 3.24/1.80 % (13947)Time elapsed: 0.002 s
% 3.24/1.80 % (13947)Instructions burned: 3 (million)
% 3.24/1.80 % (13947)------------------------------
% 3.24/1.80 % (13947)------------------------------
% 3.24/1.80 % (13946)Instruction limit reached!
% 3.24/1.80 % (13946)------------------------------
% 3.24/1.80 % (13946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.80 % (13946)Termination reason: Unknown
% 3.24/1.80 % (13946)Termination phase: shuffling
% 3.24/1.80
% 3.24/1.80 % (13946)Memory used [KB]: 1791
% 3.24/1.80 % (13946)Time elapsed: 0.003 s
% 3.24/1.80 % (13946)Instructions burned: 7 (million)
% 3.24/1.80 % (13946)------------------------------
% 3.24/1.80 % (13946)------------------------------
% 3.24/1.80 % (13943)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_13596 for (2999ds/3Mi)
% 3.24/1.80 % (13943)Instruction limit reached!
% 3.24/1.80 % (13943)------------------------------
% 3.24/1.80 % (13943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.80 % (13943)Termination reason: Unknown
% 3.24/1.80 % (13943)Termination phase: shuffling
% 3.24/1.80
% 3.24/1.80 % (13943)Memory used [KB]: 1663
% 3.24/1.80 % (13943)Time elapsed: 0.003 s
% 3.24/1.80 % (13943)Instructions burned: 3 (million)
% 3.24/1.80 % (13943)------------------------------
% 3.24/1.80 % (13943)------------------------------
% 3.24/1.80 % (13949)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/4Mi)
% 3.24/1.81 % (13951)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/18Mi)
% 3.24/1.81 % (13949)Instruction limit reached!
% 3.24/1.81 % (13949)------------------------------
% 3.24/1.81 % (13949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (13949)Termination reason: Unknown
% 3.24/1.81 % (13949)Termination phase: shuffling
% 3.24/1.81
% 3.24/1.81 % (13949)Memory used [KB]: 1791
% 3.24/1.81 % (13949)Time elapsed: 0.003 s
% 3.24/1.81 % (13949)Instructions burned: 5 (million)
% 3.24/1.81 % (13949)------------------------------
% 3.24/1.81 % (13949)------------------------------
% 3.24/1.81 % (13952)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/710Mi)
% 3.24/1.81 % (13953)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/6Mi)
% 3.24/1.81 % (13951)Instruction limit reached!
% 3.24/1.81 % (13951)------------------------------
% 3.24/1.81 % (13951)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (13951)Termination reason: Unknown
% 3.24/1.81 % (13951)Termination phase: shuffling
% 3.24/1.81
% 3.24/1.81 % (13951)Memory used [KB]: 2046
% 3.24/1.81 % (13951)Time elapsed: 0.007 s
% 3.24/1.81 % (13951)Instructions burned: 20 (million)
% 3.24/1.81 % (13951)------------------------------
% 3.24/1.81 % (13951)------------------------------
% 3.24/1.81 % (13953)Instruction limit reached!
% 3.24/1.81 % (13953)------------------------------
% 3.24/1.81 % (13953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.81 % (13953)Termination reason: Unknown
% 3.24/1.81 % (13953)Termination phase: shuffling
% 3.24/1.81
% 3.24/1.81 % (13953)Memory used [KB]: 1791
% 3.24/1.81 % (13953)Time elapsed: 0.004 s
% 3.24/1.81 % (13953)Instructions burned: 8 (million)
% 3.24/1.81 % (13953)------------------------------
% 3.24/1.81 % (13953)------------------------------
% 3.24/1.82 % (13959)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_13596 for (2999ds/902Mi)
% 3.24/1.82 % (13962)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_13596 for (2999ds/21Mi)
% 3.24/1.82 % (13963)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13596 for (2999ds/5Mi)
% 3.24/1.82 % (13963)Instruction limit reached!
% 3.24/1.82 % (13963)------------------------------
% 3.24/1.82 % (13963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.82 % (13963)Termination reason: Unknown
% 3.24/1.82 % (13963)Termination phase: shuffling
% 3.24/1.82
% 3.24/1.82 % (13963)Memory used [KB]: 1791
% 3.24/1.82 % (13963)Time elapsed: 0.003 s
% 3.24/1.82 % (13963)Instructions burned: 6 (million)
% 3.24/1.82 % (13963)------------------------------
% 3.24/1.82 % (13963)------------------------------
% 3.24/1.83 % (13913)Instruction limit reached!
% 3.24/1.83 % (13913)------------------------------
% 3.24/1.83 % (13913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.83 % (13913)Termination reason: Unknown
% 3.24/1.83 % (13913)Termination phase: Property scanning
% 3.24/1.83
% 3.24/1.83 % (13913)Memory used [KB]: 3198
% 3.24/1.83 % (13913)Time elapsed: 0.067 s
% 3.24/1.83 % (13913)Instructions burned: 186 (million)
% 3.24/1.83 % (13913)------------------------------
% 3.24/1.83 % (13913)------------------------------
% 3.24/1.83 % (13962)Instruction limit reached!
% 3.24/1.83 % (13962)------------------------------
% 3.24/1.83 % (13962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.83 % (13962)Termination reason: Unknown
% 3.24/1.83 % (13962)Termination phase: shuffling
% 3.24/1.83
% 3.24/1.83 % (13962)Memory used [KB]: 2046
% 3.24/1.83 % (13962)Time elapsed: 0.007 s
% 3.24/1.83 % (13962)Instructions burned: 21 (million)
% 3.24/1.83 % (13962)------------------------------
% 3.24/1.83 % (13962)------------------------------
% 3.24/1.84 % (13970)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_13596 for (2998ds/6Mi)
% 3.24/1.84 % (13971)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13596 for (2998ds/377Mi)
% 3.24/1.84 % (13972)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_13596 for (2998ds/779Mi)
% 3.24/1.84 % (13970)Instruction limit reached!
% 3.24/1.84 % (13970)------------------------------
% 3.24/1.84 % (13970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.84 % (13970)Termination reason: Unknown
% 3.24/1.84 % (13970)Termination phase: shuffling
% 3.24/1.84
% 3.24/1.84 % (13970)Memory used [KB]: 1791
% 3.24/1.84 % (13970)Time elapsed: 0.003 s
% 3.24/1.84 % (13970)Instructions burned: 6 (million)
% 3.24/1.84 % (13970)------------------------------
% 3.24/1.84 % (13970)------------------------------
% 3.24/1.85 % (13976)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_13596 for (2998ds/19Mi)
% 3.24/1.85 % (13976)Instruction limit reached!
% 3.24/1.85 % (13976)------------------------------
% 3.24/1.85 % (13976)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.85 % (13976)Termination reason: Unknown
% 3.24/1.85 % (13976)Termination phase: shuffling
% 3.24/1.85
% 3.24/1.85 % (13976)Memory used [KB]: 2046
% 3.24/1.85 % (13976)Time elapsed: 0.007 s
% 3.24/1.85 % (13976)Instructions burned: 20 (million)
% 3.24/1.85 % (13976)------------------------------
% 3.24/1.85 % (13976)------------------------------
% 3.24/1.86 % (13983)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13596 for (2998ds/879Mi)
% 3.24/1.89 % (13918)Instruction limit reached!
% 3.24/1.89 % (13918)------------------------------
% 3.24/1.89 % (13918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.89 % (13918)Termination reason: Unknown
% 3.24/1.89 % (13918)Termination phase: Property scanning
% 3.24/1.89
% 3.24/1.89 % (13918)Memory used [KB]: 6012
% 3.24/1.89 % (13918)Time elapsed: 0.130 s
% 3.24/1.89 % (13918)Instructions burned: 277 (million)
% 3.24/1.89 % (13918)------------------------------
% 3.24/1.89 % (13918)------------------------------
% 3.24/1.90 % (14009)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13596 for (2998ds/17Mi)
% 3.24/1.91 % (14009)Instruction limit reached!
% 3.24/1.91 % (14009)------------------------------
% 3.24/1.91 % (14009)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.91 % (14009)Termination reason: Unknown
% 3.24/1.91 % (14009)Termination phase: shuffling
% 3.24/1.91
% 3.24/1.91 % (14009)Memory used [KB]: 2046
% 3.24/1.91 % (14009)Time elapsed: 0.007 s
% 3.24/1.91 % (14009)Instructions burned: 18 (million)
% 3.24/1.91 % (14009)------------------------------
% 3.24/1.91 % (14009)------------------------------
% 3.24/1.92 % (14015)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13596 for (2998ds/3Mi)
% 3.24/1.92 % (14015)Instruction limit reached!
% 3.24/1.92 % (14015)------------------------------
% 3.24/1.92 % (14015)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.92 % (14015)Termination reason: Unknown
% 3.24/1.92 % (14015)Termination phase: shuffling
% 3.24/1.92
% 3.24/1.92 % (14015)Memory used [KB]: 1791
% 3.24/1.92 % (14015)Time elapsed: 0.003 s
% 3.24/1.92 % (14015)Instructions burned: 5 (million)
% 3.24/1.92 % (14015)------------------------------
% 3.24/1.92 % (14015)------------------------------
% 3.24/1.93 % (14020)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_13596 for (2998ds/30Mi)
% 3.24/1.94 % (14020)Instruction limit reached!
% 3.24/1.94 % (14020)------------------------------
% 3.24/1.94 % (14020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.94 % (14020)Termination reason: Unknown
% 3.24/1.94 % (14020)Termination phase: shuffling
% 3.24/1.94
% 3.24/1.94 % (14020)Memory used [KB]: 2430
% 3.24/1.94 % (14020)Time elapsed: 0.011 s
% 3.24/1.94 % (14020)Instructions burned: 32 (million)
% 3.24/1.94 % (14020)------------------------------
% 3.24/1.94 % (14020)------------------------------
% 3.24/1.94 % (13971)Instruction limit reached!
% 3.24/1.94 % (13971)------------------------------
% 3.24/1.94 % (13971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.94 % (13971)Termination reason: Unknown
% 3.24/1.94 % (13971)Termination phase: Saturation
% 3.24/1.94
% 3.24/1.94 % (13971)Memory used [KB]: 8571
% 3.24/1.94 % (13971)Time elapsed: 0.109 s
% 3.24/1.94 % (13971)Instructions burned: 380 (million)
% 3.24/1.94 % (13971)------------------------------
% 3.24/1.94 % (13971)------------------------------
% 3.24/1.95 % (14021)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_13596 for (2997ds/127Mi)
% 3.24/1.95 % (14022)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_13596 for (2997ds/100Mi)
% 3.24/1.98 % (14022)Instruction limit reached!
% 3.24/1.98 % (14022)------------------------------
% 3.24/1.98 % (14022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.98 % (14022)Termination reason: Unknown
% 3.24/1.98 % (14022)Termination phase: Property scanning
% 3.24/1.98
% 3.24/1.98 % (14022)Memory used [KB]: 2814
% 3.24/1.98 % (14022)Time elapsed: 0.029 s
% 3.24/1.98 % (14022)Instructions burned: 100 (million)
% 3.24/1.98 % (14022)------------------------------
% 3.24/1.98 % (14022)------------------------------
% 3.24/1.99 % (14021)Instruction limit reached!
% 3.24/1.99 % (14021)------------------------------
% 3.24/1.99 % (14021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.99 % (14021)Termination reason: Unknown
% 3.24/1.99 % (14021)Termination phase: Clausification
% 3.24/1.99
% 3.24/1.99 % (14021)Memory used [KB]: 3070
% 3.24/1.99 % (14021)Time elapsed: 0.037 s
% 3.24/1.99 % (14021)Instructions burned: 129 (million)
% 3.24/1.99 % (14021)------------------------------
% 3.24/1.99 % (14021)------------------------------
% 3.24/1.99 % (14023)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_13596 for (2997ds/3Mi)
% 3.24/1.99 % (14023)Instruction limit reached!
% 3.24/1.99 % (14023)------------------------------
% 3.24/1.99 % (14023)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.99 % (14023)Termination reason: Unknown
% 3.24/1.99 % (14023)Termination phase: shuffling
% 3.24/1.99
% 3.24/1.99 % (14023)Memory used [KB]: 1663
% 3.24/1.99 % (14023)Time elapsed: 0.002 s
% 3.24/1.99 % (14023)Instructions burned: 3 (million)
% 3.24/1.99 % (14023)------------------------------
% 3.24/1.99 % (14023)------------------------------
% 3.24/2.00 % (14025)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_13596 for (2997ds/86Mi)
% 3.24/2.01 % (14024)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_13596 for (2997ds/20Mi)
% 5.09/2.02 % (14024)Instruction limit reached!
% 5.09/2.02 % (14024)------------------------------
% 5.09/2.02 % (14024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.02 % (14024)Termination reason: Unknown
% 5.09/2.02 % (14024)Termination phase: shuffling
% 5.09/2.02
% 5.09/2.02 % (14024)Memory used [KB]: 2046
% 5.09/2.02 % (14024)Time elapsed: 0.010 s
% 5.09/2.02 % (14024)Instructions burned: 21 (million)
% 5.09/2.02 % (14024)------------------------------
% 5.09/2.02 % (14024)------------------------------
% 5.09/2.02 % (13952)Instruction limit reached!
% 5.09/2.02 % (13952)------------------------------
% 5.09/2.02 % (13952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.02 % (13952)Termination reason: Unknown
% 5.09/2.02 % (13952)Termination phase: Saturation
% 5.09/2.02
% 5.09/2.02 % (13952)Memory used [KB]: 11897
% 5.09/2.02 % (13952)Time elapsed: 0.219 s
% 5.09/2.02 % (13952)Instructions burned: 711 (million)
% 5.09/2.02 % (13952)------------------------------
% 5.09/2.02 % (13952)------------------------------
% 5.09/2.03 % (14025)Instruction limit reached!
% 5.09/2.03 % (14025)------------------------------
% 5.09/2.03 % (14025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.03 % (14025)Termination reason: Unknown
% 5.09/2.03 % (14025)Termination phase: Property scanning
% 5.09/2.03
% 5.09/2.03 % (14025)Memory used [KB]: 2814
% 5.09/2.03 % (14025)Time elapsed: 0.026 s
% 5.09/2.03 % (14025)Instructions burned: 86 (million)
% 5.09/2.03 % (14025)------------------------------
% 5.09/2.03 % (14025)------------------------------
% 5.09/2.03 % (13972)Instruction limit reached!
% 5.09/2.03 % (13972)------------------------------
% 5.09/2.03 % (13972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.03 % (13972)Termination reason: Unknown
% 5.09/2.03 % (13972)Termination phase: Saturation
% 5.09/2.03
% 5.09/2.03 % (13972)Memory used [KB]: 8059
% 5.09/2.03 % (13972)Time elapsed: 0.194 s
% 5.09/2.03 % (13972)Instructions burned: 779 (million)
% 5.09/2.03 % (13972)------------------------------
% 5.09/2.03 % (13972)------------------------------
% 5.09/2.03 % (14026)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_13596 for (2997ds/107Mi)
% 5.09/2.03 % (14027)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_13596 for (2997ds/39Mi)
% 5.09/2.04 % (13935)Instruction limit reached!
% 5.09/2.04 % (13935)------------------------------
% 5.09/2.04 % (13935)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.04 % (13935)Termination reason: Unknown
% 5.09/2.04 % (13935)Termination phase: Saturation
% 5.09/2.04
% 5.09/2.04 % (13935)Memory used [KB]: 17782
% 5.09/2.04 % (13935)Time elapsed: 0.267 s
% 5.09/2.04 % (13935)Instructions burned: 1043 (million)
% 5.09/2.04 % (13935)------------------------------
% 5.09/2.04 % (13935)------------------------------
% 5.09/2.04 % (14029)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_13596 for (2997ds/46Mi)
% 5.09/2.05 % (14027)Instruction limit reached!
% 5.09/2.05 % (14027)------------------------------
% 5.09/2.05 % (14027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.05 % (14027)Termination reason: Unknown
% 5.09/2.05 % (14027)Termination phase: shuffling
% 5.09/2.05
% 5.09/2.05 % (14027)Memory used [KB]: 2430
% 5.09/2.05 % (14027)Time elapsed: 0.012 s
% 5.09/2.05 % (14027)Instructions burned: 40 (million)
% 5.09/2.05 % (14027)------------------------------
% 5.09/2.05 % (14027)------------------------------
% 5.09/2.05 % (14028)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_13596 for (2997ds/448Mi)
% 5.09/2.05 % (14030)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_13596 for (2996ds/98Mi)
% 5.09/2.05 % (14029)Instruction limit reached!
% 5.09/2.05 % (14029)------------------------------
% 5.09/2.05 % (14029)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.05 % (14029)Termination reason: Unknown
% 5.09/2.05 % (14029)Termination phase: shuffling
% 5.09/2.05
% 5.09/2.05 % (14029)Memory used [KB]: 2686
% 5.09/2.05 % (14029)Time elapsed: 0.014 s
% 5.09/2.05 % (14029)Instructions burned: 46 (million)
% 5.09/2.05 % (14029)------------------------------
% 5.09/2.05 % (14029)------------------------------
% 5.09/2.06 % (14031)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_13596 for (2996ds/507Mi)
% 5.09/2.06 % (14032)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_13596 for (2996ds/149Mi)
% 5.09/2.06 % (14026)Instruction limit reached!
% 5.09/2.06 % (14026)------------------------------
% 5.09/2.06 % (14026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.06 % (14026)Termination reason: Unknown
% 5.09/2.06 % (14026)Termination phase: SInE selection
% 5.09/2.06
% 5.09/2.06 % (14026)Memory used [KB]: 2942
% 5.09/2.06 % (14026)Time elapsed: 0.032 s
% 5.09/2.06 % (14026)Instructions burned: 109 (million)
% 5.09/2.06 % (14026)------------------------------
% 5.09/2.06 % (14026)------------------------------
% 5.09/2.07 % (14033)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_13596 for (2996ds/39Mi)
% 5.09/2.08 % (14030)Instruction limit reached!
% 5.09/2.08 % (14030)------------------------------
% 5.09/2.08 % (14030)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.08 % (14030)Termination reason: Unknown
% 5.09/2.08 % (14030)Termination phase: Property scanning
% 5.09/2.08
% 5.09/2.08 % (14030)Memory used [KB]: 2814
% 5.09/2.08 % (14030)Time elapsed: 0.027 s
% 5.09/2.08 % (14030)Instructions burned: 101 (million)
% 5.09/2.08 % (14030)------------------------------
% 5.09/2.08 % (14030)------------------------------
% 5.09/2.08 % (14033)Instruction limit reached!
% 5.09/2.08 % (14033)------------------------------
% 5.09/2.08 % (14033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.08 % (14033)Termination reason: Unknown
% 5.09/2.08 % (14033)Termination phase: shuffling
% 5.09/2.08
% 5.09/2.08 % (14033)Memory used [KB]: 2558
% 5.09/2.08 % (14033)Time elapsed: 0.013 s
% 5.09/2.08 % (14033)Instructions burned: 41 (million)
% 5.09/2.08 % (14033)------------------------------
% 5.09/2.08 % (14033)------------------------------
% 5.09/2.09 % (14034)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_13596 for (2996ds/6Mi)
% 5.09/2.09 % (14034)Instruction limit reached!
% 5.09/2.09 % (14034)------------------------------
% 5.09/2.09 % (14034)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.09 % (14034)Termination reason: Unknown
% 5.09/2.09 % (14034)Termination phase: shuffling
% 5.09/2.09
% 5.09/2.09 % (14034)Memory used [KB]: 1791
% 5.09/2.09 % (14034)Time elapsed: 0.003 s
% 5.09/2.09 % (14034)Instructions burned: 6 (million)
% 5.09/2.09 % (14034)------------------------------
% 5.09/2.09 % (14034)------------------------------
% 5.09/2.10 % (14035)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_13596 for (2996ds/1087Mi)
% 5.09/2.10 % (14036)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_13596 for (2996ds/47Mi)
% 5.09/2.10 % (13959)Instruction limit reached!
% 5.09/2.10 % (13959)------------------------------
% 5.09/2.10 % (13959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.10 % (13959)Termination reason: Unknown
% 5.09/2.10 % (13959)Termination phase: Saturation
% 5.09/2.10
% 5.09/2.10 % (13959)Memory used [KB]: 13560
% 5.09/2.10 % (13959)Time elapsed: 0.281 s
% 5.09/2.10 % (13959)Instructions burned: 903 (million)
% 5.09/2.10 % (13959)------------------------------
% 5.09/2.10 % (13959)------------------------------
% 5.09/2.10 % (14032)Instruction limit reached!
% 5.09/2.10 % (14032)------------------------------
% 5.09/2.10 % (14032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.10 % (14032)Termination reason: Unknown
% 5.09/2.10 % (14032)Termination phase: Function definition elimination
% 5.09/2.10
% 5.09/2.10 % (14032)Memory used [KB]: 3070
% 5.09/2.10 % (14032)Time elapsed: 0.039 s
% 5.09/2.10 % (14032)Instructions burned: 149 (million)
% 5.09/2.10 % (14032)------------------------------
% 5.09/2.10 % (14032)------------------------------
% 5.09/2.11 % (14037)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_13596 for (2996ds/44Mi)
% 5.09/2.11 % (14036)Instruction limit reached!
% 5.09/2.11 % (14036)------------------------------
% 5.09/2.11 % (14036)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.09/2.11 % (14036)Termination reason: Unknown
% 5.09/2.11 % (14036)Termination phase: shuffling
% 5.09/2.11
% 5.09/2.11 % (14036)Memory used [KB]: 2558
% 5.09/2.11 % (14036)Time elapsed: 0.014 s
% 5.09/2.11 % (14036)Instructions burned: 47 (million)
% 5.09/2.11 % (14036)------------------------------
% 5.09/2.11 % (14036)------------------------------
% 5.09/2.11 % (14038)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_13596 for (2996ds/432Mi)
% 5.62/2.12 % (14037)Instruction limit reached!
% 5.62/2.12 % (14037)------------------------------
% 5.62/2.12 % (14037)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.62/2.12 % (14037)Termination reason: Unknown
% 5.62/2.12 % (14037)Termination phase: shuffling
% 5.62/2.12
% 5.62/2.12 % (14037)Memory used [KB]: 2558
% 5.62/2.12 % (14037)Time elapsed: 0.014 s
% 5.62/2.12 % (14037)Instructions burned: 46 (million)
% 5.62/2.12 % (14037)------------------------------
% 5.62/2.12 % (14037)------------------------------
% 5.62/2.12 % (14039)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_13596 for (2995ds/62Mi)
% 5.62/2.13 % (14040)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_13596 for (2995ds/31Mi)
% 5.62/2.14 % (14039)Instruction limit reached!
% 5.62/2.14 % (14039)------------------------------
% 5.62/2.14 % (14039)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.62/2.14 % (14039)Termination reason: Unknown
% 5.62/2.14 % (14039)Termination phase: shuffling
% 5.62/2.14
% 5.62/2.14 % (14039)Memory used [KB]: 2814
% 5.62/2.14 % (14039)Time elapsed: 0.019 s
% 5.62/2.14 % (14039)Instructions burned: 62 (million)
% 5.62/2.14 % (14039)------------------------------
% 5.62/2.14 % (14039)------------------------------
% 5.62/2.14 % (14040)Instruction limit reached!
% 5.62/2.14 % (14040)------------------------------
% 5.62/2.14 % (14040)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.62/2.14 % (14040)Termination reason: Unknown
% 5.62/2.14 % (14040)Termination phase: shuffling
% 5.62/2.14
% 5.62/2.14 % (14040)Memory used [KB]: 2430
% 5.62/2.14 % (14040)Time elapsed: 0.010 s
% 5.62/2.14 % (14040)Instructions burned: 32 (million)
% 5.62/2.14 % (14040)------------------------------
% 5.62/2.14 % (14040)------------------------------
% 5.62/2.14 % (13983)Instruction limit reached!
% 5.62/2.14 % (13983)------------------------------
% 5.62/2.14 % (13983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.62/2.14 % (13983)Termination reason: Unknown
% 5.62/2.14 % (13983)Termination phase: Saturation
% 5.62/2.14
% 5.62/2.14 % (13983)Memory used [KB]: 10874
% 5.62/2.14 % (13983)Time elapsed: 0.276 s
% 5.62/2.14 % (13983)Instructions burned: 880 (million)
% 5.62/2.14 % (13983)------------------------------
% 5.62/2.14 % (13983)------------------------------
% 5.62/2.15 % (14041)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_13596 for (2995ds/96Mi)
% 5.62/2.15 % (14042)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_13596 for (2995ds/453Mi)
% 5.62/2.15 % (14043)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_13596 for (2995ds/71Mi)
% 5.62/2.17 % (14043)Instruction limit reached!
% 5.62/2.17 % (14043)------------------------------
% 5.62/2.17 % (14043)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.62/2.17 % (14043)Termination reason: Unknown
% 5.62/2.17 % (14043)Termination phase: Property scanning
% 5.62/2.17
% 5.62/2.17 % (14043)Memory used [KB]: 2814
% 5.62/2.17 % (14043)Time elapsed: 0.021 s
% 5.62/2.17 % (14043)Instructions burned: 74 (million)
% 5.62/2.17 % (14043)------------------------------
% 5.62/2.17 % (14043)------------------------------
% 5.62/2.17 % (14041)Instruction limit reached!
% 5.62/2.17 % (14041)------------------------------
% 5.62/2.17 % (14041)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.62/2.17 % (14041)Termination reason: Unknown
% 5.62/2.17 % (14041)Termination phase: Property scanning
% 5.62/2.17
% 5.62/2.17 % (14041)Memory used [KB]: 2942
% 5.62/2.17 % (14041)Time elapsed: 0.027 s
% 5.62/2.17 % (14041)Instructions burned: 99 (million)
% 5.62/2.17 % (14041)------------------------------
% 5.62/2.17 % (14041)------------------------------
% 5.62/2.18 % (14044)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_13596 for (2995ds/83Mi)
% 5.62/2.18 % (14045)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_13596 for (2995ds/11Mi)
% 5.62/2.19 % (14045)Instruction limit reached!
% 5.62/2.19 % (14045)------------------------------
% 5.62/2.19 % (14045)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.62/2.19 % (14045)Termination reason: Unknown
% 5.62/2.19 % (14045)Termination phase: shuffling
% 5.62/2.19
% 5.62/2.19 % (14045)Memory used [KB]: 1918
% 5.62/2.19 % (14045)Time elapsed: 0.005 s
% 5.62/2.19 % (14045)Instructions burned: 13 (million)
% 5.62/2.19 % (14045)------------------------------
% 5.62/2.19 % (14045)------------------------------
% 5.80/2.20 % (14046)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_13596 for (2995ds/41Mi)
% 5.80/2.20 % (14031)Instruction limit reached!
% 5.80/2.20 % (14031)------------------------------
% 5.80/2.20 % (14031)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.20 % (14031)Termination reason: Unknown
% 5.80/2.20 % (14031)Termination phase: Saturation
% 5.80/2.20
% 5.80/2.20 % (14031)Memory used [KB]: 8443
% 5.80/2.20 % (14031)Time elapsed: 0.168 s
% 5.80/2.20 % (14031)Instructions burned: 507 (million)
% 5.80/2.20 % (14031)------------------------------
% 5.80/2.20 % (14031)------------------------------
% 5.80/2.20 % (14044)Instruction limit reached!
% 5.80/2.20 % (14044)------------------------------
% 5.80/2.20 % (14044)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.20 % (14044)Termination reason: Unknown
% 5.80/2.20 % (14044)Termination phase: Property scanning
% 5.80/2.20
% 5.80/2.20 % (14044)Memory used [KB]: 2814
% 5.80/2.20 % (14044)Time elapsed: 0.024 s
% 5.80/2.20 % (14044)Instructions burned: 86 (million)
% 5.80/2.20 % (14044)------------------------------
% 5.80/2.20 % (14044)------------------------------
% 5.80/2.21 % (14046)Instruction limit reached!
% 5.80/2.21 % (14046)------------------------------
% 5.80/2.21 % (14046)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.21 % (14046)Termination reason: Unknown
% 5.80/2.21 % (14046)Termination phase: shuffling
% 5.80/2.21
% 5.80/2.21 % (14046)Memory used [KB]: 2430
% 5.80/2.21 % (14046)Time elapsed: 0.012 s
% 5.80/2.21 % (14046)Instructions burned: 42 (million)
% 5.80/2.21 % (14046)------------------------------
% 5.80/2.21 % (14046)------------------------------
% 5.80/2.21 % (14047)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13596 for (2995ds/6Mi)
% 5.80/2.21 % (14047)Instruction limit reached!
% 5.80/2.21 % (14047)------------------------------
% 5.80/2.21 % (14047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.21 % (14047)Termination reason: Unknown
% 5.80/2.21 % (14047)Termination phase: shuffling
% 5.80/2.21
% 5.80/2.21 % (14047)Memory used [KB]: 1791
% 5.80/2.21 % (14047)Time elapsed: 0.003 s
% 5.80/2.21 % (14047)Instructions burned: 6 (million)
% 5.80/2.21 % (14047)------------------------------
% 5.80/2.21 % (14047)------------------------------
% 5.80/2.21 % (14028)Instruction limit reached!
% 5.80/2.21 % (14028)------------------------------
% 5.80/2.21 % (14028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.21 % (14028)Termination reason: Unknown
% 5.80/2.21 % (14028)Termination phase: Saturation
% 5.80/2.21
% 5.80/2.21 % (14028)Memory used [KB]: 7931
% 5.80/2.21 % (14028)Time elapsed: 0.164 s
% 5.80/2.21 % (14028)Instructions burned: 448 (million)
% 5.80/2.21 % (14028)------------------------------
% 5.80/2.21 % (14028)------------------------------
% 5.80/2.21 % (14048)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_13596 for (2995ds/150Mi)
% 5.80/2.22 % (14049)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_13596 for (2995ds/343Mi)
% 5.80/2.22 % (14038)Instruction limit reached!
% 5.80/2.22 % (14038)------------------------------
% 5.80/2.22 % (14038)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.22 % (14038)Termination reason: Unknown
% 5.80/2.22 % (14038)Termination phase: Saturation
% 5.80/2.22
% 5.80/2.22 % (14038)Memory used [KB]: 3454
% 5.80/2.22 % (14038)Time elapsed: 0.109 s
% 5.80/2.22 % (14038)Instructions burned: 434 (million)
% 5.80/2.22 % (14038)------------------------------
% 5.80/2.22 % (14038)------------------------------
% 5.80/2.22 % (14050)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_13596 for (2995ds/109Mi)
% 5.80/2.22 % (14051)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_13596 for (2995ds/253Mi)
% 5.80/2.23 % (14052)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_13596 for (2994ds/140Mi)
% 5.80/2.25 % (14050)Instruction limit reached!
% 5.80/2.25 % (14050)------------------------------
% 5.80/2.25 % (14050)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.25 % (14050)Termination reason: Unknown
% 5.80/2.25 % (14050)Termination phase: SInE selection
% 5.80/2.25
% 5.80/2.25 % (14050)Memory used [KB]: 2942
% 5.80/2.25 % (14050)Time elapsed: 0.030 s
% 5.80/2.25 % (14050)Instructions burned: 111 (million)
% 5.80/2.25 % (14050)------------------------------
% 5.80/2.25 % (14050)------------------------------
% 5.80/2.26 % (14048)Instruction limit reached!
% 5.80/2.26 % (14048)------------------------------
% 5.80/2.26 % (14048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.26 % (14048)Termination reason: Unknown
% 5.80/2.26 % (14048)Termination phase: Preprocessing 3
% 5.80/2.26
% 5.80/2.26 % (14048)Memory used [KB]: 5245
% 5.80/2.26 % (14048)Time elapsed: 0.045 s
% 5.80/2.26 % (14048)Instructions burned: 150 (million)
% 5.80/2.26 % (14048)------------------------------
% 5.80/2.26 % (14048)------------------------------
% 5.80/2.26 % (14053)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_13596 for (2994ds/614Mi)
% 5.80/2.26 % (14042)Instruction limit reached!
% 5.80/2.26 % (14042)------------------------------
% 5.80/2.26 % (14042)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.26 % (14042)Termination reason: Unknown
% 5.80/2.26 % (14042)Termination phase: Saturation
% 5.80/2.26
% 5.80/2.26 % (14042)Memory used [KB]: 7931
% 5.80/2.26 % (14042)Time elapsed: 0.113 s
% 5.80/2.26 % (14042)Instructions burned: 454 (million)
% 5.80/2.26 % (14042)------------------------------
% 5.80/2.26 % (14042)------------------------------
% 5.80/2.26 % (14052)Instruction limit reached!
% 5.80/2.26 % (14052)------------------------------
% 5.80/2.26 % (14052)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.26 % (14052)Termination reason: Unknown
% 5.80/2.26 % (14052)Termination phase: Property scanning
% 5.80/2.26
% 5.80/2.26 % (14052)Memory used [KB]: 3070
% 5.80/2.26 % (14052)Time elapsed: 0.037 s
% 5.80/2.26 % (14052)Instructions burned: 140 (million)
% 5.80/2.26 % (14052)------------------------------
% 5.80/2.26 % (14052)------------------------------
% 5.80/2.27 % (14054)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_13596 for (2994ds/202Mi)
% 5.80/2.27 % (14055)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_13596 for (2994ds/339Mi)
% 5.80/2.27 % (14056)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=283:si=on:rtra=on_0 on DTF2THF_13596 for (2994ds/283Mi)
% 6.58/2.30 % (14051)Instruction limit reached!
% 6.58/2.30 % (14051)------------------------------
% 6.58/2.30 % (14051)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.30 % (14051)Termination reason: Unknown
% 6.58/2.30 % (14051)Termination phase: Property scanning
% 6.58/2.30
% 6.58/2.30 % (14051)Memory used [KB]: 6012
% 6.58/2.30 % (14051)Time elapsed: 0.078 s
% 6.58/2.30 % (14051)Instructions burned: 253 (million)
% 6.58/2.30 % (14051)------------------------------
% 6.58/2.30 % (14051)------------------------------
% 6.58/2.31 % (14049)Instruction limit reached!
% 6.58/2.31 % (14049)------------------------------
% 6.58/2.31 % (14049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.31 % (14049)Termination reason: Unknown
% 6.58/2.31 % (14049)Termination phase: Saturation
% 6.58/2.31
% 6.58/2.31 % (14049)Memory used [KB]: 8187
% 6.58/2.31 % (14049)Time elapsed: 0.089 s
% 6.58/2.31 % (14049)Instructions burned: 346 (million)
% 6.58/2.31 % (14049)------------------------------
% 6.58/2.31 % (14049)------------------------------
% 6.58/2.31 % (14057)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_13596 for (2994ds/134Mi)
% 6.58/2.31 % (14058)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_13596 for (2994ds/298Mi)
% 6.58/2.32 % (14054)Instruction limit reached!
% 6.58/2.32 % (14054)------------------------------
% 6.58/2.32 % (14054)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.32 % (14054)Termination reason: Unknown
% 6.58/2.32 % (14054)Termination phase: Property scanning
% 6.58/2.32
% 6.58/2.32 % (14054)Memory used [KB]: 5756
% 6.58/2.32 % (14054)Time elapsed: 0.059 s
% 6.58/2.32 % (14054)Instructions burned: 204 (million)
% 6.58/2.32 % (14054)------------------------------
% 6.58/2.32 % (14054)------------------------------
% 6.58/2.33 % (14059)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_13596 for (2993ds/192Mi)
% 6.58/2.34 % (14056)Instruction limit reached!
% 6.58/2.34 % (14056)------------------------------
% 6.58/2.34 % (14056)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.34 % (14056)Termination reason: Unknown
% 6.58/2.34 % (14056)Termination phase: Saturation
% 6.58/2.34
% 6.58/2.34 % (14056)Memory used [KB]: 7931
% 6.58/2.34 % (14056)Time elapsed: 0.071 s
% 6.58/2.34 % (14056)Instructions burned: 283 (million)
% 6.58/2.34 % (14056)------------------------------
% 6.58/2.34 % (14056)------------------------------
% 6.58/2.35 % (14057)Instruction limit reached!
% 6.58/2.35 % (14057)------------------------------
% 6.58/2.35 % (14057)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.35 % (14057)Termination reason: Unknown
% 6.58/2.35 % (14057)Termination phase: Property scanning
% 6.58/2.35
% 6.58/2.35 % (14057)Memory used [KB]: 3070
% 6.58/2.35 % (14057)Time elapsed: 0.039 s
% 6.58/2.35 % (14057)Instructions burned: 136 (million)
% 6.58/2.35 % (14057)------------------------------
% 6.58/2.35 % (14057)------------------------------
% 6.58/2.35 % (14060)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_13596 for (2993ds/258Mi)
% 6.58/2.36 % (14061)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_13596 for (2993ds/90Mi)
% 6.58/2.36 % (14055)Instruction limit reached!
% 6.58/2.36 % (14055)------------------------------
% 6.58/2.36 % (14055)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.36 % (14055)Termination reason: Unknown
% 6.58/2.36 % (14055)Termination phase: Saturation
% 6.58/2.36
% 6.58/2.36 % (14055)Memory used [KB]: 8187
% 6.58/2.36 % (14055)Time elapsed: 0.092 s
% 6.58/2.36 % (14055)Instructions burned: 341 (million)
% 6.58/2.36 % (14055)------------------------------
% 6.58/2.36 % (14055)------------------------------
% 6.58/2.37 % (14062)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_13596 for (2993ds/231Mi)
% 6.58/2.38 % (14061)Instruction limit reached!
% 6.58/2.38 % (14061)------------------------------
% 6.58/2.38 % (14061)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.38 % (14061)Termination reason: Unknown
% 6.58/2.38 % (14061)Termination phase: Property scanning
% 6.58/2.38
% 6.58/2.38 % (14061)Memory used [KB]: 2814
% 6.58/2.38 % (14061)Time elapsed: 0.027 s
% 6.58/2.38 % (14061)Instructions burned: 91 (million)
% 6.58/2.38 % (14061)------------------------------
% 6.58/2.38 % (14061)------------------------------
% 6.58/2.38 % (14059)Instruction limit reached!
% 6.58/2.38 % (14059)------------------------------
% 6.58/2.38 % (14059)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.38 % (14059)Termination reason: Unknown
% 6.58/2.38 % (14059)Termination phase: Property scanning
% 6.58/2.38
% 6.58/2.38 % (14059)Memory used [KB]: 3198
% 6.58/2.38 % (14059)Time elapsed: 0.051 s
% 6.58/2.38 % (14059)Instructions burned: 194 (million)
% 6.58/2.38 % (14059)------------------------------
% 6.58/2.38 % (14059)------------------------------
% 6.58/2.39 % (14035)First to succeed.
% 6.58/2.39 % (14058)Instruction limit reached!
% 6.58/2.39 % (14058)------------------------------
% 6.58/2.39 % (14058)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.39 % (14058)Termination reason: Unknown
% 6.58/2.39 % (14058)Termination phase: Saturation
% 6.58/2.39
% 6.58/2.39 % (14058)Memory used [KB]: 7931
% 6.58/2.39 % (14058)Time elapsed: 0.076 s
% 6.58/2.39 % (14058)Instructions burned: 299 (million)
% 6.58/2.39 % (14058)------------------------------
% 6.58/2.39 % (14058)------------------------------
% 6.58/2.39 % (14035)Refutation found. Thanks to Tanya!
% 6.58/2.39 % SZS status Theorem for DTF2THF_13596
% 6.58/2.39 % SZS output start Proof for DTF2THF_13596
% 6.58/2.39 thf(func_def_0, type, in: $i > $i > $o).
% 6.58/2.39 thf(func_def_1, type, exu: ($i > $o) > $o).
% 6.58/2.39 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 6.58/2.39 thf(func_def_8, type, powerset: $i > $i).
% 6.58/2.39 thf(func_def_10, type, setunion: $i > $i).
% 6.58/2.39 thf(func_def_19, type, descr: ($i > $o) > $i).
% 6.58/2.39 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_26, type, prop2set: $o > $i).
% 6.58/2.39 thf(func_def_36, type, nonempty: $i > $o).
% 6.58/2.39 thf(func_def_69, type, set2prop: $i > $o).
% 6.58/2.39 thf(func_def_88, type, subset: $i > $i > $o).
% 6.58/2.39 thf(func_def_89, type, disjoint: $i > $i > $o).
% 6.58/2.39 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 6.58/2.39 thf(func_def_114, type, binunion: $i > $i > $i).
% 6.58/2.39 thf(func_def_122, type, binintersect: $i > $i > $i).
% 6.58/2.39 thf(func_def_135, type, regular: $i > $o).
% 6.58/2.39 thf(func_def_136, type, setminus: $i > $i > $i).
% 6.58/2.39 thf(func_def_147, type, symdiff: $i > $i > $i).
% 6.58/2.39 thf(func_def_153, type, iskpair: $i > $o).
% 6.58/2.39 thf(func_def_158, type, kpair: $i > $i > $i).
% 6.58/2.39 thf(func_def_160, type, cartprod: $i > $i > $i).
% 6.58/2.39 thf(func_def_177, type, singleton: $i > $o).
% 6.58/2.39 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 6.58/2.39 thf(func_def_184, type, atmost1p: $i > $o).
% 6.58/2.39 thf(func_def_185, type, atleast2p: $i > $o).
% 6.58/2.39 thf(func_def_186, type, atmost2p: $i > $o).
% 6.58/2.39 thf(func_def_187, type, upairsetp: $i > $o).
% 6.58/2.39 thf(func_def_191, type, kfst: $i > $i).
% 6.58/2.39 thf(func_def_203, type, ksnd: $i > $i).
% 6.58/2.39 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 6.58/2.39 thf(func_def_222, type, func: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_223, type, funcSet: $i > $i > $i).
% 6.58/2.39 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 6.58/2.39 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 6.58/2.39 thf(func_def_323, type, breln1: $i > $i > $o).
% 6.58/2.39 thf(func_def_325, type, breln1Set: $i > $i).
% 6.58/2.39 thf(func_def_327, type, transitive: $i > $i > $o).
% 6.58/2.39 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 6.58/2.39 thf(func_def_329, type, reflexive: $i > $i > $o).
% 6.58/2.39 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 6.58/2.39 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 6.58/2.39 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 6.58/2.39 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 6.58/2.39 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 6.58/2.39 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 6.58/2.39 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 6.58/2.39 thf(func_def_378, type, omegaS: $i > $i).
% 6.58/2.39 thf(func_def_382, type, peano3: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_383, type, peano4: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_384, type, peano5: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_386, type, peano: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_387, type, transitiveset: $i > $o).
% 6.58/2.39 thf(func_def_392, type, stricttotalorderedByIn: $i > $o).
% 6.58/2.39 thf(func_def_393, type, wellorderedByIn: $i > $o).
% 6.58/2.39 thf(func_def_394, type, ordinal: $i > $o).
% 6.58/2.39 thf(func_def_395, type, limitOrdinal: $i > $o).
% 6.58/2.39 thf(func_def_412, type, sP0: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_414, type, sP2: $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_415, type, sP3: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_417, type, sP5: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_418, type, sP6: $i > ($i > $i) > $i > $o).
% 6.58/2.39 thf(func_def_420, type, sP8: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_421, type, sP9: $i > $i > $o > $o).
% 6.58/2.39 thf(func_def_422, type, sP10: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_423, type, sP11: $i > $o).
% 6.58/2.39 thf(func_def_424, type, sP12: $i > $o).
% 6.58/2.39 thf(func_def_427, type, sP15: ($i > $o) > $o).
% 6.58/2.39 thf(func_def_428, type, sP16: ($i > $i > $o) > $i > $i > $o).
% 6.58/2.39 thf(func_def_429, type, sP17: ($i > $i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_432, type, sP20: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_434, type, sP22: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_435, type, sP23: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_436, type, sP24: $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_437, type, sP25: $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_438, type, sP26: ($i > $o) > $o).
% 6.58/2.39 thf(func_def_439, type, sP27: $i > ($i > $o) > $o).
% 6.58/2.39 thf(func_def_440, type, sP28: $i > ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_441, type, sP29: $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_442, type, sP30: ($i > $i) > $i > $i > $o).
% 6.58/2.39 thf(func_def_444, type, sP32: ($i > $o) > $o).
% 6.58/2.39 thf(func_def_447, type, sP35: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_448, type, sP36: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_449, type, sP37: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_450, type, sP38: $i > $i > $i > $o > $o).
% 6.58/2.39 thf(func_def_451, type, sP39: $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_453, type, sP41: $i > $i > ($i > $o) > $o).
% 6.58/2.39 thf(func_def_454, type, sP42: $i > $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_455, type, sP43: $i > $i > $i > $o > $o).
% 6.58/2.39 thf(func_def_456, type, sP44: $o > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_457, type, sP45: $i > $i > $o > $o).
% 6.58/2.39 thf(func_def_458, type, sP46: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_459, type, sP47: $i > ($i > $o) > ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_460, type, sP48: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_461, type, sP49: $i > $i > ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_464, type, sP52: $i > $i > $o).
% 6.58/2.39 thf(func_def_466, type, sP54: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_467, type, sP55: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_468, type, sP56: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_469, type, sP57: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_470, type, sP58: ($i > $i) > $i > $i > $o).
% 6.58/2.39 thf(func_def_474, type, sP62: $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_475, type, sP63: $o > $i > $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_477, type, sP65: $i > $i > $o).
% 6.58/2.39 thf(func_def_479, type, sP67: $i > $i > ($i > $i) > $o).
% 6.58/2.39 thf(func_def_481, type, sP69: $i > ($i > $o) > $o).
% 6.58/2.39 thf(func_def_482, type, sP70: $i > ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_483, type, sP71: $i > ($i > $o) > $o).
% 6.58/2.39 thf(func_def_485, type, sP73: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_488, type, sP76: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_489, type, sP77: $i > $i > ($i > $i > $o) > $o).
% 6.58/2.39 thf(func_def_490, type, sP78: ($i > $i > $o) > $i > $i > $o).
% 6.58/2.39 thf(func_def_491, type, sP79: ($i > $o) > $o).
% 6.58/2.39 thf(func_def_492, type, sP80: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_493, type, sP81: $i > $i > $o).
% 6.58/2.39 thf(func_def_494, type, sP82: $i > $i > $o).
% 6.58/2.39 thf(func_def_495, type, sP83: $i > $o).
% 6.58/2.39 thf(func_def_496, type, sP84: $i > $i > $o).
% 6.58/2.39 thf(func_def_497, type, sP85: $i > $i > $o).
% 6.58/2.39 thf(func_def_498, type, sP86: $i > $o > $i > $o).
% 6.58/2.39 thf(func_def_499, type, sP87: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_501, type, sP89: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_502, type, sP90: $i > ($i > $i) > $i > $o).
% 6.58/2.39 thf(func_def_504, type, sP92: $o > $i > $i > $o).
% 6.58/2.39 thf(func_def_506, type, sP94: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_508, type, sP96: ($i > $o) > $o).
% 6.58/2.39 thf(func_def_509, type, sP97: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_511, type, sP99: $i > ($i > $o) > $o).
% 6.58/2.39 thf(func_def_514, type, sP102: ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_515, type, sP103: $i > $i > ($i > $o) > $i > $o).
% 6.58/2.39 thf(func_def_516, type, sP104: $i > $o).
% 6.58/2.39 thf(func_def_517, type, sP105: $i > $i > $o).
% 6.58/2.39 thf(func_def_521, type, sK109: $i > $o).
% 6.58/2.39 thf(func_def_542, type, sK130: $i > $o).
% 6.58/2.39 thf(func_def_544, type, sK132: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_545, type, sK133: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_546, type, sK134: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_547, type, sK135: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_548, type, sK136: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_549, type, sK137: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_578, type, sK166: $i > $o).
% 6.58/2.39 thf(func_def_580, type, sK168: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_602, type, sK190: $i > $i).
% 6.58/2.39 thf(func_def_609, type, sK197: $i > $i).
% 6.58/2.39 thf(func_def_622, type, sK210: $i > $o).
% 6.58/2.39 thf(func_def_625, type, sK213: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_626, type, sK214: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_644, type, sK232: $i > $o).
% 6.58/2.39 thf(func_def_650, type, sK238: $i > $o).
% 6.58/2.39 thf(func_def_652, type, sK240: $i > $o).
% 6.58/2.39 thf(func_def_662, type, sK250: $i > $i).
% 6.58/2.39 thf(func_def_664, type, sK252: $i > ($i > $i) > $i > $i).
% 6.58/2.39 thf(func_def_665, type, sK253: $i > ($i > $i) > $i > $i).
% 6.58/2.39 thf(func_def_684, type, sK272: $i > $i).
% 6.58/2.39 thf(func_def_706, type, sK294: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_707, type, sK295: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_713, type, sK301: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_717, type, sK305: $i > $i).
% 6.58/2.39 thf(func_def_718, type, sK306: $i > $i).
% 6.58/2.39 thf(func_def_728, type, sK316: $i > $o).
% 6.58/2.39 thf(func_def_729, type, sK317: $i > $o).
% 6.58/2.39 thf(func_def_730, type, sK318: ($i > $o) > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_731, type, sK319: ($i > $o) > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_743, type, sK331: $i > $i).
% 6.58/2.39 thf(func_def_744, type, sK332: ($i > $i) > $i > $i > $i).
% 6.58/2.39 thf(func_def_746, type, sK334: $i > $i > $o).
% 6.58/2.39 thf(func_def_751, type, sK339: $i > $i > $o).
% 6.58/2.39 thf(func_def_755, type, sK343: $i > $o).
% 6.58/2.39 thf(func_def_756, type, sK344: ($i > $o) > $i).
% 6.58/2.39 thf(func_def_771, type, sK359: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_772, type, sK360: ($i > $o) > $i).
% 6.58/2.39 thf(func_def_773, type, sK361: $i > $o).
% 6.58/2.39 thf(func_def_777, type, sK365: $i > $i > $o).
% 6.58/2.39 thf(func_def_779, type, sK367: $i > ($i > $i > $o) > $i).
% 6.58/2.39 thf(func_def_780, type, sK368: $i > ($i > $i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_781, type, sK369: ($i > $i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_782, type, sK370: ($i > $i > $o) > $i > $i > $i).
% 6.58/2.39 thf(func_def_792, type, sK380: $i > $o).
% 6.58/2.39 thf(func_def_793, type, sK381: $i > $o).
% 6.58/2.39 thf(func_def_794, type, sK382: ($i > $o) > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_795, type, sK383: ($i > $o) > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_796, type, sK384: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_802, type, sK390: $i > $o).
% 6.58/2.39 thf(func_def_809, type, sK397: $i > $i > $i).
% 6.58/2.39 thf(func_def_810, type, sK398: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_811, type, sK399: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_812, type, sK400: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_813, type, sK401: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_824, type, sK412: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_825, type, sK413: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_826, type, sK414: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_827, type, sK415: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_832, type, sK420: ($i > $o) > $i).
% 6.58/2.39 thf(func_def_833, type, sK421: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_834, type, sK422: $i > $o).
% 6.58/2.39 thf(func_def_837, type, sK425: $i > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_838, type, sK426: $i > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_839, type, sK427: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_842, type, sK430: $i > $o).
% 6.58/2.39 thf(func_def_843, type, sK431: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_844, type, sK432: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_877, type, sK465: $i > $i).
% 6.58/2.39 thf(func_def_878, type, sK466: ($i > $i) > $i > $i > $i).
% 6.58/2.39 thf(func_def_881, type, sK469: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_882, type, sK470: ($i > $o) > $i).
% 6.58/2.39 thf(func_def_883, type, sK471: $i > $o).
% 6.58/2.39 thf(func_def_887, type, sK475: $i > $i > $i).
% 6.58/2.39 thf(func_def_890, type, sK478: $i > $i).
% 6.58/2.39 thf(func_def_891, type, sK479: ($i > $i) > $i > $i > $i).
% 6.58/2.39 thf(func_def_894, type, sK482: $i > $i).
% 6.58/2.39 thf(func_def_896, type, sK484: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_912, type, sK500: $i > $i > $o).
% 6.58/2.39 thf(func_def_914, type, sK502: $i > $o).
% 6.58/2.39 thf(func_def_920, type, sK508: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_921, type, sK509: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_922, type, sK510: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_924, type, sK512: $i > $o).
% 6.58/2.39 thf(func_def_927, type, sK515: $i > $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_939, type, sK527: $i > $i > $i).
% 6.58/2.39 thf(func_def_947, type, sK535: $i > $i).
% 6.58/2.39 thf(func_def_948, type, sK536: ($i > $i) > $i > $i > $i).
% 6.58/2.39 thf(func_def_968, type, sK556: $i > $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_971, type, sK559: $i > $o).
% 6.58/2.39 thf(func_def_973, type, sK561: $i > $i).
% 6.58/2.39 thf(func_def_981, type, sK569: $i > $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1005, type, sK593: $i > $i > $i > $o).
% 6.58/2.39 thf(func_def_1006, type, sK594: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1020, type, sK608: $i > ($i > $o) > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1021, type, sK609: $i > ($i > $o) > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1024, type, sK612: $i > $o).
% 6.58/2.39 thf(func_def_1025, type, sK613: $i > $o).
% 6.58/2.39 thf(func_def_1026, type, sK614: $i > $i > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1027, type, sK615: $i > $i > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1028, type, sK616: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1032, type, sK620: $i > $o).
% 6.58/2.39 thf(func_def_1039, type, sK627: $i > $i).
% 6.58/2.39 thf(func_def_1042, type, sK630: $i > $i > ($i > $i) > $i).
% 6.58/2.39 thf(func_def_1048, type, sK636: $i > $i > $i).
% 6.58/2.39 thf(func_def_1077, type, sK665: $i > $i > $i).
% 6.58/2.39 thf(func_def_1078, type, sK666: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1079, type, sK667: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1080, type, sK668: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1081, type, sK669: $i > $o).
% 6.58/2.39 thf(func_def_1083, type, sK671: $i > $o).
% 6.58/2.39 thf(func_def_1094, type, sK682: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1095, type, sK683: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1138, type, sK726: $i > $o).
% 6.58/2.39 thf(func_def_1141, type, sK729: $i > $i).
% 6.58/2.39 thf(func_def_1144, type, sK732: $i > $i > ($i > $i) > $i).
% 6.58/2.39 thf(func_def_1145, type, sK733: ($i > $i) > $i > $i > $i).
% 6.58/2.39 thf(func_def_1146, type, sK734: $i > ($i > $i) > $i > $i > $i).
% 6.58/2.39 thf(func_def_1149, type, sK737: $i > $i > $i).
% 6.58/2.39 thf(func_def_1151, type, sK739: $i > $i).
% 6.58/2.39 thf(func_def_1164, type, sK752: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1169, type, sK757: $o > $i > $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1180, type, sK768: $i > $i).
% 6.58/2.39 thf(func_def_1186, type, sK774: $i > $i > $i).
% 6.58/2.39 thf(func_def_1187, type, sK775: $i > $i > $i).
% 6.58/2.39 thf(func_def_1195, type, sK783: $i > $i).
% 6.58/2.39 thf(func_def_1198, type, sK786: $i > $i > ($i > $i) > $i).
% 6.58/2.39 thf(func_def_1199, type, sK787: $i > $i > ($i > $i) > $i).
% 6.58/2.39 thf(func_def_1206, type, sK794: $i > $i > $o).
% 6.58/2.39 thf(func_def_1214, type, sK802: $i > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1215, type, sK803: $i > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1216, type, sK804: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1219, type, sK807: $i > $o).
% 6.58/2.39 thf(func_def_1224, type, sK812: $i > $o).
% 6.58/2.39 thf(func_def_1225, type, sK813: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1226, type, sK814: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1227, type, sK815: $i > $o).
% 6.58/2.39 thf(func_def_1229, type, sK817: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1230, type, sK818: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1231, type, sK819: $i > $i > $o).
% 6.58/2.39 thf(func_def_1253, type, sK841: $i > $i > $i).
% 6.58/2.39 thf(func_def_1269, type, sK857: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1270, type, sK858: $i > $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1278, type, sK866: $i > ($i > $i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1279, type, sK867: ($i > $i > $o) > $i > $i > $i).
% 6.58/2.39 thf(func_def_1280, type, sK868: $i > $i > ($i > $i > $o) > $i).
% 6.58/2.39 thf(func_def_1281, type, sK869: $i > $i > ($i > $i > $o) > $i).
% 6.58/2.39 thf(func_def_1284, type, sK872: $i > $i > $o).
% 6.58/2.39 thf(func_def_1325, type, sK913: ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1326, type, sK914: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1327, type, sK915: $i > $o).
% 6.58/2.39 thf(func_def_1333, type, sK921: $i > $i > $i).
% 6.58/2.39 thf(func_def_1334, type, sK922: $i > $i > $i).
% 6.58/2.39 thf(func_def_1335, type, sK923: $i > $i > $i).
% 6.58/2.39 thf(func_def_1336, type, sK924: $i > $i > $i).
% 6.58/2.39 thf(func_def_1337, type, sK925: $i > $i > $i).
% 6.58/2.39 thf(func_def_1338, type, sK926: $i > $i).
% 6.58/2.39 thf(func_def_1339, type, sK927: $i > $i).
% 6.58/2.39 thf(func_def_1340, type, sK928: $i > $i).
% 6.58/2.39 thf(func_def_1341, type, sK929: $i > $i).
% 6.58/2.39 thf(func_def_1342, type, sK930: $i > $i > $i).
% 6.58/2.39 thf(func_def_1343, type, sK931: $i > $i > $i).
% 6.58/2.39 thf(func_def_1344, type, sK932: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1345, type, sK933: $i > $i > $i).
% 6.58/2.39 thf(func_def_1346, type, sK934: $i > $i > $i).
% 6.58/2.39 thf(func_def_1347, type, sK935: $i > $i > $i).
% 6.58/2.39 thf(func_def_1348, type, sK936: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1349, type, sK937: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1351, type, sK939: $i > $i).
% 6.58/2.39 thf(func_def_1352, type, sK940: $i > $o > $i > $i).
% 6.58/2.39 thf(func_def_1356, type, sK944: $i > $o).
% 6.58/2.39 thf(func_def_1358, type, sK946: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1359, type, sK947: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1369, type, sK957: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1370, type, sK958: $i > $i > $i > $i).
% 6.58/2.39 thf(func_def_1388, type, sK976: $i > $o).
% 6.58/2.39 thf(func_def_1393, type, sK981: $i > $i).
% 6.58/2.39 thf(func_def_1394, type, sK982: $i > $i).
% 6.58/2.39 thf(func_def_1395, type, sK983: ($i > $i) > $i > $i).
% 6.58/2.39 thf(func_def_1396, type, sK984: $i > ($i > $i) > $i > $i).
% 6.58/2.39 thf(func_def_1423, type, sK1011: $i > $o).
% 6.58/2.39 thf(func_def_1431, type, sK1019: $i > $o).
% 6.58/2.39 thf(func_def_1452, type, sK1040: ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1453, type, sK1041: ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1454, type, sK1042: $i > $o).
% 6.58/2.39 thf(func_def_1472, type, sK1060: $i > $o).
% 6.58/2.39 thf(func_def_1474, type, sK1062: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1475, type, sK1063: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1487, type, sK1075: $i > $o).
% 6.58/2.39 thf(func_def_1488, type, sK1076: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1489, type, sK1077: $i > ($i > $o) > $i).
% 6.58/2.39 thf(func_def_1509, type, sK1097: $i > $o).
% 6.58/2.39 thf(func_def_1511, type, sK1099: $i > $i).
% 6.58/2.39 thf(func_def_1513, type, sK1101: $i > ($i > $i) > $i > $i).
% 6.58/2.39 thf(func_def_1525, type, sK1113: $i > $i > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1526, type, sK1114: $i > $i > ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1527, type, sK1115: ($i > $o) > $i > $i).
% 6.58/2.39 thf(func_def_1531, type, sK1119: $i > $o).
% 6.58/2.39 thf(func_def_1548, type, sK1136: $i > $i > $o).
% 6.58/2.39 thf(func_def_1566, type, sK1154: $i > $i > $o).
% 6.58/2.39 thf(func_def_1571, type, sK1159: $i > $i).
% 6.58/2.39 thf(func_def_1573, type, sK1161: $i > $i > $i).
% 6.58/2.39 thf(func_def_1574, type, sK1162: $i > $i).
% 6.58/2.39 thf(func_def_1592, type, sK1180: $i > $i > $i).
% 6.58/2.39 thf(func_def_1601, type, sK1189: $i > $i > $o).
% 6.58/2.39 thf(func_def_1605, type, ph1193: !>[X0: $tType]:(X0)).
% 6.58/2.39 thf(f9908,plain,(
% 6.58/2.39 $false),
% 6.58/2.39 inference(subsumption_resolution,[status(thm)],[f9907,f6886])).
% 6.58/2.39 thf(f6886,plain,(
% 6.58/2.39 ((transitiveset @ sK963) = $true)),
% 6.58/2.39 inference(binary_proxy_clausification,[status(thm)],[f6885])).
% 6.58/2.39 thf(f6885,plain,(
% 6.58/2.39 (((wellorderedByIn @ sK963) & (transitiveset @ sK963)) = $true)),
% 6.58/2.39 inference(beta_eta_normalization,[status(thm)],[f6533])).
% 6.58/2.39 thf(f6533,plain,(
% 6.58/2.39 (((^[Y0 : $i]: ((wellorderedByIn @ Y0) & (transitiveset @ Y0))) @ sK963) = $true)),
% 6.58/2.39 inference(definition_unfolding,[status(thm)],[f4971,f3811])).
% 6.58/2.39 thf(f3811,plain,(
% 6.58/2.39 (ordinal = (^[Y0 : $i]: ((wellorderedByIn @ Y0) & (transitiveset @ Y0))))),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f563])).
% 6.58/2.39 thf(f563,plain,(
% 6.58/2.39 (ordinal = (^[Y0 : $i]: ((wellorderedByIn @ Y0) & (transitiveset @ Y0))))),
% 6.58/2.39 inference(fool_elimination,[status(thm)],[f562])).
% 6.58/2.39 thf(f562,plain,(
% 6.58/2.39 ((^[X0 : $i] : ((transitiveset @ X0) & (wellorderedByIn @ X0))) = ordinal)),
% 6.58/2.39 inference(rectify,[status(thm)],[f338])).
% 6.58/2.39 thf(f338,axiom,(
% 6.58/2.39 ((^[X1 : $i] : ((transitiveset @ X1) & (wellorderedByIn @ X1))) = ordinal)),
% 6.58/2.39 file('/export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2THF_13596.p',ordinal)).
% 6.58/2.39 thf(f4971,plain,(
% 6.58/2.39 ((ordinal @ sK963) = $true)),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f3188])).
% 6.58/2.39 thf(f3188,plain,(
% 6.58/2.39 (setextsub = $true) & (inComplementUnionImpNotIn1 = $true) & (notequalI1 = $true) & (woz13rule2 = $true) & (setunionsingleton = $true) & (singletonprop = $true) & (theeq = $true) & (breln1all1 = $true) & (emptyE1 = $true) & (upairinpowunion = $true) & (setunionTransitive = $true) & (ex1E1 = $true) & (emptyinPowerset = $true) & (complementTE1 = $true) & (upairsubunion = $true) & (exuI1 = $true) & (ifSingleton = $true) & (breln1invI = $true) & (cartprodmempair = $true) & (image1Ex = $true) & (symdiffIneg1 = $true) & (lamProp = $true) & (replAx = $true) & (ex1E2 = $true) & (notequalI2 = $true) & (binintersectSubset4 = $true) & (image1Equiv = $true) & (complementSubsetComplementIntersect = $true) & (breln1invprop = $true) & (iftrueProp1 = $true) & (binintersectTELcontra = $true) & (breln1SetBreln1 = $true) & (exuE1 = $true) & (woz2A = $true) & (inPowerset = $true) & (inCongP = $true) & (singletonsubset = $true) & (brelnall1 = $true) & (binintersectER = $true) & (app = $true) & (upairset2E = $true) & (emptysetAx = $true) & (cartprodpairin = $true) & (woz1_5 = $true) & (setbeta = $true) & (funcext = $true) & (breln1unionIL = $true) & (woz13rule1 = $true) & (setukpairinjR2 = $true) & (powersetTI1 = $true) & (cartprodpairmemER = $true) & (kpairp = $true) & (doubleComplementSub1 = $true) & (subsetI2 = $true) & (vacuousDall = $true) & (setukpairinjL = $true) & (image1E = $true) & (emptyI = $true) & (woz13rule0 = $true) & (inIntersectImpInUnion2 = $true) & (funcext2 = $true) & (apProp = $true) & (setOfPairsIsBReln1 = $true) & (descrp = $true) & (breln1compE = $true) & (funcGraphProp4 = $true) & (setukpairIL = $true) & (setoftrueEq = $true) & (powersetT_lem = $true) & (funcextLem = $true) & (powersetsubset = $true) & (breln1unionIR = $true) & (emptyinunitempty = $true) & (binunionTIRcontra = $true) & (lam2lamEq = $true) & (setOfPairsIsBReln = $true) & (setunionI = $true) & (doubleComplementEq = $true) & (emptysetimpfalse = $true) & (noeltsimpempty = $true) & (descr__Cong = $true) & (beta2 = $true) & (binintersectTERcontra = $true) & (funcGraphProp3 = $true) & (kpairiskpair = $true) & (setextT = $true) & (breln1compI = $true) & (setunionsingleton1 = $true) & (image1I = $true) & (sepInPowerset = $true) & (quantDeMorgan1 = $true) & (cartprodpairsurjEq = $true) & (exuE2 = $true) & (setadjoinSub = $true) & (powersetE = $true) & (setunionE = $true) & (image1Ex1 = $true) & (emptyset__Cong = $true) & (eta2 = $true) & (breln1unionprop = $true) & (omegaSAx = $true) & (complementTnotintersectT = $true) & (cartprodmempaircEq = $true) & (kfstpairEq = $true) & (lamp = $true) & (dsetconstrI = $true) & (contrasubsetT = $true) & (peano0notS = $true) & (demorgan2b = $true) & (powersetAx = $true) & (surjFuncSetFuncIn = $true) & (binunionEcases = $true) & (nonemptyI = $true) & (binunionTILcontra = $true) & (breln1unionI = $true) & (nonemptyE1 = $true) & (binunionRsub = $true) & (binunionTE = $true) & (cartprodsndpairEq = $true) & (breln1invE = $true) & (transitivesetOp2 = $true) & (transitivesetOp1 = $true) & (funcGraphProp1 = $true) & (setukpairinjL2 = $true) & (iftrue = $true) & (binunionIL = $true) & (binunionE = $true) & (setminusIRneg = $true) & (upairequniteq = $true) & (woz1_1 = $true) & (dpsetconstrER = $true) & (setext = $true) & (notinself = $true) & (sepSubset = $true) & (injFuncSetFuncIn = $true) & (binintersectSubset3 = $true) & (setadjoin__Cong = $true) & (woz1_4 = $true) & (dsetconstrEL = $true) & (omegaIndAx = $true) & (contraSubsetComplement = $true) & (disjointsetsI1 = $true) & (contrasubsetT2 = $true) & (cartprodsndin = $true) & (contrasubsetT1 = $true) & (demorgan2b2 = $true) & (ap2apEq1 = $true) & (subset2powerset = $true) & (symdiffE = $true) & (symdiffIneg2 = $true) & (complementT_lem = $true) & (binintersectLsub = $true) & (woz1_2 = $true) & (iffalseProp1 = $true) & (omegaSp = $true) & (setminusILneg = $true) & (emptysetsubset = $true) & (dpsetconstrEL1 = $true) & (notdexE = $true) & (iftrueorfalse = $true) & (singletonsswitch = $true) & (setminusEL = $true) & (injFuncSetFuncInj = $true) & (singletoninpowerset = $true) & (setminusERneg = $true) & (eqbreln = $true) & (quantDeMorgan3 = $true) & (ordinalMinLem1 = $true) & (setadjoinSub2 = $true) & (lam2p = $true) & (theprop = $true) & (intersectInPowersetIntersectUnions = $true) & (brelnall2 = $true) & (quantDeMorgan4 = $true) & (doubleComplementE1 = $true) & (notinemptyset = $true) & (subsetE2 = $true) & (setunion__Cong = $true) & (contrasubsetT3 = $true) & (setunionsingleton2 = $true) & (breln1compEex = $true) & (binintersectI = $true) & (exuEu = $true) & (complementTI1 = $true) & (setadjoinIL = $true) & (setminusLsub = $true) & (setadjoinE = $true) & (eqimpsubset1 = $true) & (prop2setI = $true) & (powersetE1 = $true) & (upairsetIR = $true) & (doubleComplementSub2 = $true) & (woz2W = $true) & (demorgan2 = $true) & (ksndpairEq = $true) & (exuE3e = $true) & (binintTransitive = $true) & (demorgan1a = $true) & (breln1compprop = $true) & (binunionTEcontra = $true) & (setminusI = $true) & (injFuncInInjFuncSet = $true) & (ubforcartprodlem2 = $true) & (exuI3 = $true) & (setextAx = $true) & (notsubsetI = $true) & (subbreln = $true) & (peanoSinj = $true) & (foundation2 = $true) & (demorgan2a1 = $true) & (emptysetE = $true) & (binintersectSubset2 = $true) & (breln1all2 = $true) & (uniqinunit = $true) & (subsetTrans = $true) & (setminusT_lem = $true) & (subsetTI = $true) & (binintersectRsub = $true) & (iffalse = $true) & (omega0Ax = $true) & (powersetTE1 = $true) & (demorgan1b = $true) & (powerset__Cong = $true) & (subsetRefl = $true) & (binunionIR = $true) & (emptyInPowerset = $true) & (cartprodpairmemEL = $true) & (choice2fnsingleton = $true) & (binintersectSubset1 = $true) & (quantDeMorgan2 = $true) & (exu__Cong = $true) & (subbreln1 = $true) & (ubforcartprodlem3 = $true) & (woz1_3 = $true) & (complementUnionInPowersetComplement = $true) & (nonemptyI1 = $true) & (symdiffI2 = $true) & (demorgan2a = $true) & (cartprodfstin = $true) & (complementTcontraSubset = $true) & (kpairsurjEq = $true) & (dpsetconstrEL2 = $true) & (ex1I = $true) & (subsetE = $true) & (breln1unionE = $true) & (beta1 = $true) & (bs114d = $true) & (leftInvIsSurj = $true) & (binintersectEL = $true) & (omegaSclos = $true) & (binintersectSubset5 = $true) & (breln1unionEcases = $true) & (upairset2IR = $true) & (kfstsingleton = $true) & (foundationAx = $true) & (funcImageSingleton = $true) & (prop2set2propI = $true) & (in__Cong = $true) & (demorgan2a2 = $true) & (ap2apEq2 = $true) & (nonemptyImpWitness = $true) & (dpsetconstrERa = $true) & (funcGraphProp2 = $true) & (exuI2 = $true) & (eqinunit = $true) & (subsetI1 = $true) & (binunionT_lem = $true) & (surjCantorThm = $true) & (woz2Ex = $true) & (setunionAx = $true) & (complementInPowersetComplementIntersect = $true) & (dsetconstrER = $true) & (inComplementUnionImpInComplement1 = $true) & (woz13rule4 = $true) & (((ordinal @ sK963) = $true) & (((in @ sK965 @ sK963) = $true) & ((in @ sK964 @ sK965) = $true) & ($true != (in @ sK964 @ sK963)))) & (subPowSU = $true) & (dsetconstr__Cong = $true) & (symdiffI1 = $true) & (setukpairinjR11 = $true) & (funcinfuncset = $true) & (upairsetE = $true) & (demorgan1 = $true) & (cartprodfstpairEq = $true) & (setminusER = $true) & (setadjoinIR = $true) & (setadjoinOr = $true) & (setminusSubset1 = $true) & (setunionE2 = $true) & (setminusELneg = $true) & (ifp = $true) & (eta1 = $true) & (secondinupair = $true) & (powersetI1 = $true) & (eqimpsubset2 = $true) & (inIntersectImpInIntersectUnions = $true) & (exuE3u = $true) & (setukpairinjR12 = $true) & (binintersectT_lem = $true) & (notinsingleton = $true) & (singletoninpowunion = $true) & (doubleComplementI1 = $true) & (setadjoinAx = $true) & (ksndsingleton = $true) & (setukpairinjL1 = $true) & (breln1unionCommutes = $true) & (infuncsetfunc = $true) & (ap2p = $true) & (dpsetconstrI = $true) & (prop2setE = $true) & (cartprodmempair1 = $true) & (iffalseProp2 = $true) & (ubforcartprodlem1 = $true) & (woz13rule3 = $true) & (setminusSubset2 = $true) & (eqbreln1 = $true) & (notdallE = $true) & (binunionLsub = $true) & (dpsetconstrSub = $true) & (subsetemptysetimpeq = $true) & (inIntersectImpInUnion = $true) & (omega__Cong = $true) & (woz2B = $true) & (setukpairinjR = $true) & (surjFuncSetFuncSurj = $true) & (complementImpComplementIntersect = $true) & (singletonsuniq = $true) & (wellorderingAx = $true) & (setukpairinjR1 = $true) & (ex1I2 = $true) & (iftrueProp2 = $true) & (notinself2 = $true) & (setukpairIR = $true) & (powersetI = $true) & (upairsetIL = $true)),
% 6.58/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK963,sK964,sK965])],[f1311,f3187,f3186])).
% 6.58/2.39 thf(f3186,plain,(
% 6.58/2.39 ? [X0 : $i] : (((ordinal @ X0) = $true) & ? [X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X1 @ X2) = $true) & ((in @ X1 @ X0) != $true))) => (((ordinal @ sK963) = $true) & ? [X2 : $i,X1 : $i] : (($true = (in @ X2 @ sK963)) & ((in @ X1 @ X2) = $true) & ((in @ X1 @ sK963) != $true)))),
% 6.58/2.39 introduced(definition,[],[choice_axiom])).
% 6.58/2.39 thf(f3187,plain,(
% 6.58/2.39 ? [X2 : $i,X1 : $i] : (($true = (in @ X2 @ sK963)) & ((in @ X1 @ X2) = $true) & ((in @ X1 @ sK963) != $true)) => (((in @ sK965 @ sK963) = $true) & ((in @ sK964 @ sK965) = $true) & ($true != (in @ sK964 @ sK963)))),
% 6.58/2.39 introduced(definition,[],[choice_axiom])).
% 6.58/2.39 thf(f1311,plain,(
% 6.58/2.39 (setextsub = $true) & (inComplementUnionImpNotIn1 = $true) & (notequalI1 = $true) & (woz13rule2 = $true) & (setunionsingleton = $true) & (singletonprop = $true) & (theeq = $true) & (breln1all1 = $true) & (emptyE1 = $true) & (upairinpowunion = $true) & (setunionTransitive = $true) & (ex1E1 = $true) & (emptyinPowerset = $true) & (complementTE1 = $true) & (upairsubunion = $true) & (exuI1 = $true) & (ifSingleton = $true) & (breln1invI = $true) & (cartprodmempair = $true) & (image1Ex = $true) & (symdiffIneg1 = $true) & (lamProp = $true) & (replAx = $true) & (ex1E2 = $true) & (notequalI2 = $true) & (binintersectSubset4 = $true) & (image1Equiv = $true) & (complementSubsetComplementIntersect = $true) & (breln1invprop = $true) & (iftrueProp1 = $true) & (binintersectTELcontra = $true) & (breln1SetBreln1 = $true) & (exuE1 = $true) & (woz2A = $true) & (inPowerset = $true) & (inCongP = $true) & (singletonsubset = $true) & (brelnall1 = $true) & (binintersectER = $true) & (app = $true) & (upairset2E = $true) & (emptysetAx = $true) & (cartprodpairin = $true) & (woz1_5 = $true) & (setbeta = $true) & (funcext = $true) & (breln1unionIL = $true) & (woz13rule1 = $true) & (setukpairinjR2 = $true) & (powersetTI1 = $true) & (cartprodpairmemER = $true) & (kpairp = $true) & (doubleComplementSub1 = $true) & (subsetI2 = $true) & (vacuousDall = $true) & (setukpairinjL = $true) & (image1E = $true) & (emptyI = $true) & (woz13rule0 = $true) & (inIntersectImpInUnion2 = $true) & (funcext2 = $true) & (apProp = $true) & (setOfPairsIsBReln1 = $true) & (descrp = $true) & (breln1compE = $true) & (funcGraphProp4 = $true) & (setukpairIL = $true) & (setoftrueEq = $true) & (powersetT_lem = $true) & (funcextLem = $true) & (powersetsubset = $true) & (breln1unionIR = $true) & (emptyinunitempty = $true) & (binunionTIRcontra = $true) & (lam2lamEq = $true) & (setOfPairsIsBReln = $true) & (setunionI = $true) & (doubleComplementEq = $true) & (emptysetimpfalse = $true) & (noeltsimpempty = $true) & (descr__Cong = $true) & (beta2 = $true) & (binintersectTERcontra = $true) & (funcGraphProp3 = $true) & (kpairiskpair = $true) & (setextT = $true) & (breln1compI = $true) & (setunionsingleton1 = $true) & (image1I = $true) & (sepInPowerset = $true) & (quantDeMorgan1 = $true) & (cartprodpairsurjEq = $true) & (exuE2 = $true) & (setadjoinSub = $true) & (powersetE = $true) & (setunionE = $true) & (image1Ex1 = $true) & (emptyset__Cong = $true) & (eta2 = $true) & (breln1unionprop = $true) & (omegaSAx = $true) & (complementTnotintersectT = $true) & (cartprodmempaircEq = $true) & (kfstpairEq = $true) & (lamp = $true) & (dsetconstrI = $true) & (contrasubsetT = $true) & (peano0notS = $true) & (demorgan2b = $true) & (powersetAx = $true) & (surjFuncSetFuncIn = $true) & (binunionEcases = $true) & (nonemptyI = $true) & (binunionTILcontra = $true) & (breln1unionI = $true) & (nonemptyE1 = $true) & (binunionRsub = $true) & (binunionTE = $true) & (cartprodsndpairEq = $true) & (breln1invE = $true) & (transitivesetOp2 = $true) & (transitivesetOp1 = $true) & (funcGraphProp1 = $true) & (setukpairinjL2 = $true) & (iftrue = $true) & (binunionIL = $true) & (binunionE = $true) & (setminusIRneg = $true) & (upairequniteq = $true) & (woz1_1 = $true) & (dpsetconstrER = $true) & (setext = $true) & (notinself = $true) & (sepSubset = $true) & (injFuncSetFuncIn = $true) & (binintersectSubset3 = $true) & (setadjoin__Cong = $true) & (woz1_4 = $true) & (dsetconstrEL = $true) & (omegaIndAx = $true) & (contraSubsetComplement = $true) & (disjointsetsI1 = $true) & (contrasubsetT2 = $true) & (cartprodsndin = $true) & (contrasubsetT1 = $true) & (demorgan2b2 = $true) & (ap2apEq1 = $true) & (subset2powerset = $true) & (symdiffE = $true) & (symdiffIneg2 = $true) & (complementT_lem = $true) & (binintersectLsub = $true) & (woz1_2 = $true) & (iffalseProp1 = $true) & (omegaSp = $true) & (setminusILneg = $true) & (emptysetsubset = $true) & (dpsetconstrEL1 = $true) & (notdexE = $true) & (iftrueorfalse = $true) & (singletonsswitch = $true) & (setminusEL = $true) & (injFuncSetFuncInj = $true) & (singletoninpowerset = $true) & (setminusERneg = $true) & (eqbreln = $true) & (quantDeMorgan3 = $true) & (ordinalMinLem1 = $true) & (setadjoinSub2 = $true) & (lam2p = $true) & (theprop = $true) & (intersectInPowersetIntersectUnions = $true) & (brelnall2 = $true) & (quantDeMorgan4 = $true) & (doubleComplementE1 = $true) & (notinemptyset = $true) & (subsetE2 = $true) & (setunion__Cong = $true) & (contrasubsetT3 = $true) & (setunionsingleton2 = $true) & (breln1compEex = $true) & (binintersectI = $true) & (exuEu = $true) & (complementTI1 = $true) & (setadjoinIL = $true) & (setminusLsub = $true) & (setadjoinE = $true) & (eqimpsubset1 = $true) & (prop2setI = $true) & (powersetE1 = $true) & (upairsetIR = $true) & (doubleComplementSub2 = $true) & (woz2W = $true) & (demorgan2 = $true) & (ksndpairEq = $true) & (exuE3e = $true) & (binintTransitive = $true) & (demorgan1a = $true) & (breln1compprop = $true) & (binunionTEcontra = $true) & (setminusI = $true) & (injFuncInInjFuncSet = $true) & (ubforcartprodlem2 = $true) & (exuI3 = $true) & (setextAx = $true) & (notsubsetI = $true) & (subbreln = $true) & (peanoSinj = $true) & (foundation2 = $true) & (demorgan2a1 = $true) & (emptysetE = $true) & (binintersectSubset2 = $true) & (breln1all2 = $true) & (uniqinunit = $true) & (subsetTrans = $true) & (setminusT_lem = $true) & (subsetTI = $true) & (binintersectRsub = $true) & (iffalse = $true) & (omega0Ax = $true) & (powersetTE1 = $true) & (demorgan1b = $true) & (powerset__Cong = $true) & (subsetRefl = $true) & (binunionIR = $true) & (emptyInPowerset = $true) & (cartprodpairmemEL = $true) & (choice2fnsingleton = $true) & (binintersectSubset1 = $true) & (quantDeMorgan2 = $true) & (exu__Cong = $true) & (subbreln1 = $true) & (ubforcartprodlem3 = $true) & (woz1_3 = $true) & (complementUnionInPowersetComplement = $true) & (nonemptyI1 = $true) & (symdiffI2 = $true) & (demorgan2a = $true) & (cartprodfstin = $true) & (complementTcontraSubset = $true) & (kpairsurjEq = $true) & (dpsetconstrEL2 = $true) & (ex1I = $true) & (subsetE = $true) & (breln1unionE = $true) & (beta1 = $true) & (bs114d = $true) & (leftInvIsSurj = $true) & (binintersectEL = $true) & (omegaSclos = $true) & (binintersectSubset5 = $true) & (breln1unionEcases = $true) & (upairset2IR = $true) & (kfstsingleton = $true) & (foundationAx = $true) & (funcImageSingleton = $true) & (prop2set2propI = $true) & (in__Cong = $true) & (demorgan2a2 = $true) & (ap2apEq2 = $true) & (nonemptyImpWitness = $true) & (dpsetconstrERa = $true) & (funcGraphProp2 = $true) & (exuI2 = $true) & (eqinunit = $true) & (subsetI1 = $true) & (binunionT_lem = $true) & (surjCantorThm = $true) & (woz2Ex = $true) & (setunionAx = $true) & (complementInPowersetComplementIntersect = $true) & (dsetconstrER = $true) & (inComplementUnionImpInComplement1 = $true) & (woz13rule4 = $true) & ? [X0 : $i] : (((ordinal @ X0) = $true) & ? [X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X1 @ X2) = $true) & ((in @ X1 @ X0) != $true))) & (subPowSU = $true) & (dsetconstr__Cong = $true) & (symdiffI1 = $true) & (setukpairinjR11 = $true) & (funcinfuncset = $true) & (upairsetE = $true) & (demorgan1 = $true) & (cartprodfstpairEq = $true) & (setminusER = $true) & (setadjoinIR = $true) & (setadjoinOr = $true) & (setminusSubset1 = $true) & (setunionE2 = $true) & (setminusELneg = $true) & (ifp = $true) & (eta1 = $true) & (secondinupair = $true) & (powersetI1 = $true) & (eqimpsubset2 = $true) & (inIntersectImpInIntersectUnions = $true) & (exuE3u = $true) & (setukpairinjR12 = $true) & (binintersectT_lem = $true) & (notinsingleton = $true) & (singletoninpowunion = $true) & (doubleComplementI1 = $true) & (setadjoinAx = $true) & (ksndsingleton = $true) & (setukpairinjL1 = $true) & (breln1unionCommutes = $true) & (infuncsetfunc = $true) & (ap2p = $true) & (dpsetconstrI = $true) & (prop2setE = $true) & (cartprodmempair1 = $true) & (iffalseProp2 = $true) & (ubforcartprodlem1 = $true) & (woz13rule3 = $true) & (setminusSubset2 = $true) & (eqbreln1 = $true) & (notdallE = $true) & (binunionLsub = $true) & (dpsetconstrSub = $true) & (subsetemptysetimpeq = $true) & (inIntersectImpInUnion = $true) & (omega__Cong = $true) & (woz2B = $true) & (setukpairinjR = $true) & (surjFuncSetFuncSurj = $true) & (complementImpComplementIntersect = $true) & (singletonsuniq = $true) & (wellorderingAx = $true) & (setukpairinjR1 = $true) & (ex1I2 = $true) & (iftrueProp2 = $true) & (notinself2 = $true) & (setukpairIR = $true) & (powersetI = $true) & (upairsetIL = $true)),
% 6.58/2.39 inference(flattening,[status(thm)],[f1310])).
% 6.58/2.39 thf(f1310,plain,(
% 6.58/2.39 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (? [X1 : $i,X2 : $i] : ((((in @ X1 @ X0) != $true) & ((in @ X1 @ X2) = $true)) & ((in @ X2 @ X0) = $true)) & ((ordinal @ X0) = $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)),
% 6.58/2.39 inference(ennf_transformation,[status(thm)],[f363])).
% 6.58/2.39 thf(f363,plain,(
% 6.58/2.39 ~((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) => ! [X0 : $i] : (((ordinal @ X0) = $true) => ! [X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) => (((in @ X1 @ X2) = $true) => ((in @ X1 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.58/2.39 inference(fool_elimination,[status(thm)],[f362])).
% 6.58/2.39 thf(f362,plain,(
% 6.58/2.39 ~(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 => ! [X0 : $i] : ((ordinal @ X0) => ! [X1 : $i,X2 : $i] : ((in @ X2 @ X0) => ((in @ X1 @ X2) => (in @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.58/2.39 inference(rectify,[status(thm)],[f341])).
% 6.58/2.39 thf(f341,negated_conjecture,(
% 6.58/2.39 ~(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 => ! [X11 : $i] : ((ordinal @ X11) => ! [X1 : $i,X3 : $i] : ((in @ X3 @ X11) => ((in @ X1 @ X3) => (in @ X1 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.58/2.39 inference(negated_conjecture,[status(cth)],[f340])).
% 6.58/2.39 thf(f340,conjecture,(
% 6.58/2.39 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 => ! [X11 : $i] : ((ordinal @ X11) => ! [X1 : $i,X3 : $i] : ((in @ X3 @ X11) => ((in @ X1 @ X3) => (in @ X1 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.58/2.39 file('/export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2THF_13596.p',ordinalTransSet)).
% 6.58/2.39 thf(f9907,plain,(
% 6.58/2.39 ((transitiveset @ sK963) != $true)),
% 6.58/2.39 inference(subsumption_resolution,[status(thm)],[f9904,f4970])).
% 6.58/2.39 thf(f4970,plain,(
% 6.58/2.39 ((in @ sK965 @ sK963) = $true)),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f3188])).
% 6.58/2.39 thf(f9904,plain,(
% 6.58/2.39 ((in @ sK965 @ sK963) != $true) | ((transitiveset @ sK963) != $true)),
% 6.58/2.39 inference(trivial_inequality_removal,[status(thm)],[f9898])).
% 6.58/2.39 thf(f9898,plain,(
% 6.58/2.39 ((in @ sK965 @ sK963) != $true) | ((transitiveset @ sK963) != $true) | ($true != $true)),
% 6.58/2.39 inference(superposition,[status(thm)],[f9862,f7025])).
% 6.58/2.39 thf(f7025,plain,(
% 6.58/2.39 ( ! [X2 : $i,X3 : $i] : (((subset @ X3 @ X2) = $true) | ($true != (transitiveset @ X2)) | ((in @ X3 @ X2) != $true)) )),
% 6.58/2.39 inference(trivial_inequality_removal,[status(thm)],[f6767])).
% 6.58/2.39 thf(f6767,plain,(
% 6.58/2.39 ( ! [X2 : $i,X3 : $i] : (($true != (transitiveset @ X2)) | ((in @ X3 @ X2) != $true) | ((subset @ X3 @ X2) = $true) | ($true != $true)) )),
% 6.58/2.39 inference(definition_unfolding,[status(thm)],[f5533,f5124])).
% 6.58/2.39 thf(f5124,plain,(
% 6.58/2.39 (transitivesetOp1 = $true)),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f3188])).
% 6.58/2.39 thf(f5533,plain,(
% 6.58/2.39 ( ! [X2 : $i,X3 : $i] : (((subset @ X3 @ X2) = $true) | ((in @ X3 @ X2) != $true) | ($true != (transitiveset @ X2)) | (transitivesetOp1 != $true)) )),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f3526])).
% 6.58/2.39 thf(f3526,plain,(
% 6.58/2.39 ((transitivesetOp1 = $true) | ((((subset @ sK1174 @ sK1173) != $true) & ((in @ sK1174 @ sK1173) = $true)) & ($true = (transitiveset @ sK1173)))) & (! [X2 : $i] : (! [X3 : $i] : (((subset @ X3 @ X2) = $true) | ((in @ X3 @ X2) != $true)) | ($true != (transitiveset @ X2))) | (transitivesetOp1 != $true))),
% 6.58/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1173,sK1174])],[f3523,f3525,f3524])).
% 6.58/2.39 thf(f3524,plain,(
% 6.58/2.39 ? [X0 : $i] : (? [X1 : $i] : (($true != (subset @ X1 @ X0)) & ((in @ X1 @ X0) = $true)) & ($true = (transitiveset @ X0))) => (? [X1 : $i] : (($true != (subset @ X1 @ sK1173)) & ((in @ X1 @ sK1173) = $true)) & ($true = (transitiveset @ sK1173)))),
% 6.58/2.39 introduced(definition,[],[choice_axiom])).
% 6.58/2.39 thf(f3525,plain,(
% 6.58/2.39 ? [X1 : $i] : (($true != (subset @ X1 @ sK1173)) & ((in @ X1 @ sK1173) = $true)) => (((subset @ sK1174 @ sK1173) != $true) & ((in @ sK1174 @ sK1173) = $true))),
% 6.58/2.39 introduced(definition,[],[choice_axiom])).
% 6.58/2.39 thf(f3523,plain,(
% 6.58/2.39 ((transitivesetOp1 = $true) | ? [X0 : $i] : (? [X1 : $i] : (($true != (subset @ X1 @ X0)) & ((in @ X1 @ X0) = $true)) & ($true = (transitiveset @ X0)))) & (! [X2 : $i] : (! [X3 : $i] : (((subset @ X3 @ X2) = $true) | ((in @ X3 @ X2) != $true)) | ($true != (transitiveset @ X2))) | (transitivesetOp1 != $true))),
% 6.58/2.39 inference(rectify,[status(thm)],[f3522])).
% 6.58/2.39 thf(f3522,plain,(
% 6.58/2.39 ((transitivesetOp1 = $true) | ? [X0 : $i] : (? [X1 : $i] : (($true != (subset @ X1 @ X0)) & ((in @ X1 @ X0) = $true)) & ($true = (transitiveset @ X0)))) & (! [X0 : $i] : (! [X1 : $i] : (($true = (subset @ X1 @ X0)) | ((in @ X1 @ X0) != $true)) | ($true != (transitiveset @ X0))) | (transitivesetOp1 != $true))),
% 6.58/2.39 inference(nnf_transformation,[status(thm)],[f1131])).
% 6.58/2.39 thf(f1131,plain,(
% 6.58/2.39 (transitivesetOp1 = $true) <=> ! [X0 : $i] : (! [X1 : $i] : (($true = (subset @ X1 @ X0)) | ((in @ X1 @ X0) != $true)) | ($true != (transitiveset @ X0)))),
% 6.58/2.39 inference(ennf_transformation,[status(thm)],[f976])).
% 6.58/2.39 thf(f976,plain,(
% 6.58/2.39 (transitivesetOp1 = $true) <=> ! [X0 : $i] : (($true = (transitiveset @ X0)) => ! [X1 : $i] : (((in @ X1 @ X0) = $true) => ($true = (subset @ X1 @ X0))))),
% 6.58/2.39 inference(fool_elimination,[status(thm)],[f975])).
% 6.58/2.39 thf(f975,plain,(
% 6.58/2.39 (transitivesetOp1 = ! [X0 : $i] : ((transitiveset @ X0) => ! [X1 : $i] : ((in @ X1 @ X0) => (subset @ X1 @ X0))))),
% 6.58/2.39 inference(rectify,[status(thm)],[f332])).
% 6.58/2.39 thf(f332,axiom,(
% 6.58/2.39 (transitivesetOp1 = ! [X11 : $i] : ((transitiveset @ X11) => ! [X3 : $i] : ((in @ X3 @ X11) => (subset @ X3 @ X11))))),
% 6.58/2.39 file('/export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2THF_13596.p',transitivesetOp1)).
% 6.58/2.39 thf(f9862,plain,(
% 6.58/2.39 ($true != (subset @ sK965 @ sK963))),
% 6.58/2.39 inference(trivial_inequality_removal,[status(thm)],[f9850])).
% 6.58/2.39 thf(f9850,plain,(
% 6.58/2.39 ($true != $true) | ($true != (subset @ sK965 @ sK963))),
% 6.58/2.39 inference(superposition,[status(thm)],[f8933,f4969])).
% 6.58/2.39 thf(f4969,plain,(
% 6.58/2.39 ((in @ sK964 @ sK965) = $true)),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f3188])).
% 6.58/2.39 thf(f8933,plain,(
% 6.58/2.39 ( ! [X0 : $i] : (($true != (in @ sK964 @ X0)) | ((subset @ X0 @ sK963) != $true)) )),
% 6.58/2.39 inference(trivial_inequality_removal,[status(thm)],[f8926])).
% 6.58/2.39 thf(f8926,plain,(
% 6.58/2.39 ( ! [X0 : $i] : (((subset @ X0 @ sK963) != $true) | ($true != $true) | ($true != (in @ sK964 @ X0))) )),
% 6.58/2.39 inference(superposition,[status(thm)],[f4968,f6841])).
% 6.58/2.39 thf(f6841,plain,(
% 6.58/2.39 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true) | ($true != (subset @ X2 @ X0))) )),
% 6.58/2.39 inference(trivial_inequality_removal,[status(thm)],[f6551])).
% 6.58/2.39 thf(f6551,plain,(
% 6.58/2.39 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != $true) | ((in @ X1 @ X2) != $true) | ($true != (subset @ X2 @ X0)) | ((in @ X1 @ X0) = $true)) )),
% 6.58/2.39 inference(definition_unfolding,[status(thm)],[f5266,f5040])).
% 6.58/2.39 thf(f5040,plain,(
% 6.58/2.39 (notsubsetI = $true)),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f3188])).
% 6.58/2.39 thf(f5266,plain,(
% 6.58/2.39 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (subset @ X2 @ X0)) | ((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true) | (notsubsetI != $true)) )),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f3210])).
% 6.58/2.39 thf(f3210,plain,(
% 6.58/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X2 @ X0)) | ((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true)) | (notsubsetI != $true)) & ((notsubsetI = $true) | (((subset @ sK979 @ sK977) = $true) & ((in @ sK978 @ sK977) != $true) & ((in @ sK978 @ sK979) = $true)))),
% 6.58/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK977,sK978,sK979])],[f3208,f3209])).
% 6.58/2.39 thf(f3209,plain,(
% 6.58/2.39 ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X3) = $true) & ((in @ X4 @ X3) != $true) & ($true = (in @ X4 @ X5))) => (((subset @ sK979 @ sK977) = $true) & ((in @ sK978 @ sK977) != $true) & ((in @ sK978 @ sK979) = $true))),
% 6.58/2.39 introduced(definition,[],[choice_axiom])).
% 6.58/2.39 thf(f3208,plain,(
% 6.58/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X2 @ X0)) | ((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true)) | (notsubsetI != $true)) & ((notsubsetI = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X3) = $true) & ((in @ X4 @ X3) != $true) & ($true = (in @ X4 @ X5))))),
% 6.58/2.39 inference(rectify,[status(thm)],[f3207])).
% 6.58/2.39 thf(f3207,plain,(
% 6.58/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X2 @ X0)) | ((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true)) | (notsubsetI != $true)) & ((notsubsetI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X2 @ X0)) & ((in @ X1 @ X0) != $true) & ((in @ X1 @ X2) = $true)))),
% 6.58/2.39 inference(nnf_transformation,[status(thm)],[f1321])).
% 6.58/2.39 thf(f1321,plain,(
% 6.58/2.39 ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X2 @ X0)) | ((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true)) <=> (notsubsetI = $true)),
% 6.58/2.39 inference(flattening,[status(thm)],[f1320])).
% 6.58/2.39 thf(f1320,plain,(
% 6.58/2.39 (notsubsetI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((($true != (subset @ X2 @ X0)) | ((in @ X1 @ X0) = $true)) | ((in @ X1 @ X2) != $true))),
% 6.58/2.39 inference(ennf_transformation,[status(thm)],[f1040])).
% 6.58/2.39 thf(f1040,plain,(
% 6.58/2.39 (notsubsetI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => (((in @ X1 @ X0) != $true) => ($true != (subset @ X2 @ X0))))),
% 6.58/2.39 inference(flattening,[status(thm)],[f475])).
% 6.58/2.39 thf(f475,plain,(
% 6.58/2.39 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => (~((in @ X1 @ X0) = $true) => ~($true = (subset @ X2 @ X0)))) <=> (notsubsetI = $true)),
% 6.58/2.39 inference(fool_elimination,[status(thm)],[f474])).
% 6.58/2.39 thf(f474,plain,(
% 6.58/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => (~(in @ X1 @ X0) => ~(subset @ X2 @ X0))) = notsubsetI)),
% 6.58/2.39 inference(rectify,[status(thm)],[f85])).
% 6.58/2.39 thf(f85,axiom,(
% 6.58/2.39 (! [X4 : $i,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ X4) => ~(subset @ X3 @ X4))) = notsubsetI)),
% 6.58/2.39 file('/export/starexec/sandbox2/tmp/tmp.SZPDmTeNCX/DTF2THF_13596.p',notsubsetI)).
% 6.58/2.39 thf(f4968,plain,(
% 6.58/2.39 ($true != (in @ sK964 @ sK963))),
% 6.58/2.39 inference(cnf_transformation,[status(thm)],[f3188])).
% 6.58/2.39 % SZS output end Proof for DTF2THF_13596
% 6.58/2.39 % (14035)------------------------------
% 6.58/2.39 % (14035)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.58/2.39 % (14035)Termination reason: Refutation
% 6.58/2.39
% 6.58/2.39 % (14035)Memory used [KB]: 13176
% 6.58/2.39 % (14035)Time elapsed: 0.324 s
% 6.58/2.39 % (14035)Instructions burned: 909 (million)
% 6.58/2.39 % (14035)------------------------------
% 6.58/2.39 % (14035)------------------------------
% 6.58/2.39 % (13899)Success in time 0.67 s
%------------------------------------------------------------------------------