%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU799^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n008.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 09:37:18 AM UTC 2026
% Result : Theorem 7.65s 2.38s
% Output : Refutation 7.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 5
% Syntax : Number of formulae : 32 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 4966 (1324 equ; 0 cnn)
% Maximal formula atoms : 317 ( 155 avg)
% Number of connectives : 2520 ( 28 ~; 17 |; 958 &; 246 @)
% ( 2 <=>;1269 =>; 0 <=; 0 <~>)
% Maximal formula depth : 324 ( 73 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 647 ( 644 usr; 322 con; 0-6 aty)
% Number of variables : 93 ( 30 ^; 45 !; 18 ?; 93 :)
% 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_382,type,
sP1: $i > $o > $i > $i > $o ).
thf(func_def_383,type,
sP2: $i > $i > $i > $o > $o ).
thf(func_def_384,type,
sP3: $i > $o ).
thf(func_def_385,type,
sP4: $i > ( $i > $i ) > $i > $o ).
thf(func_def_387,type,
sP6: ( $i > $o ) > $o ).
thf(func_def_388,type,
sP7: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_389,type,
sP8: $i > ( $i > $i > $o ) > $o ).
thf(func_def_391,type,
sP10: $i > ( $i > $o ) > $o ).
thf(func_def_393,type,
sP12: ( $i > $o ) > $i > $o ).
thf(func_def_395,type,
sP14: ( $i > $o ) > ( $i > $o ) > $i > $i > $o ).
thf(func_def_396,type,
sP15: ( $i > $o ) > $o ).
thf(func_def_397,type,
sP16: $i > ( $i > $o ) > $o ).
thf(func_def_398,type,
sP17: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_399,type,
sP18: ( $i > $o ) > $o ).
thf(func_def_402,type,
sP21: $i > $i > $i > $o ).
thf(func_def_403,type,
sP22: ( $i > $o ) > $o ).
thf(func_def_404,type,
sP23: $i > ( $i > $o ) > $o ).
thf(func_def_406,type,
sP25: $i > $i > $i > $o ).
thf(func_def_409,type,
sP28: $i > $i > $o ).
thf(func_def_411,type,
sP30: $i > $i > $o > $o ).
thf(func_def_412,type,
sP31: $i > $i > $i > $o ).
thf(func_def_416,type,
sP35: $i > $i > ( $i > $o ) > $o ).
thf(func_def_417,type,
sP36: $i > $i > $i > $i > $o ).
thf(func_def_418,type,
sP37: $i > $i > $i > $i > $o ).
thf(func_def_419,type,
sP38: $i > $i > $i > $i > $o ).
thf(func_def_420,type,
sP39: $i > $i > $o ).
thf(func_def_422,type,
sP41: $i > $i > $i > $i > $o ).
thf(func_def_423,type,
sP42: $i > $i > $i > $o ).
thf(func_def_425,type,
sP44: $i > $i > $i > $o ).
thf(func_def_426,type,
sP45: $i > $i > $i > $o ).
thf(func_def_427,type,
sP46: $i > $i > $i > $o ).
thf(func_def_429,type,
sP48: ( $i > $o ) > $o ).
thf(func_def_430,type,
sP49: $i > $i > $i > $o ).
thf(func_def_432,type,
sP51: ( $i > $o ) > $i > $o ).
thf(func_def_434,type,
sP53: $o > $i > $i > $o ).
thf(func_def_436,type,
sP55: $i > $i > ( $i > $i ) > $o ).
thf(func_def_438,type,
sP57: $i > $i > $i > $o ).
thf(func_def_439,type,
sP58: $i > $i > $i > $o ).
thf(func_def_440,type,
sP59: $i > ( $i > $o ) > $o ).
thf(func_def_441,type,
sP60: $i > ( $i > $o ) > $i > $o ).
thf(func_def_442,type,
sP61: ( $i > $o ) > $i > $o ).
thf(func_def_443,type,
sP62: $i > ( $i > $o ) > $o ).
thf(func_def_444,type,
sP63: $i > ( $i > $o ) > $o ).
thf(func_def_446,type,
sP65: ( $i > $o ) > $i > $o ).
thf(func_def_447,type,
sP66: ( $i > $o ) > $i > $o ).
thf(func_def_448,type,
sP67: $i > $i > $i > $o > $o ).
thf(func_def_449,type,
sP68: $i > $i > $i > $i > $o ).
thf(func_def_451,type,
sP70: $i > $i > $i > $o ).
thf(func_def_452,type,
sP71: $i > ( $i > $o ) > $o ).
thf(func_def_453,type,
sP72: $i > ( $i > $o ) > $i > $o ).
thf(func_def_454,type,
sP73: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_455,type,
sP74: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_456,type,
sP75: $i > $i > $o ).
thf(func_def_459,type,
sP78: $i > ( $i > $o ) > $o ).
thf(func_def_461,type,
sP80: $i > ( $i > $o ) > $o ).
thf(func_def_462,type,
sP81: $i > $i > $i > ( $i > $o ) > $o ).
thf(func_def_463,type,
sP82: $o > $i > $i > $o ).
thf(func_def_464,type,
sP83: $i > $i > $i > $o ).
thf(func_def_465,type,
sP84: ( $i > $i ) > $i > $i > $o ).
thf(func_def_467,type,
sP86: $i > $i > $i > $i > $i > $o ).
thf(func_def_468,type,
sP87: ( $i > $i ) > $i > $i > $o ).
thf(func_def_470,type,
sP89: $i > $i > $i > $o ).
thf(func_def_471,type,
sP90: $i > $i > $o ).
thf(func_def_472,type,
sP91: $i > $o ).
thf(func_def_473,type,
sP92: $i > $i > $o ).
thf(func_def_474,type,
sP93: $i > $i > $o ).
thf(func_def_475,type,
sP94: $i > $i > $o ).
thf(func_def_476,type,
sP95: $i > $o > $i > $i > $i > $i > $o ).
thf(func_def_477,type,
sP96: $i > $i > $i > $o ).
thf(func_def_478,type,
sP97: $i > $o ).
thf(func_def_479,type,
sP98: $i > $o ).
thf(func_def_480,type,
sP99: $o > $i > $i > $o ).
thf(func_def_481,type,
sP100: $i > $i > $i > $o ).
thf(func_def_483,type,
sP102: $i > $i > $i > $i > $o ).
thf(func_def_493,type,
sK112: $i > $i > $i ).
thf(func_def_522,type,
sK141: $i > $o ).
thf(func_def_530,type,
sK149: $i > $o ).
thf(func_def_537,type,
sK156: $i > $i > $i ).
thf(func_def_538,type,
sK157: $i > $i ).
thf(func_def_541,type,
sK160: $i > $o ).
thf(func_def_543,type,
sK162: $i > $i > $o ).
thf(func_def_551,type,
sK170: $i > $o ).
thf(func_def_563,type,
sK182: $i > $o ).
thf(func_def_566,type,
sK185: $i > $o ).
thf(func_def_569,type,
sK188: $i > $o ).
thf(func_def_573,type,
sK192: $i > $i > $o ).
thf(func_def_580,type,
sK199: $i > $i ).
thf(func_def_581,type,
sK200: ( $i > $i ) > $i > $i > $i ).
thf(func_def_582,type,
sK201: $i > ( $i > $i ) > $i > $i ).
thf(func_def_592,type,
sK211: ( $i > $o ) > $i ).
thf(func_def_593,type,
sK212: $i > ( $i > $o ) > $i ).
thf(func_def_594,type,
sK213: $i > $o ).
thf(func_def_598,type,
sK217: $i > $i > $o ).
thf(func_def_600,type,
sK219: $i > ( $i > $i > $o ) > $i ).
thf(func_def_601,type,
sK220: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_602,type,
sK221: $i > ( $i > $i > $o ) > $i ).
thf(func_def_603,type,
sK222: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_608,type,
sK227: $i > $o ).
thf(func_def_610,type,
sK229: $i > ( $i > $o ) > $i ).
thf(func_def_611,type,
sK230: $i > ( $i > $o ) > $i ).
thf(func_def_617,type,
sK236: $i > $o ).
thf(func_def_619,type,
sK238: $i > ( $i > $o ) > $i ).
thf(func_def_620,type,
sK239: ( $i > $o ) > $i > $i ).
thf(func_def_626,type,
sK245: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_627,type,
sK246: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_630,type,
sK249: $i > $o ).
thf(func_def_631,type,
sK250: $i > $o ).
thf(func_def_642,type,
sK261: ( $i > $o ) > $i ).
thf(func_def_643,type,
sK262: ( $i > $o ) > $i ).
thf(func_def_644,type,
sK263: $i > $o ).
thf(func_def_646,type,
sK265: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_647,type,
sK266: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_648,type,
sK267: $i > ( $i > $o ) > $i ).
thf(func_def_652,type,
sK271: $i > $o ).
thf(func_def_662,type,
sK281: ( $i > $o ) > $i ).
thf(func_def_663,type,
sK282: $i > ( $i > $o ) > $i ).
thf(func_def_664,type,
sK283: $i > $o ).
thf(func_def_668,type,
sK287: $i > $o ).
thf(func_def_675,type,
sK294: $i > $i ).
thf(func_def_676,type,
sK295: $i > $i ).
thf(func_def_687,type,
sK306: $i > $o ).
thf(func_def_688,type,
sK307: ( $i > $o ) > $i > $i ).
thf(func_def_691,type,
sK310: $i > $i > $o ).
thf(func_def_699,type,
sK318: $i > $i > $i ).
thf(func_def_701,type,
sK320: $i > $i > $i > $i ).
thf(func_def_702,type,
sK321: $i > $i > $i > $i > $i ).
thf(func_def_706,type,
sK325: $i > ( $i > $o ) > $i ).
thf(func_def_707,type,
sK326: ( $i > $o ) > $i ).
thf(func_def_708,type,
sK327: $i > $o ).
thf(func_def_721,type,
sK340: $i > $o ).
thf(func_def_723,type,
sK342: $i > ( $i > $o ) > $i ).
thf(func_def_724,type,
sK343: $i > ( $i > $o ) > $i ).
thf(func_def_760,type,
sK379: $i > $i ).
thf(func_def_766,type,
sK385: $i > $i ).
thf(func_def_767,type,
sK386: ( $i > $i ) > $i > $i > $i ).
thf(func_def_778,type,
sK397: $i > $i > $i ).
thf(func_def_779,type,
sK398: $i > $i > $i ).
thf(func_def_785,type,
sK404: $i > $i > $i ).
thf(func_def_795,type,
sK414: $i > $i > $o ).
thf(func_def_797,type,
sK416: $i > $i > $i > $o ).
thf(func_def_817,type,
sK436: $i > $o ).
thf(func_def_818,type,
sK437: $i > $o ).
thf(func_def_819,type,
sK438: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_820,type,
sK439: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_838,type,
sK457: $i > $i ).
thf(func_def_840,type,
sK459: $i > ( $i > $i ) > $i > $i ).
thf(func_def_860,type,
sK479: $i > $i > $i ).
thf(func_def_868,type,
sK487: $i > $i > ( $i > $o ) > $i ).
thf(func_def_869,type,
sK488: $i > $o ).
thf(func_def_872,type,
sK491: $i > $i > $i > $i > $i ).
thf(func_def_873,type,
sK492: $i > $i > $i > $i > $i ).
thf(func_def_874,type,
sK493: $i > $i > $i > $i > $i ).
thf(func_def_875,type,
sK494: $i > $i > $i > $i > $i ).
thf(func_def_880,type,
sK499: $i > $i > $i > $i > $i ).
thf(func_def_881,type,
sK500: $i > $i > $i > $i > $i ).
thf(func_def_888,type,
sK507: $i > $i > $i ).
thf(func_def_889,type,
sK508: $i > $i > $i > $i ).
thf(func_def_890,type,
sK509: $i > $i > $i > $i > $i ).
thf(func_def_891,type,
sK510: $i > $i > $i > $i > $i ).
thf(func_def_892,type,
sK511: $i > $i > $i > $i > $i ).
thf(func_def_897,type,
sK516: $i > $i ).
thf(func_def_899,type,
sK518: $i > ( $i > $i ) > $i > $i ).
thf(func_def_957,type,
sK576: $i > $o ).
thf(func_def_958,type,
sK577: $i > $i > $i > $i ).
thf(func_def_959,type,
sK578: $i > $i > $i > $i > $i ).
thf(func_def_969,type,
sK588: $i > $i > $i > $i ).
thf(func_def_970,type,
sK589: $i > $i > $i > $i ).
thf(func_def_976,type,
sK595: $i > $i > $i ).
thf(func_def_985,type,
sK604: $i > $o ).
thf(func_def_995,type,
sK614: $i > ( $i > $o ) > $i ).
thf(func_def_996,type,
sK615: ( $i > $o ) > $i ).
thf(func_def_997,type,
sK616: $i > $o ).
thf(func_def_1006,type,
sK625: $i > $i > $i > $i ).
thf(func_def_1007,type,
sK626: $i > $i > $i > $i ).
thf(func_def_1011,type,
sK630: $i > $i ).
thf(func_def_1014,type,
sK633: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1018,type,
sK637: $i > $i > $o ).
thf(func_def_1052,type,
sK671: $i > $o ).
thf(func_def_1053,type,
sK672: ( $i > $o ) > $i > $i ).
thf(func_def_1054,type,
sK673: ( $i > $o ) > $i > $i ).
thf(func_def_1076,type,
sK695: $i > $i ).
thf(func_def_1079,type,
sK698: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1080,type,
sK699: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1086,type,
sK705: $i > $i > $o ).
thf(func_def_1101,type,
sK720: $i > $i > $i > $i ).
thf(func_def_1102,type,
sK721: $i > $i > $i > $i ).
thf(func_def_1103,type,
sK722: $i > $i > $i > $i ).
thf(func_def_1104,type,
sK723: $i > $i > $i > $i ).
thf(func_def_1112,type,
sK731: $i > $o ).
thf(func_def_1115,type,
sK734: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1116,type,
sK735: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1117,type,
sK736: $i > ( $i > $o ) > $i ).
thf(func_def_1120,type,
sK739: $i > $o ).
thf(func_def_1130,type,
sK749: $i > $i > $o ).
thf(func_def_1140,type,
sK759: $i > ( $i > $o ) > $i ).
thf(func_def_1141,type,
sK760: $i > ( $i > $o ) > $i ).
thf(func_def_1142,type,
sK761: ( $i > $o ) > $i > $i ).
thf(func_def_1144,type,
sK763: $i > $o ).
thf(func_def_1149,type,
sK768: $i > $o ).
thf(func_def_1150,type,
sK769: ( $i > $o ) > $i > $i ).
thf(func_def_1151,type,
sK770: $i > ( $i > $o ) > $i ).
thf(func_def_1167,type,
sK786: $i > $i > $o ).
thf(func_def_1176,type,
sK795: ( $i > $o ) > $i > $i ).
thf(func_def_1177,type,
sK796: ( $i > $o ) > $i > $i ).
thf(func_def_1178,type,
sK797: ( $i > $o ) > $i > $i ).
thf(func_def_1179,type,
sK798: $i > $o ).
thf(func_def_1181,type,
sK800: $i > $o ).
thf(func_def_1187,type,
sK806: $i > $i > $i > $i > $o ).
thf(func_def_1195,type,
sK814: $i > $o ).
thf(func_def_1196,type,
sK815: $i > $o ).
thf(func_def_1197,type,
sK816: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1198,type,
sK817: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1199,type,
sK818: $i > $i > $i > $i ).
thf(func_def_1234,type,
sK853: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1235,type,
sK854: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1236,type,
sK855: $i > ( $i > $o ) > $i ).
thf(func_def_1239,type,
sK858: $i > $o ).
thf(func_def_1246,type,
sK865: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1247,type,
sK866: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1248,type,
sK867: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_1249,type,
sK868: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_1250,type,
sK869: $i > $i > $o ).
thf(func_def_1274,type,
sK893: $i > $i ).
thf(func_def_1297,type,
sK916: $i > $i > $i ).
thf(func_def_1303,type,
sK922: $i > $i > $i ).
thf(func_def_1305,type,
sK924: $i > $o ).
thf(func_def_1306,type,
sK925: ( $i > $o ) > $i > $i ).
thf(func_def_1307,type,
sK926: $i > ( $i > $o ) > $i ).
thf(func_def_1312,type,
sK931: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1313,type,
sK932: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1314,type,
sK933: $i > ( $i > $o ) > $i ).
thf(func_def_1318,type,
sK937: $i > $o ).
thf(func_def_1322,type,
sK941: $i > $i ).
thf(func_def_1323,type,
sK942: $i > $i > $i > $o ).
thf(func_def_1324,type,
sK943: $i > $i > $i > $i ).
thf(func_def_1353,type,
sK972: $i > $i ).
thf(func_def_1359,type,
sK978: $i > $i ).
thf(func_def_1360,type,
sK979: $i > $i ).
thf(func_def_1361,type,
sK980: ( $i > $i ) > $i > $i ).
thf(func_def_1362,type,
sK981: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1385,type,
sK1004: $i > $i > $i > $i > $i > $i ).
thf(func_def_1392,type,
sK1011: $i > $i ).
thf(func_def_1394,type,
sK1013: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1395,type,
sK1014: $i > $i > $i ).
thf(func_def_1396,type,
sK1015: $i > $i > $i ).
thf(func_def_1397,type,
sK1016: $i > $i > $i ).
thf(func_def_1398,type,
sK1017: $i > $i > $i ).
thf(func_def_1399,type,
sK1018: $i > $i > $i ).
thf(func_def_1400,type,
sK1019: $i > $i > $i ).
thf(func_def_1401,type,
sK1020: $i > $i > $i ).
thf(func_def_1402,type,
sK1021: $i > $i > $i > $i ).
thf(func_def_1403,type,
sK1022: $i > $i ).
thf(func_def_1404,type,
sK1023: $i > $i ).
thf(func_def_1405,type,
sK1024: $i > $i ).
thf(func_def_1406,type,
sK1025: $i > $i ).
thf(func_def_1407,type,
sK1026: $i > $i > $i ).
thf(func_def_1408,type,
sK1027: $i > $i > $i ).
thf(func_def_1409,type,
sK1028: $i > $i > $i ).
thf(func_def_1410,type,
sK1029: $i > $i > $i > $i ).
thf(func_def_1411,type,
sK1030: $i > $i > $i > $i ).
thf(func_def_1413,type,
sK1032: $i > $i ).
thf(func_def_1420,type,
sK1039: $i > $o > $i > $i > $i > $i > $i ).
thf(func_def_1438,type,
sK1057: $i > $i > $i > $i ).
thf(func_def_1445,type,
sK1064: $i > $i ).
thf(func_def_1446,type,
sK1065: $i > $i ).
thf(func_def_1453,type,
sK1072: $o > $i > $i > $i ).
thf(func_def_1457,type,
sK1076: $i > $i > $i > $i ).
thf(func_def_1458,type,
sK1077: $i > $i > $i > $i ).
thf(func_def_1462,type,
sK1081: $i > $o ).
thf(func_def_1486,type,
sK1105: $i > $i ).
thf(func_def_1488,type,
sK1107: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1489,type,
sK1108: ( $i > $o ) > $i ).
thf(func_def_1490,type,
sK1109: $i > $o ).
thf(func_def_1508,type,
sK1127: $i > $i > $i > $i > $i ).
thf(func_def_1529,type,
ph1148:
!>[X0: $tType] : X0 ).
thf(f9344,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f9333,f4563]) ).
thf(f4563,plain,
( $true
!= ( in @ sK702 @ ( funcSet @ sK700 @ sK701 ) ) ),
inference(cnf_transformation,[status(thm)],[f2603]) ).
thf(f2603,plain,
( ( setukpairinjL1 = $true )
& ( vacuousDall = $true )
& ( dsetconstrER = $true )
& ( demorgan2b = $true )
& ( funcext = $true )
& ( breln1invprop = $true )
& ( symdiffIneg2 = $true )
& ( setminusLsub = $true )
& ( setadjoinIL = $true )
& ( nonemptyI = $true )
& ( cartprodpairmemEL = $true )
& ( exuE1 = $true )
& ( setminusSubset2 = $true )
& ( binintersectER = $true )
& ( breln1unionCommutes = $true )
& ( cartprodmempaircEq = $true )
& ( demorgan2b2 = $true )
& ( contrasubsetT1 = $true )
& ( image1Equiv = $true )
& ( doubleComplementE1 = $true )
& ( lamp = $true )
& ( wellorderingAx = $true )
& ( symdiffE = $true )
& ( setbeta = $true )
& ( breln1unionE = $true )
& ( binintersectSubset3 = $true )
& ( notequalI1 = $true )
& ( setukpairinjR = $true )
& ( complementTnotintersectT = $true )
& ( singletonsuniq = $true )
& ( breln1unionI = $true )
& ( funcext2 = $true )
& ( brelnall2 = $true )
& ( kfstpairEq = $true )
& ( setunion__Cong = $true )
& ( apProp = $true )
& ( infuncsetfunc = $true )
& ( $true
!= ( in @ sK702 @ ( funcSet @ sK700 @ sK701 ) ) )
& ( $true
= ( in @ sK702 @ ( injFuncSet @ sK700 @ sK701 ) ) )
& ( quantDeMorgan2 = $true )
& ( omegaIndAx = $true )
& ( setoftrueEq = $true )
& ( singletonsubset = $true )
& ( setunionE = $true )
& ( emptysetsubset = $true )
& ( subbreln1 = $true )
& ( powersetI = $true )
& ( contrasubsetT = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrI = $true )
& ( contrasubsetT3 = $true )
& ( emptyinPowerset = $true )
& ( subPowSU = $true )
& ( breln1unionIL = $true )
& ( in__Cong = $true )
& ( binintersectLsub = $true )
& ( emptyInPowerset = $true )
& ( setextAx = $true )
& ( singletonsswitch = $true )
& ( exuE3e = $true )
& ( woz2B = $true )
& ( complementImpComplementIntersect = $true )
& ( iffalse = $true )
& ( beta2 = $true )
& ( breln1invE = $true )
& ( ifSingleton = $true )
& ( setadjoinE = $true )
& ( funcGraphProp4 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( symdiffIneg1 = $true )
& ( iftrueProp1 = $true )
& ( upairsubunion = $true )
& ( exuI3 = $true )
& ( setminusER = $true )
& ( emptysetAx = $true )
& ( ifp = $true )
& ( setukpairIL = $true )
& ( bs114d = $true )
& ( omega0Ax = $true )
& ( nonemptyI1 = $true )
& ( demorgan2a1 = $true )
& ( lam2lamEq = $true )
& ( powersetT_lem = $true )
& ( complementT_lem = $true )
& ( kpairp = $true )
& ( exuE3u = $true )
& ( notinsingleton = $true )
& ( eqimpsubset1 = $true )
& ( breln1all1 = $true )
& ( woz13rule4 = $true )
& ( kpairiskpair = $true )
& ( ex1I = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR2 = $true )
& ( exuI1 = $true )
& ( binunionTIRcontra = $true )
& ( upairset2IR = $true )
& ( lam2p = $true )
& ( binunionTE = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( setminusSubset1 = $true )
& ( setextsub = $true )
& ( dsetconstrEL = $true )
& ( setOfPairsIsBReln1 = $true )
& ( setunionsingleton1 = $true )
& ( binunionRsub = $true )
& ( binunionIR = $true )
& ( dsetconstrI = $true )
& ( prop2setE = $true )
& ( demorgan2a = $true )
& ( powerset__Cong = $true )
& ( exuEu = $true )
& ( choice2fnsingleton = $true )
& ( binintersectEL = $true )
& ( breln1compI = $true )
& ( symdiffI1 = $true )
& ( secondinupair = $true )
& ( woz1_5 = $true )
& ( complementTI1 = $true )
& ( cartprodfstin = $true )
& ( upairsetE = $true )
& ( inIntersectImpInUnion2 = $true )
& ( descrp = $true )
& ( setminusELneg = $true )
& ( doubleComplementEq = $true )
& ( dpsetconstrEL1 = $true )
& ( eta1 = $true )
& ( eta2 = $true )
& ( eqimpsubset2 = $true )
& ( quantDeMorgan3 = $true )
& ( funcImageSingleton = $true )
& ( subsetemptysetimpeq = $true )
& ( nonemptyE1 = $true )
& ( prop2setI = $true )
& ( binintersectSubset4 = $true )
& ( woz1_2 = $true )
& ( binunionEcases = $true )
& ( ksndpairEq = $true )
& ( subsetTrans = $true )
& ( cartprodmempair = $true )
& ( woz13rule0 = $true )
& ( setadjoinSub2 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( setminusERneg = $true )
& ( singletoninpowunion = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( sepInPowerset = $true )
& ( funcGraphProp3 = $true )
& ( omegaSAx = $true )
& ( prop2set2propI = $true )
& ( disjointsetsI1 = $true )
& ( ubforcartprodlem3 = $true )
& ( app = $true )
& ( powersetsubset = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( ex1E2 = $true )
& ( emptysetimpfalse = $true )
& ( setukpairinjL = $true )
& ( kpairsurjEq = $true )
& ( dpsetconstrSub = $true )
& ( binintersectRsub = $true )
& ( woz13rule1 = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( breln1compprop = $true )
& ( woz2Ex = $true )
& ( demorgan2 = $true )
& ( subsetI1 = $true )
& ( cartprodfstpairEq = $true )
& ( setextT = $true )
& ( quantDeMorgan4 = $true )
& ( contraSubsetComplement = $true )
& ( demorgan1b = $true )
& ( doubleComplementSub2 = $true )
& ( demorgan1 = $true )
& ( setukpairinjL2 = $true )
& ( setunionsingleton2 = $true )
& ( quantDeMorgan1 = $true )
& ( dsetconstr__Cong = $true )
& ( iftrue = $true )
& ( binunionTEcontra = $true )
& ( setunionE2 = $true )
& ( binintersectTERcontra = $true )
& ( upairsetIR = $true )
& ( ex1I2 = $true )
& ( complementTcontraSubset = $true )
& ( ex1E1 = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectSubset5 = $true )
& ( image1I = $true )
& ( cartprodpairin = $true )
& ( upairset2E = $true )
& ( ksndsingleton = $true )
& ( upairinpowunion = $true )
& ( binintersectT_lem = $true )
& ( dpsetconstrEL2 = $true )
& ( setminusT_lem = $true )
& ( woz1_1 = $true )
& ( breln1unionEcases = $true )
& ( iffalseProp2 = $true )
& ( sepSubset = $true )
& ( singletoninpowerset = $true )
& ( notinemptyset = $true )
& ( subsetI2 = $true )
& ( upairsetIL = $true )
& ( descr__Cong = $true )
& ( binintersectTELcontra = $true )
& ( subsetE = $true )
& ( breln1SetBreln1 = $true )
& ( binintersectSubset2 = $true )
& ( emptyset__Cong = $true )
& ( binunionTILcontra = $true )
& ( setminusIRneg = $true )
& ( setunionAx = $true )
& ( setadjoinSub = $true )
& ( eqbreln1 = $true )
& ( eqbreln = $true )
& ( image1Ex1 = $true )
& ( setadjoinAx = $true )
& ( ubforcartprodlem1 = $true )
& ( setminusEL = $true )
& ( subsetRefl = $true )
& ( theeq = $true )
& ( notdexE = $true )
& ( woz1_3 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( powersetTE1 = $true )
& ( image1Ex = $true )
& ( subsetTI = $true )
& ( inCongP = $true )
& ( funcGraphProp1 = $true )
& ( funcGraphProp2 = $true )
& ( cartprodpairmemER = $true )
& ( omega__Cong = $true )
& ( setunionI = $true )
& ( complementUnionInPowersetComplement = $true )
& ( inIntersectImpInUnion = $true )
& ( brelnall1 = $true )
& ( woz2A = $true )
& ( uniqinunit = $true )
& ( contrasubsetT2 = $true )
& ( noeltsimpempty = $true )
& ( subbreln = $true )
& ( cartprodmempair1 = $true )
& ( setext = $true )
& ( singletonprop = $true )
& ( powersetE1 = $true )
& ( breln1unionprop = $true )
& ( breln1unionIR = $true )
& ( binunionIL = $true )
& ( cartprodsndpairEq = $true )
& ( exuE2 = $true )
& ( woz13rule2 = $true )
& ( emptyE1 = $true )
& ( theprop = $true )
& ( setukpairinjR11 = $true )
& ( emptysetE = $true )
& ( setukpairIR = $true )
& ( emptyinunitempty = $true )
& ( notdallE = $true )
& ( ap2apEq1 = $true )
& ( exu__Cong = $true )
& ( setunionsingleton = $true )
& ( emptyI = $true )
& ( subsetE2 = $true )
& ( setminusILneg = $true )
& ( ap2p = $true )
& ( subset2powerset = $true )
& ( ap2apEq2 = $true )
& ( setukpairinjR1 = $true )
& ( powersetE = $true )
& ( image1E = $true )
& ( powersetAx = $true )
& ( binintersectI = $true )
& ( eqinunit = $true )
& ( iftrueProp2 = $true )
& ( dpsetconstrER = $true )
& ( iftrueorfalse = $true )
& ( dpsetconstrERa = $true )
& ( ubforcartprodlem2 = $true )
& ( lamProp = $true )
& ( breln1all2 = $true )
& ( upairequniteq = $true )
& ( complementSubsetComplementIntersect = $true )
& ( woz1_4 = $true )
& ( powersetI1 = $true )
& ( funcextLem = $true )
& ( symdiffI2 = $true )
& ( inPowerset = $true )
& ( woz13rule3 = $true )
& ( replAx = $true )
& ( kfstsingleton = $true )
& ( setadjoin__Cong = $true )
& ( funcinfuncset = $true )
& ( setadjoinIR = $true )
& ( breln1compEex = $true )
& ( injFuncInInjFuncSet = $true )
& ( setminusI = $true )
& ( beta1 = $true )
& ( breln1invI = $true )
& ( notequalI2 = $true )
& ( powersetTI1 = $true )
& ( nonemptyImpWitness = $true )
& ( breln1compE = $true )
& ( setadjoinOr = $true )
& ( iffalseProp1 = $true )
& ( exuI2 = $true )
& ( doubleComplementSub1 = $true )
& ( notsubsetI = $true )
& ( complementTE1 = $true )
& ( binintersectSubset1 = $true )
& ( foundationAx = $true )
& ( woz2W = $true )
& ( doubleComplementI1 = $true )
& ( binunionT_lem = $true )
& ( cartprodsndin = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK700,sK701,sK702])],[f1388,f2602]) ).
thf(f2602,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ( $true
= ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) ) ) )
=> ( ( $true
!= ( in @ sK702 @ ( funcSet @ sK700 @ sK701 ) ) )
& ( $true
= ( in @ sK702 @ ( injFuncSet @ sK700 @ sK701 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1388,plain,
( ( setukpairinjL1 = $true )
& ( vacuousDall = $true )
& ( dsetconstrER = $true )
& ( demorgan2b = $true )
& ( funcext = $true )
& ( breln1invprop = $true )
& ( symdiffIneg2 = $true )
& ( setminusLsub = $true )
& ( setadjoinIL = $true )
& ( nonemptyI = $true )
& ( cartprodpairmemEL = $true )
& ( exuE1 = $true )
& ( setminusSubset2 = $true )
& ( binintersectER = $true )
& ( breln1unionCommutes = $true )
& ( cartprodmempaircEq = $true )
& ( demorgan2b2 = $true )
& ( contrasubsetT1 = $true )
& ( image1Equiv = $true )
& ( doubleComplementE1 = $true )
& ( lamp = $true )
& ( wellorderingAx = $true )
& ( symdiffE = $true )
& ( setbeta = $true )
& ( breln1unionE = $true )
& ( binintersectSubset3 = $true )
& ( notequalI1 = $true )
& ( setukpairinjR = $true )
& ( complementTnotintersectT = $true )
& ( singletonsuniq = $true )
& ( breln1unionI = $true )
& ( funcext2 = $true )
& ( brelnall2 = $true )
& ( kfstpairEq = $true )
& ( setunion__Cong = $true )
& ( apProp = $true )
& ( infuncsetfunc = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ( $true
= ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) ) ) )
& ( quantDeMorgan2 = $true )
& ( omegaIndAx = $true )
& ( setoftrueEq = $true )
& ( singletonsubset = $true )
& ( setunionE = $true )
& ( emptysetsubset = $true )
& ( subbreln1 = $true )
& ( powersetI = $true )
& ( contrasubsetT = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrI = $true )
& ( contrasubsetT3 = $true )
& ( emptyinPowerset = $true )
& ( subPowSU = $true )
& ( breln1unionIL = $true )
& ( in__Cong = $true )
& ( binintersectLsub = $true )
& ( emptyInPowerset = $true )
& ( setextAx = $true )
& ( singletonsswitch = $true )
& ( exuE3e = $true )
& ( woz2B = $true )
& ( complementImpComplementIntersect = $true )
& ( iffalse = $true )
& ( beta2 = $true )
& ( breln1invE = $true )
& ( ifSingleton = $true )
& ( setadjoinE = $true )
& ( funcGraphProp4 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( symdiffIneg1 = $true )
& ( iftrueProp1 = $true )
& ( upairsubunion = $true )
& ( exuI3 = $true )
& ( setminusER = $true )
& ( emptysetAx = $true )
& ( ifp = $true )
& ( setukpairIL = $true )
& ( bs114d = $true )
& ( omega0Ax = $true )
& ( nonemptyI1 = $true )
& ( demorgan2a1 = $true )
& ( lam2lamEq = $true )
& ( powersetT_lem = $true )
& ( complementT_lem = $true )
& ( kpairp = $true )
& ( exuE3u = $true )
& ( notinsingleton = $true )
& ( eqimpsubset1 = $true )
& ( breln1all1 = $true )
& ( woz13rule4 = $true )
& ( kpairiskpair = $true )
& ( ex1I = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR2 = $true )
& ( exuI1 = $true )
& ( binunionTIRcontra = $true )
& ( upairset2IR = $true )
& ( lam2p = $true )
& ( binunionTE = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( setminusSubset1 = $true )
& ( setextsub = $true )
& ( dsetconstrEL = $true )
& ( setOfPairsIsBReln1 = $true )
& ( setunionsingleton1 = $true )
& ( binunionRsub = $true )
& ( binunionIR = $true )
& ( dsetconstrI = $true )
& ( prop2setE = $true )
& ( demorgan2a = $true )
& ( powerset__Cong = $true )
& ( exuEu = $true )
& ( choice2fnsingleton = $true )
& ( binintersectEL = $true )
& ( breln1compI = $true )
& ( symdiffI1 = $true )
& ( secondinupair = $true )
& ( woz1_5 = $true )
& ( complementTI1 = $true )
& ( cartprodfstin = $true )
& ( upairsetE = $true )
& ( inIntersectImpInUnion2 = $true )
& ( descrp = $true )
& ( setminusELneg = $true )
& ( doubleComplementEq = $true )
& ( dpsetconstrEL1 = $true )
& ( eta1 = $true )
& ( eta2 = $true )
& ( eqimpsubset2 = $true )
& ( quantDeMorgan3 = $true )
& ( funcImageSingleton = $true )
& ( subsetemptysetimpeq = $true )
& ( nonemptyE1 = $true )
& ( prop2setI = $true )
& ( binintersectSubset4 = $true )
& ( woz1_2 = $true )
& ( binunionEcases = $true )
& ( ksndpairEq = $true )
& ( subsetTrans = $true )
& ( cartprodmempair = $true )
& ( woz13rule0 = $true )
& ( setadjoinSub2 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( setminusERneg = $true )
& ( singletoninpowunion = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( sepInPowerset = $true )
& ( funcGraphProp3 = $true )
& ( omegaSAx = $true )
& ( prop2set2propI = $true )
& ( disjointsetsI1 = $true )
& ( ubforcartprodlem3 = $true )
& ( app = $true )
& ( powersetsubset = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( ex1E2 = $true )
& ( emptysetimpfalse = $true )
& ( setukpairinjL = $true )
& ( kpairsurjEq = $true )
& ( dpsetconstrSub = $true )
& ( binintersectRsub = $true )
& ( woz13rule1 = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( breln1compprop = $true )
& ( woz2Ex = $true )
& ( demorgan2 = $true )
& ( subsetI1 = $true )
& ( cartprodfstpairEq = $true )
& ( setextT = $true )
& ( quantDeMorgan4 = $true )
& ( contraSubsetComplement = $true )
& ( demorgan1b = $true )
& ( doubleComplementSub2 = $true )
& ( demorgan1 = $true )
& ( setukpairinjL2 = $true )
& ( setunionsingleton2 = $true )
& ( quantDeMorgan1 = $true )
& ( dsetconstr__Cong = $true )
& ( iftrue = $true )
& ( binunionTEcontra = $true )
& ( setunionE2 = $true )
& ( binintersectTERcontra = $true )
& ( upairsetIR = $true )
& ( ex1I2 = $true )
& ( complementTcontraSubset = $true )
& ( ex1E1 = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectSubset5 = $true )
& ( image1I = $true )
& ( cartprodpairin = $true )
& ( upairset2E = $true )
& ( ksndsingleton = $true )
& ( upairinpowunion = $true )
& ( binintersectT_lem = $true )
& ( dpsetconstrEL2 = $true )
& ( setminusT_lem = $true )
& ( woz1_1 = $true )
& ( breln1unionEcases = $true )
& ( iffalseProp2 = $true )
& ( sepSubset = $true )
& ( singletoninpowerset = $true )
& ( notinemptyset = $true )
& ( subsetI2 = $true )
& ( upairsetIL = $true )
& ( descr__Cong = $true )
& ( binintersectTELcontra = $true )
& ( subsetE = $true )
& ( breln1SetBreln1 = $true )
& ( binintersectSubset2 = $true )
& ( emptyset__Cong = $true )
& ( binunionTILcontra = $true )
& ( setminusIRneg = $true )
& ( setunionAx = $true )
& ( setadjoinSub = $true )
& ( eqbreln1 = $true )
& ( eqbreln = $true )
& ( image1Ex1 = $true )
& ( setadjoinAx = $true )
& ( ubforcartprodlem1 = $true )
& ( setminusEL = $true )
& ( subsetRefl = $true )
& ( theeq = $true )
& ( notdexE = $true )
& ( woz1_3 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( powersetTE1 = $true )
& ( image1Ex = $true )
& ( subsetTI = $true )
& ( inCongP = $true )
& ( funcGraphProp1 = $true )
& ( funcGraphProp2 = $true )
& ( cartprodpairmemER = $true )
& ( omega__Cong = $true )
& ( setunionI = $true )
& ( complementUnionInPowersetComplement = $true )
& ( inIntersectImpInUnion = $true )
& ( brelnall1 = $true )
& ( woz2A = $true )
& ( uniqinunit = $true )
& ( contrasubsetT2 = $true )
& ( noeltsimpempty = $true )
& ( subbreln = $true )
& ( cartprodmempair1 = $true )
& ( setext = $true )
& ( singletonprop = $true )
& ( powersetE1 = $true )
& ( breln1unionprop = $true )
& ( breln1unionIR = $true )
& ( binunionIL = $true )
& ( cartprodsndpairEq = $true )
& ( exuE2 = $true )
& ( woz13rule2 = $true )
& ( emptyE1 = $true )
& ( theprop = $true )
& ( setukpairinjR11 = $true )
& ( emptysetE = $true )
& ( setukpairIR = $true )
& ( emptyinunitempty = $true )
& ( notdallE = $true )
& ( ap2apEq1 = $true )
& ( exu__Cong = $true )
& ( setunionsingleton = $true )
& ( emptyI = $true )
& ( subsetE2 = $true )
& ( setminusILneg = $true )
& ( ap2p = $true )
& ( subset2powerset = $true )
& ( ap2apEq2 = $true )
& ( setukpairinjR1 = $true )
& ( powersetE = $true )
& ( image1E = $true )
& ( powersetAx = $true )
& ( binintersectI = $true )
& ( eqinunit = $true )
& ( iftrueProp2 = $true )
& ( dpsetconstrER = $true )
& ( iftrueorfalse = $true )
& ( dpsetconstrERa = $true )
& ( ubforcartprodlem2 = $true )
& ( lamProp = $true )
& ( breln1all2 = $true )
& ( upairequniteq = $true )
& ( complementSubsetComplementIntersect = $true )
& ( woz1_4 = $true )
& ( powersetI1 = $true )
& ( funcextLem = $true )
& ( symdiffI2 = $true )
& ( inPowerset = $true )
& ( woz13rule3 = $true )
& ( replAx = $true )
& ( kfstsingleton = $true )
& ( setadjoin__Cong = $true )
& ( funcinfuncset = $true )
& ( setadjoinIR = $true )
& ( breln1compEex = $true )
& ( injFuncInInjFuncSet = $true )
& ( setminusI = $true )
& ( beta1 = $true )
& ( breln1invI = $true )
& ( notequalI2 = $true )
& ( powersetTI1 = $true )
& ( nonemptyImpWitness = $true )
& ( breln1compE = $true )
& ( setadjoinOr = $true )
& ( iffalseProp1 = $true )
& ( exuI2 = $true )
& ( doubleComplementSub1 = $true )
& ( notsubsetI = $true )
& ( complementTE1 = $true )
& ( binintersectSubset1 = $true )
& ( foundationAx = $true )
& ( woz2W = $true )
& ( doubleComplementI1 = $true )
& ( binunionT_lem = $true )
& ( cartprodsndin = $true ) ),
inference(flattening,[status(thm)],[f1387]) ).
thf(f1387,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ( $true
= ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) ) ) )
& ( 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)],[f365]) ).
thf(f365,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) ) )
=> ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f364]) ).
thf(f364,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) )
=> ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f320]) ).
thf(f320,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
=> ! [X3: $i,X4: $i,X12: $i] :
( ( in @ X12 @ ( injFuncSet @ X3 @ X4 ) )
=> ( in @ X12 @ ( funcSet @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f319]) ).
thf(f319,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
=> ! [X3: $i,X4: $i,X12: $i] :
( ( in @ X12 @ ( injFuncSet @ X3 @ X4 ) )
=> ( in @ X12 @ ( funcSet @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.an1PBkiMQH/DTF2THF_18125.p',injFuncSetFuncIn) ).
thf(f9333,plain,
( $true
= ( in @ sK702 @ ( funcSet @ sK700 @ sK701 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f9321]) ).
thf(f9321,plain,
( ( $true != $true )
| ( $true
= ( in @ sK702 @ ( funcSet @ sK700 @ sK701 ) ) ) ),
inference(superposition,[status(thm)],[f6742,f6541]) ).
thf(f6541,plain,
( $true
= ( in @ sK702 @ ( dsetconstr @ ( funcSet @ sK700 @ sK701 ) @ ( injective @ sK700 @ sK701 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f5964]) ).
thf(f5964,plain,
( $true
= ( in @ sK702
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) )
@ sK700
@ sK701 ) ) ),
inference(definition_unfolding,[status(thm)],[f4562,f3862]) ).
thf(f3862,plain,
( injFuncSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f618]) ).
thf(f618,plain,
( injFuncSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f617]) ).
thf(f617,plain,
( injFuncSet
= ( ^ [X0: $i,X1: $i] :
( dsetconstr @ ( funcSet @ X0 @ X1 )
@ ^ [X2: $i] : ( injective @ X0 @ X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f317]) ).
thf(f317,axiom,
( injFuncSet
= ( ^ [X3: $i,X4: $i] :
( dsetconstr @ ( funcSet @ X3 @ X4 )
@ ^ [X12: $i] : ( injective @ X3 @ X4 @ X12 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.an1PBkiMQH/DTF2THF_18125.p',injFuncSet) ).
thf(f4562,plain,
( $true
= ( in @ sK702 @ ( injFuncSet @ sK700 @ sK701 ) ) ),
inference(cnf_transformation,[status(thm)],[f2603]) ).
thf(f6742,plain,
! [X2: $i > $o,X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( dsetconstr @ X0 @ X2 ) ) )
| ( $true
= ( in @ X1 @ X0 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f6741]) ).
thf(f6741,plain,
! [X2: $i > $o,X0: $i,X1: $i] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( $true
= ( in @ X1 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5824]) ).
thf(f5824,plain,
! [X2: $i > $o,X0: $i,X1: $i] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( $true
= ( in @ X1 @ X0 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f4093,f4497]) ).
thf(f4497,plain,
dsetconstrEL = $true,
inference(cnf_transformation,[status(thm)],[f2603]) ).
thf(f4093,plain,
! [X2: $i > $o,X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( dsetconstrEL != $true ) ),
inference(cnf_transformation,[status(thm)],[f2378]) ).
thf(f2378,plain,
( ( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) ) )
| ( dsetconstrEL != $true ) )
& ( ( dsetconstrEL = $true )
| ( ( $true
!= ( in @ sK575 @ sK574 ) )
& ( $true
= ( in @ sK575
@ ( dsetconstr @ sK574
@ ^ [Y0: $i] : ( sK576 @ Y0 ) ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK574,sK575,sK576])],[f2376,f2377]) ).
thf(f2377,plain,
( ? [X3: $i,X4: $i,X5: $i > $o] :
( ( ( in @ X4 @ X3 )
!= $true )
& ( $true
= ( in @ X4
@ ( dsetconstr @ X3
@ ^ [Y0: $i] : ( X5 @ Y0 ) ) ) ) )
=> ( ( $true
!= ( in @ sK575 @ sK574 ) )
& ( $true
= ( in @ sK575
@ ( dsetconstr @ sK574
@ ^ [Y0: $i] : ( sK576 @ Y0 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2376,plain,
( ( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) ) )
| ( dsetconstrEL != $true ) )
& ( ( dsetconstrEL = $true )
| ? [X3: $i,X4: $i,X5: $i > $o] :
( ( ( in @ X4 @ X3 )
!= $true )
& ( $true
= ( in @ X4
@ ( dsetconstr @ X3
@ ^ [Y0: $i] : ( X5 @ Y0 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2375]) ).
thf(f2375,plain,
( ( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) ) )
| ( dsetconstrEL != $true ) )
& ( ( dsetconstrEL = $true )
| ? [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( $true
= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1154]) ).
thf(f1154,plain,
( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) ) )
<=> ( dsetconstrEL = $true ) ),
inference(ennf_transformation,[status(thm)],[f682]) ).
thf(f682,plain,
( ( dsetconstrEL = $true )
<=> ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
=> ( $true
= ( in @ X1 @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f681]) ).
thf(f681,plain,
! [X0: $i,X1: $i,X2: $i > $o] :
( ( ( in @ X1
@ ( dsetconstr @ X0
@ ^ [X3: $i] : ( X2 @ X3 ) ) )
=> ( in @ X1 @ X0 ) )
= dsetconstrEL ),
inference(rectify,[status(thm)],[f15]) ).
thf(f15,axiom,
! [X3: $i,X1: $i,X0: $i > $o] :
( ( ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) )
=> ( in @ X1 @ X3 ) )
= dsetconstrEL ),
file('/export/starexec/sandbox/tmp/tmp.an1PBkiMQH/DTF2THF_18125.p',dsetconstrEL) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU799^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n008.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:34:24 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.45 thf(spec,logic,$$dhol).
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % File : SEU799^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.45 % Domain : Set Theory
% 0.30/0.45 % Problem : More about Functions - Injective Functions
% 0.30/0.45 % Version : Especial.
% 0.30/0.45 % English : (! A:i.! B:i.! f:i.in f (injFuncSet A B) -> in f (funcSet A B))
% 0.30/0.45
% 0.30/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.45 % Source : [Bro08]
% 0.30/0.45 % Names : ZFC301g [Bro08]
% 0.30/0.45
% 0.30/0.45 % Status : Theorem
% 0.30/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.43 v6.1.0, 0.71 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.30/0.45 % Syntax : Number of formulae : 685 ( 318 unt; 366 typ; 318 def)
% 0.30/0.45 % Number of atoms : 1961 ( 458 equ; 0 cnn)
% 0.30/0.45 % Maximal formula atoms : 317 ( 6 avg)
% 0.30/0.45 % Number of connectives : 4317 ( 73 ~; 14 |; 53 &;3144 @)
% 0.30/0.45 % ( 17 <=>;1016 =>; 0 <=; 0 <~>)
% 0.30/0.45 % Maximal formula depth : 323 ( 2 avg)
% 0.30/0.45 % Number of types : 2 ( 0 usr)
% 0.30/0.45 % Number of type conns : 170 ( 170 >; 0 *; 0 +; 0 <<)
% 0.30/0.45 % Number of symbols : 369 ( 366 usr; 319 con; 0-4 aty)
% 0.30/0.45 % Number of variables : 1127 ( 78 ^;1006 !; 43 ?;1127 :)
% 0.30/0.45 % SPC : TH0_THM_EQU_NAR
% 0.30/0.45
% 0.30/0.45 % Comments : http://mathgate.info/detsetitem.php?id=424
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 thf(in_type,type,
% 0.30/0.45 in: $i > $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(exu_type,type,
% 0.30/0.45 exu: ( $i > $o ) > $o ).
% 0.30/0.45
% 0.30/0.45 thf(exu,definition,
% 0.30/0.45 ( exu
% 0.30/0.45 = ( ^ [Xphi: $i > $o] :
% 0.30/0.45 ? [Xx: $i] :
% 0.30/0.45 ( ( Xphi @ Xx )
% 0.30/0.45 & ! [Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setextAx_type,type,
% 0.30/0.45 setextAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setextAx,definition,
% 0.30/0.45 ( setextAx
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 <=> ( in @ Xx @ B ) )
% 0.30/0.45 => ( A = B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyset_type,type,
% 0.30/0.45 emptyset: $i ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetAx_type,type,
% 0.30/0.45 emptysetAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetAx,definition,
% 0.30/0.45 ( emptysetAx
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoin_type,type,
% 0.30/0.45 setadjoin: $i > $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinAx_type,type,
% 0.30/0.45 setadjoinAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinAx,definition,
% 0.30/0.45 ( setadjoinAx
% 0.30/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.45 <=> ( ( Xy = Xx )
% 0.30/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(powerset_type,type,
% 0.30/0.45 powerset: $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(powersetAx_type,type,
% 0.30/0.45 powersetAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(powersetAx,definition,
% 0.30/0.45 ( powersetAx
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.45 <=> ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setunion_type,type,
% 0.30/0.45 setunion: $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(setunionAx_type,type,
% 0.30/0.45 setunionAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setunionAx,definition,
% 0.30/0.45 ( setunionAx
% 0.30/0.45 = ( ! [A: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.45 <=> ? [B: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(omega_type,type,
% 0.30/0.45 omega: $i ).
% 0.30/0.45
% 0.30/0.45 thf(omega0Ax_type,type,
% 0.30/0.45 omega0Ax: $o ).
% 0.30/0.45
% 0.30/0.45 thf(omega0Ax,definition,
% 0.30/0.45 ( omega0Ax
% 0.30/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(omegaSAx_type,type,
% 0.30/0.45 omegaSAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(omegaSAx,definition,
% 0.30/0.45 ( omegaSAx
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ omega )
% 0.30/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(omegaIndAx_type,type,
% 0.30/0.45 omegaIndAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(omegaIndAx,definition,
% 0.30/0.45 ( omegaIndAx
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ( ( in @ emptyset @ A )
% 0.30/0.45 & ! [Xx: $i] :
% 0.30/0.45 ( ( ( in @ Xx @ omega )
% 0.30/0.45 & ( in @ Xx @ A ) )
% 0.30/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.45 => ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ omega )
% 0.30/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(replAx_type,type,
% 0.30/0.45 replAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(replAx,definition,
% 0.30/0.45 ( replAx
% 0.30/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( exu
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.45 => ? [B: $i] :
% 0.30/0.45 ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 <=> ? [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ A )
% 0.30/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(foundationAx_type,type,
% 0.30/0.45 foundationAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(foundationAx,definition,
% 0.30/0.45 ( foundationAx
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.45 => ? [B: $i] :
% 0.30/0.45 ( ( in @ B @ A )
% 0.30/0.45 & ~ ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(wellorderingAx_type,type,
% 0.30/0.45 wellorderingAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(wellorderingAx,definition,
% 0.30/0.45 ( wellorderingAx
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ? [B: $i] :
% 0.30/0.45 ( ! [C: $i] :
% 0.30/0.45 ( ( in @ C @ B )
% 0.30/0.45 => ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ C )
% 0.30/0.45 => ( in @ Xx @ A ) ) )
% 0.30/0.45 & ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( ( in @ Xx @ A )
% 0.30/0.45 & ( in @ Xy @ A ) )
% 0.30/0.45 => ( ! [C: $i] :
% 0.30/0.45 ( ( in @ C @ B )
% 0.30/0.45 => ( ( in @ Xx @ C )
% 0.30/0.45 <=> ( in @ Xy @ C ) ) )
% 0.30/0.45 => ( Xx = Xy ) ) )
% 0.30/0.45 & ! [C: $i,D: $i] :
% 0.30/0.45 ( ( ( in @ C @ B )
% 0.30/0.45 & ( in @ D @ B ) )
% 0.30/0.45 => ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ C )
% 0.30/0.45 => ( in @ Xx @ D ) )
% 0.30/0.45 | ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ D )
% 0.30/0.45 => ( in @ Xx @ C ) ) ) )
% 0.30/0.45 & ! [C: $i] :
% 0.30/0.45 ( ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ C )
% 0.30/0.45 => ( in @ Xx @ A ) )
% 0.30/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.45 => ? [D: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ D @ B )
% 0.30/0.45 & ( in @ Xx @ C )
% 0.30/0.45 & ~ ? [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ D )
% 0.30/0.45 & ( in @ Xy @ C ) )
% 0.30/0.45 & ! [E: $i] :
% 0.30/0.45 ( ( in @ E @ B )
% 0.30/0.45 => ( ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ E )
% 0.30/0.45 => ( in @ Xy @ D ) )
% 0.30/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(descr_type,type,
% 0.30/0.45 descr: ( $i > $o ) > $i ).
% 0.30/0.45
% 0.30/0.45 thf(descrp_type,type,
% 0.30/0.45 descrp: $o ).
% 0.30/0.45
% 0.30/0.45 thf(descrp,definition,
% 0.30/0.45 ( descrp
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ( Xphi
% 0.30/0.45 @ ( descr
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstr_type,type,
% 0.30/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrI_type,type,
% 0.30/0.45 dsetconstrI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrI,definition,
% 0.30/0.45 ( dsetconstrI
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( ( Xphi @ Xx )
% 0.30/0.45 => ( in @ Xx
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrEL_type,type,
% 0.30/0.45 dsetconstrEL: $o ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrEL,definition,
% 0.30/0.45 ( dsetconstrEL
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrER_type,type,
% 0.30/0.45 dsetconstrER: $o ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrER,definition,
% 0.30/0.45 ( dsetconstrER
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuE1_type,type,
% 0.30/0.45 exuE1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuE1,definition,
% 0.30/0.45 ( exuE1
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ( ( Xphi @ Xx )
% 0.30/0.45 & ! [Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(prop2set_type,type,
% 0.30/0.45 prop2set: $o > $i ).
% 0.30/0.45
% 0.30/0.45 thf(prop2setE_type,type,
% 0.30/0.45 prop2setE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(prop2setE,definition,
% 0.30/0.45 ( prop2setE
% 0.30/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.45 => Xphi ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetE_type,type,
% 0.30/0.45 emptysetE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetE,definition,
% 0.30/0.45 ( emptysetE
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ emptyset )
% 0.30/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetimpfalse_type,type,
% 0.30/0.45 emptysetimpfalse: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetimpfalse,definition,
% 0.30/0.45 ( emptysetimpfalse
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ emptyset )
% 0.30/0.45 => $false ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(notinemptyset_type,type,
% 0.30/0.45 notinemptyset: $o ).
% 0.30/0.45
% 0.30/0.45 thf(notinemptyset,definition,
% 0.30/0.45 ( notinemptyset
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuE3e_type,type,
% 0.30/0.45 exuE3e: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuE3e,definition,
% 0.30/0.45 ( exuE3e
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setext_type,type,
% 0.30/0.45 setext: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setext,definition,
% 0.30/0.45 ( setext
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( in @ Xx @ B ) )
% 0.30/0.45 => ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 => ( in @ Xx @ A ) )
% 0.30/0.45 => ( A = B ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyI_type,type,
% 0.30/0.45 emptyI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptyI,definition,
% 0.30/0.45 ( emptyI
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ~ ( in @ Xx @ A )
% 0.30/0.45 => ( A = emptyset ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(noeltsimpempty_type,type,
% 0.30/0.45 noeltsimpempty: $o ).
% 0.30/0.45
% 0.30/0.45 thf(noeltsimpempty,definition,
% 0.30/0.45 ( noeltsimpempty
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ~ ( in @ Xx @ A )
% 0.30/0.45 => ( A = emptyset ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setbeta_type,type,
% 0.30/0.45 setbeta: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setbeta,definition,
% 0.30/0.45 ( setbeta
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( ( in @ Xx
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(nonempty_type,type,
% 0.30/0.45 nonempty: $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(nonempty,definition,
% 0.30/0.45 ( nonempty
% 0.30/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyE1_type,type,
% 0.30/0.45 nonemptyE1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyE1,definition,
% 0.30/0.45 ( nonemptyE1
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ( nonempty @ A )
% 0.30/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyI_type,type,
% 0.30/0.45 nonemptyI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyI,definition,
% 0.30/0.45 ( nonemptyI
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( ( Xphi @ Xx )
% 0.30/0.45 => ( nonempty
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyI1_type,type,
% 0.30/0.45 nonemptyI1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyI1,definition,
% 0.30/0.45 ( nonemptyI1
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinIL_type,type,
% 0.30/0.45 setadjoinIL: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinIL,definition,
% 0.30/0.45 ( setadjoinIL
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyinunitempty_type,type,
% 0.30/0.45 emptyinunitempty: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptyinunitempty,definition,
% 0.30/0.45 ( emptyinunitempty
% 0.30/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinIR_type,type,
% 0.30/0.45 setadjoinIR: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinIR,definition,
% 0.30/0.45 ( setadjoinIR
% 0.30/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ A )
% 0.30/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinE_type,type,
% 0.30/0.45 setadjoinE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinE,definition,
% 0.30/0.45 ( setadjoinE
% 0.30/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.45 => ! [Xphi: $o] :
% 0.30/0.45 ( ( ( Xy = Xx )
% 0.30/0.45 => Xphi )
% 0.30/0.45 => ( ( ( in @ Xy @ A )
% 0.30/0.45 => Xphi )
% 0.30/0.45 => Xphi ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinOr_type,type,
% 0.30/0.45 setadjoinOr: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinOr,definition,
% 0.30/0.45 ( setadjoinOr
% 0.30/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.45 => ( ( Xy = Xx )
% 0.30/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setoftrueEq_type,type,
% 0.30/0.45 setoftrueEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setoftrueEq,definition,
% 0.30/0.45 ( setoftrueEq
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xx: $i] : $true )
% 0.30/0.45 = A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(powersetI_type,type,
% 0.30/0.45 powersetI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(powersetI,definition,
% 0.30/0.45 ( powersetI
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 => ( in @ Xx @ A ) )
% 0.30/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyinPowerset_type,type,
% 0.30/0.45 emptyinPowerset: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptyinPowerset,definition,
% 0.30/0.45 ( emptyinPowerset
% 0.30/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyInPowerset_type,type,
% 0.30/0.45 emptyInPowerset: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptyInPowerset,definition,
% 0.30/0.45 ( emptyInPowerset
% 0.30/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(powersetE_type,type,
% 0.30/0.45 powersetE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(powersetE,definition,
% 0.30/0.45 ( powersetE
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.45 => ( ( in @ Xx @ B )
% 0.30/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setunionI_type,type,
% 0.30/0.45 setunionI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setunionI,definition,
% 0.30/0.45 ( setunionI
% 0.30/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 => ( ( in @ B @ A )
% 0.30/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setunionE_type,type,
% 0.30/0.45 setunionE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setunionE,definition,
% 0.30/0.45 ( setunionE
% 0.30/0.45 = ( ! [A: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.45 => ! [Xphi: $o] :
% 0.30/0.45 ( ! [B: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 => ( ( in @ B @ A )
% 0.30/0.45 => Xphi ) )
% 0.30/0.45 => Xphi ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(subPowSU_type,type,
% 0.30/0.45 subPowSU: $o ).
% 0.30/0.45
% 0.30/0.45 thf(subPowSU,definition,
% 0.30/0.45 ( subPowSU
% 0.30/0.45 = ( ! [A: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuE2_type,type,
% 0.30/0.45 exuE2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuE2,definition,
% 0.30/0.45 ( exuE2
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ! [Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xy )
% 0.30/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyImpWitness_type,type,
% 0.30/0.45 nonemptyImpWitness: $o ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyImpWitness,definition,
% 0.30/0.45 ( nonemptyImpWitness
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ( nonempty @ A )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & $true ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(uniqinunit_type,type,
% 0.30/0.45 uniqinunit: $o ).
% 0.30/0.45
% 0.30/0.45 thf(uniqinunit,definition,
% 0.30/0.45 ( uniqinunit
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(notinsingleton_type,type,
% 0.30/0.45 notinsingleton: $o ).
% 0.30/0.45
% 0.30/0.45 thf(notinsingleton,definition,
% 0.30/0.45 ( notinsingleton
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xx != Xy )
% 0.30/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(eqinunit_type,type,
% 0.30/0.45 eqinunit: $o ).
% 0.30/0.45
% 0.30/0.45 thf(eqinunit,definition,
% 0.30/0.45 ( eqinunit
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xx = Xy )
% 0.30/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(singletonsswitch_type,type,
% 0.30/0.45 singletonsswitch: $o ).
% 0.30/0.45
% 0.30/0.45 thf(singletonsswitch,definition,
% 0.30/0.45 ( singletonsswitch
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(upairsetE_type,type,
% 0.30/0.45 upairsetE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(upairsetE,definition,
% 0.30/0.45 ( upairsetE
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.45 => ( ( Xz = Xx )
% 0.30/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(upairsetIL_type,type,
% 0.30/0.45 upairsetIL: $o ).
% 0.30/0.45
% 0.30/0.45 thf(upairsetIL,definition,
% 0.30/0.45 ( upairsetIL
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(upairsetIR_type,type,
% 0.30/0.45 upairsetIR: $o ).
% 0.30/0.45
% 0.30/0.45 thf(upairsetIR,definition,
% 0.30/0.45 ( upairsetIR
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyE1_type,type,
% 0.30/0.45 emptyE1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptyE1,definition,
% 0.30/0.45 ( emptyE1
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ( Xphi @ Xx ) )
% 0.30/0.45 => ( ( ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 = emptyset )
% 0.30/0.45 => $false ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(vacuousDall_type,type,
% 0.30/0.45 vacuousDall: $o ).
% 0.30/0.45
% 0.30/0.45 thf(vacuousDall,definition,
% 0.30/0.45 ( vacuousDall
% 0.30/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ emptyset )
% 0.30/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(quantDeMorgan1_type,type,
% 0.30/0.45 quantDeMorgan1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(quantDeMorgan1,definition,
% 0.30/0.45 ( quantDeMorgan1
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ~ ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( Xphi @ Xx ) )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(quantDeMorgan2_type,type,
% 0.30/0.45 quantDeMorgan2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(quantDeMorgan2,definition,
% 0.30/0.45 ( quantDeMorgan2
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ~ ( Xphi @ Xx ) )
% 0.30/0.45 => ~ ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(quantDeMorgan3_type,type,
% 0.30/0.45 quantDeMorgan3: $o ).
% 0.30/0.45
% 0.30/0.45 thf(quantDeMorgan3,definition,
% 0.30/0.45 ( quantDeMorgan3
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ~ ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ( Xphi @ Xx ) )
% 0.30/0.45 => ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(quantDeMorgan4_type,type,
% 0.30/0.45 quantDeMorgan4: $o ).
% 0.30/0.45
% 0.30/0.45 thf(quantDeMorgan4,definition,
% 0.30/0.45 ( quantDeMorgan4
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ~ ( Xphi @ Xx ) )
% 0.30/0.45 => ~ ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(prop2setI_type,type,
% 0.30/0.45 prop2setI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(prop2setI,definition,
% 0.30/0.45 ( prop2setI
% 0.30/0.45 = ( ! [Xphi: $o] :
% 0.30/0.45 ( Xphi
% 0.30/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(set2prop_type,type,
% 0.30/0.45 set2prop: $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(prop2set2propI_type,type,
% 0.30/0.45 prop2set2propI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(prop2set2propI,definition,
% 0.30/0.45 ( prop2set2propI
% 0.30/0.45 = ( ! [Xphi: $o] :
% 0.30/0.45 ( Xphi
% 0.30/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(notdexE_type,type,
% 0.30/0.45 notdexE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(notdexE,definition,
% 0.30/0.45 ( notdexE
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ~ ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ( Xphi @ Xx ) )
% 0.30/0.45 => ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(notdallE_type,type,
% 0.30/0.45 notdallE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(notdallE,definition,
% 0.30/0.45 ( notdallE
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ~ ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( Xphi @ Xx ) )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuI1_type,type,
% 0.30/0.45 exuI1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuI1,definition,
% 0.30/0.45 ( exuI1
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ? [Xx: $i] :
% 0.30/0.45 ( ( Xphi @ Xx )
% 0.30/0.45 & ! [Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) )
% 0.30/0.45 => ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuI3_type,type,
% 0.30/0.45 exuI3: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuI3,definition,
% 0.30/0.45 ( exuI3
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.45 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xx )
% 0.30/0.45 => ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) )
% 0.30/0.45 => ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuI2_type,type,
% 0.30/0.45 exuI2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuI2,definition,
% 0.30/0.45 ( exuI2
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ? [Xx: $i] :
% 0.30/0.45 ! [Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xy )
% 0.30/0.45 <=> ( Xy = Xx ) )
% 0.30/0.45 => ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(inCongP_type,type,
% 0.30/0.45 inCongP: $o ).
% 0.30/0.45
% 0.30/0.45 thf(inCongP,definition,
% 0.30/0.45 ( inCongP
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ( A = B )
% 0.30/0.45 => ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xx = Xy )
% 0.30/0.45 => ( ( in @ Xx @ A )
% 0.30/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(in__Cong_type,type,
% 0.30/0.45 in__Cong: $o ).
% 0.30/0.45
% 0.30/0.45 thf(in__Cong,definition,
% 0.30/0.45 ( in__Cong
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ( A = B )
% 0.30/0.45 => ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xx = Xy )
% 0.30/0.45 => ( ( in @ Xx @ A )
% 0.30/0.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuE3u_type,type,
% 0.30/0.45 exuE3u: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuE3u,definition,
% 0.30/0.45 ( exuE3u
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xx )
% 0.30/0.45 => ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exu__Cong_type,type,
% 0.30/0.45 exu__Cong: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exu__Cong,definition,
% 0.30/0.45 ( exu__Cong
% 0.30/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xx = Xy )
% 0.30/0.45 => ( ( Xphi @ Xx )
% 0.30/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.45 => ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 <=> ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyset__Cong_type,type,
% 0.30/0.45 emptyset__Cong: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptyset__Cong,definition,
% 0.30/0.45 ( emptyset__Cong
% 0.30/0.45 = ( emptyset = emptyset ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoin__Cong_type,type,
% 0.30/0.45 setadjoin__Cong: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoin__Cong,definition,
% 0.30/0.45 ( setadjoin__Cong
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xx = Xy )
% 0.30/0.45 => ! [Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( Xz = Xu )
% 0.30/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong_type,type,
% 0.30/0.46 powerset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong,definition,
% 0.30/0.46 ( powerset__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( powerset @ A )
% 0.30/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong_type,type,
% 0.30/0.46 setunion__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong,definition,
% 0.30/0.46 ( setunion__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( setunion @ A )
% 0.30/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong_type,type,
% 0.30/0.46 omega__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong,definition,
% 0.30/0.46 ( omega__Cong
% 0.30/0.46 = ( omega = omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu_type,type,
% 0.30/0.46 exuEu: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu,definition,
% 0.30/0.46 ( exuEu
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong_type,type,
% 0.30/0.46 descr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong,definition,
% 0.30/0.46 ( descr__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.46 => ( ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong_type,type,
% 0.30/0.46 dsetconstr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong,definition,
% 0.30/0.46 ( dsetconstr__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset_type,type,
% 0.30/0.46 subset: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjoint_type,type,
% 0.30/0.46 disjoint: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setsmeet_type,type,
% 0.30/0.46 setsmeet: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1_type,type,
% 0.30/0.46 subsetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1,definition,
% 0.30/0.46 ( subsetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2_type,type,
% 0.30/0.46 eqimpsubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2,definition,
% 0.30/0.46 ( eqimpsubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1_type,type,
% 0.30/0.46 eqimpsubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1,definition,
% 0.30/0.46 ( eqimpsubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2_type,type,
% 0.30/0.46 subsetI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2,definition,
% 0.30/0.46 ( subsetI2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset_type,type,
% 0.30/0.46 emptysetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset,definition,
% 0.30/0.46 ( emptysetsubset
% 0.30/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE_type,type,
% 0.30/0.46 subsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE,definition,
% 0.30/0.46 ( subsetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2_type,type,
% 0.30/0.46 subsetE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2,definition,
% 0.30/0.46 ( subsetE2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI_type,type,
% 0.30/0.46 notsubsetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI,definition,
% 0.30/0.46 ( notsubsetI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1_type,type,
% 0.30/0.46 notequalI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1,definition,
% 0.30/0.46 ( notequalI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ( subset @ A @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2_type,type,
% 0.30/0.46 notequalI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2,definition,
% 0.30/0.46 ( notequalI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl_type,type,
% 0.30/0.46 subsetRefl: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl,definition,
% 0.30/0.46 ( subsetRefl
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans_type,type,
% 0.30/0.46 subsetTrans: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans,definition,
% 0.30/0.46 ( subsetTrans
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ C )
% 0.30/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub_type,type,
% 0.30/0.46 setadjoinSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub,definition,
% 0.30/0.46 ( setadjoinSub
% 0.30/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2_type,type,
% 0.30/0.46 setadjoinSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2,definition,
% 0.30/0.46 ( setadjoinSub2
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset_type,type,
% 0.30/0.46 subset2powerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset,definition,
% 0.30/0.46 ( subset2powerset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub_type,type,
% 0.30/0.46 setextsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub,definition,
% 0.30/0.46 ( setextsub
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ A )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq_type,type,
% 0.30/0.46 subsetemptysetimpeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq,definition,
% 0.30/0.46 ( subsetemptysetimpeq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( subset @ A @ emptyset )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1_type,type,
% 0.30/0.46 powersetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1,definition,
% 0.30/0.46 ( powersetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1_type,type,
% 0.30/0.46 powersetE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1,definition,
% 0.30/0.46 ( powersetE1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset_type,type,
% 0.30/0.46 inPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset,definition,
% 0.30/0.46 ( inPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset_type,type,
% 0.30/0.46 powersetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset,definition,
% 0.30/0.46 ( powersetsubset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset_type,type,
% 0.30/0.46 sepInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset,definition,
% 0.30/0.46 ( sepInPowerset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( in
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset_type,type,
% 0.30/0.46 sepSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset,definition,
% 0.30/0.46 ( sepSubset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunion_type,type,
% 0.30/0.46 binunion: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL_type,type,
% 0.30/0.46 binunionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL,definition,
% 0.30/0.46 ( binunionIL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR_type,type,
% 0.30/0.46 upairset2IR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR,definition,
% 0.30/0.46 ( upairset2IR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR_type,type,
% 0.30/0.46 binunionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR,definition,
% 0.30/0.46 ( binunionIR
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases_type,type,
% 0.30/0.46 binunionEcases: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases,definition,
% 0.30/0.46 ( binunionEcases
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( ( in @ Xx @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE_type,type,
% 0.30/0.46 binunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE,definition,
% 0.30/0.46 ( binunionE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub_type,type,
% 0.30/0.46 binunionLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub,definition,
% 0.30/0.46 ( binunionLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub_type,type,
% 0.30/0.46 binunionRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub,definition,
% 0.30/0.46 ( binunionRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersect_type,type,
% 0.30/0.46 binintersect: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI_type,type,
% 0.30/0.46 binintersectI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI,definition,
% 0.30/0.46 ( binintersectI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5_type,type,
% 0.30/0.46 binintersectSubset5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5,definition,
% 0.30/0.46 ( binintersectSubset5
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ C @ A )
% 0.30/0.46 => ( ( subset @ C @ B )
% 0.30/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL_type,type,
% 0.30/0.46 binintersectEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL,definition,
% 0.30/0.46 ( binintersectEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub_type,type,
% 0.30/0.46 binintersectLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub,definition,
% 0.30/0.46 ( binintersectLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2_type,type,
% 0.30/0.46 binintersectSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2,definition,
% 0.30/0.46 ( binintersectSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3_type,type,
% 0.30/0.46 binintersectSubset3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3,definition,
% 0.30/0.46 ( binintersectSubset3
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER_type,type,
% 0.30/0.46 binintersectER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER,definition,
% 0.30/0.46 ( binintersectER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1_type,type,
% 0.30/0.46 disjointsetsI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1,definition,
% 0.30/0.46 ( disjointsetsI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xx @ B ) )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub_type,type,
% 0.30/0.46 binintersectRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub,definition,
% 0.30/0.46 ( binintersectRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4_type,type,
% 0.30/0.46 binintersectSubset4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4,definition,
% 0.30/0.46 ( binintersectSubset4
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1_type,type,
% 0.30/0.46 binintersectSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1,definition,
% 0.30/0.46 ( binintersectSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = A )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d_type,type,
% 0.30/0.46 bs114d: $o ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d,definition,
% 0.30/0.46 ( bs114d
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(regular_type,type,
% 0.30/0.46 regular: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminus_type,type,
% 0.30/0.46 setminus: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI_type,type,
% 0.30/0.46 setminusI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI,definition,
% 0.30/0.46 ( setminusI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL_type,type,
% 0.30/0.46 setminusEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL,definition,
% 0.30/0.46 ( setminusEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER_type,type,
% 0.30/0.46 setminusER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER,definition,
% 0.30/0.46 ( setminusER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2_type,type,
% 0.30/0.46 setminusSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2,definition,
% 0.30/0.46 ( setminusSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg_type,type,
% 0.30/0.46 setminusERneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg,definition,
% 0.30/0.46 ( setminusERneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg_type,type,
% 0.30/0.46 setminusELneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg,definition,
% 0.30/0.46 ( setminusELneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg_type,type,
% 0.30/0.46 setminusILneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg,definition,
% 0.30/0.46 ( setminusILneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg_type,type,
% 0.30/0.46 setminusIRneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg,definition,
% 0.30/0.46 ( setminusIRneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub_type,type,
% 0.30/0.46 setminusLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub,definition,
% 0.30/0.46 ( setminusLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1_type,type,
% 0.30/0.46 setminusSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1,definition,
% 0.30/0.46 ( setminusSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiff_type,type,
% 0.30/0.46 symdiff: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE_type,type,
% 0.30/0.46 symdiffE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE,definition,
% 0.30/0.46 ( symdiffE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1_type,type,
% 0.30/0.46 symdiffI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1,definition,
% 0.30/0.46 ( symdiffI1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2_type,type,
% 0.30/0.46 symdiffI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2,definition,
% 0.30/0.46 ( symdiffI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1_type,type,
% 0.30/0.46 symdiffIneg1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1,definition,
% 0.30/0.46 ( symdiffIneg1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2_type,type,
% 0.30/0.46 symdiffIneg2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2,definition,
% 0.30/0.46 ( symdiffIneg2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iskpair_type,type,
% 0.30/0.46 iskpair: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair_type,type,
% 0.30/0.46 secondinupair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair,definition,
% 0.30/0.46 ( secondinupair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL_type,type,
% 0.30/0.46 setukpairIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL,definition,
% 0.30/0.46 ( setukpairIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIR_type,type,
% 0.30/0.46 setukpairIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIR,definition,
% 0.30/0.46 ( setukpairIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpairiskpair_type,type,
% 0.30/0.46 kpairiskpair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairiskpair,definition,
% 0.30/0.46 ( kpairiskpair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpair_type,type,
% 0.30/0.46 kpair: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(kpairp_type,type,
% 0.30/0.46 kpairp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairp,definition,
% 0.30/0.46 ( kpairp
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprod_type,type,
% 0.30/0.46 cartprod: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsubset_type,type,
% 0.30/0.46 singletonsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsubset,definition,
% 0.30/0.46 ( singletonsubset
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowerset_type,type,
% 0.30/0.46 singletoninpowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowerset,definition,
% 0.30/0.46 ( singletoninpowerset
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowunion_type,type,
% 0.30/0.46 singletoninpowunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowunion,definition,
% 0.30/0.46 ( singletoninpowunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2E_type,type,
% 0.30/0.46 upairset2E: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2E,definition,
% 0.30/0.46 ( upairset2E
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsubunion_type,type,
% 0.30/0.46 upairsubunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsubunion,definition,
% 0.30/0.46 ( upairsubunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairinpowunion_type,type,
% 0.30/0.46 upairinpowunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairinpowunion,definition,
% 0.30/0.46 ( upairinpowunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem1_type,type,
% 0.30/0.46 ubforcartprodlem1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem1,definition,
% 0.30/0.46 ( ubforcartprodlem1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem2_type,type,
% 0.30/0.46 ubforcartprodlem2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem2,definition,
% 0.30/0.46 ( ubforcartprodlem2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem3_type,type,
% 0.30/0.46 ubforcartprodlem3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem3,definition,
% 0.30/0.46 ( ubforcartprodlem3
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairin_type,type,
% 0.30/0.46 cartprodpairin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairin,definition,
% 0.30/0.46 ( cartprodpairin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair1_type,type,
% 0.30/0.46 cartprodmempair1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair1,definition,
% 0.30/0.46 ( cartprodmempair1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 & ( Xu
% 0.30/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair_type,type,
% 0.30/0.46 cartprodmempair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair,definition,
% 0.30/0.46 ( cartprodmempair
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE2_type,type,
% 0.30/0.46 setunionE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE2,definition,
% 0.30/0.46 ( setunionE2
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ? [X: $i] :
% 0.30/0.46 ( ( in @ X @ A )
% 0.30/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton1_type,type,
% 0.30/0.46 setunionsingleton1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton1,definition,
% 0.30/0.46 ( setunionsingleton1
% 0.30/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton2_type,type,
% 0.30/0.46 setunionsingleton2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton2,definition,
% 0.30/0.46 ( setunionsingleton2
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton_type,type,
% 0.30/0.46 setunionsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton,definition,
% 0.30/0.46 ( setunionsingleton
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/0.46 = Xx ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singleton_type,type,
% 0.30/0.46 singleton: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonprop_type,type,
% 0.30/0.46 singletonprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonprop,definition,
% 0.30/0.46 ( singletonprop
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) )
% 0.30/0.46 => ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1_type,type,
% 0.30/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E1_type,type,
% 0.30/0.46 ex1E1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E1,definition,
% 0.30/0.46 ( ex1E1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I_type,type,
% 0.30/0.46 ex1I: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I,definition,
% 0.30/0.46 ( ex1I
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xy = Xx ) ) )
% 0.30/0.46 => ( ex1 @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I2_type,type,
% 0.30/0.46 ex1I2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I2,definition,
% 0.30/0.46 ( ex1I2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) )
% 0.30/0.46 => ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsuniq_type,type,
% 0.30/0.46 singletonsuniq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsuniq,definition,
% 0.30/0.46 ( singletonsuniq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(atmost1p_type,type,
% 0.30/0.46 atmost1p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(atleast2p_type,type,
% 0.30/0.46 atleast2p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(atmost2p_type,type,
% 0.30/0.46 atmost2p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetp_type,type,
% 0.30/0.46 upairsetp: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL1_type,type,
% 0.30/0.46 setukpairinjL1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL1,definition,
% 0.30/0.46 ( setukpairinjL1
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kfstsingleton_type,type,
% 0.30/0.46 kfstsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kfstsingleton,definition,
% 0.30/0.46 ( kfstsingleton
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(theprop_type,type,
% 0.30/0.46 theprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(theprop,definition,
% 0.30/0.46 ( theprop
% 0.30/0.46 = ( ! [X: $i] :
% 0.30/0.46 ( ( singleton @ X )
% 0.30/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kfst_type,type,
% 0.30/0.46 kfst: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(kfstpairEq_type,type,
% 0.30/0.46 kfstpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kfstpairEq,definition,
% 0.30/0.46 ( kfstpairEq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xx ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstin_type,type,
% 0.30/0.46 cartprodfstin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstin,definition,
% 0.30/0.46 ( cartprodfstin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL2_type,type,
% 0.30/0.46 setukpairinjL2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL2,definition,
% 0.30/0.46 ( setukpairinjL2
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL_type,type,
% 0.30/0.46 setukpairinjL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL,definition,
% 0.30/0.46 ( setukpairinjL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xz @ Xu ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR11_type,type,
% 0.30/0.46 setukpairinjR11: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR11,definition,
% 0.30/0.46 ( setukpairinjR11
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR12_type,type,
% 0.30/0.46 setukpairinjR12: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR12,definition,
% 0.30/0.46 ( setukpairinjR12
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR1_type,type,
% 0.30/0.46 setukpairinjR1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR1,definition,
% 0.30/0.46 ( setukpairinjR1
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xu )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairequniteq_type,type,
% 0.30/0.46 upairequniteq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairequniteq,definition,
% 0.30/0.46 ( upairequniteq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR2_type,type,
% 0.30/0.46 setukpairinjR2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR2,definition,
% 0.30/0.46 ( setukpairinjR2
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR_type,type,
% 0.30/0.46 setukpairinjR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR,definition,
% 0.30/0.46 ( setukpairinjR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xz @ Xu ) )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ksndsingleton_type,type,
% 0.30/0.46 ksndsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ksndsingleton,definition,
% 0.30/0.46 ( ksndsingleton
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.46 @ ^ [Xx: $i] :
% 0.30/0.46 ( Xu
% 0.30/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ksnd_type,type,
% 0.30/0.46 ksnd: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(ksndpairEq_type,type,
% 0.30/0.46 ksndpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ksndpairEq,definition,
% 0.30/0.46 ( ksndpairEq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xy ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpairsurjEq_type,type,
% 0.30/0.46 kpairsurjEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairsurjEq,definition,
% 0.30/0.46 ( kpairsurjEq
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.46 = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndin_type,type,
% 0.30/0.46 cartprodsndin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndin,definition,
% 0.30/0.46 ( cartprodsndin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemEL_type,type,
% 0.30/0.46 cartprodpairmemEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemEL,definition,
% 0.30/0.46 ( cartprodpairmemEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemER_type,type,
% 0.30/0.46 cartprodpairmemER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemER,definition,
% 0.30/0.46 ( cartprodpairmemER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempaircEq_type,type,
% 0.30/0.46 cartprodmempaircEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempaircEq,definition,
% 0.30/0.46 ( cartprodmempaircEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstpairEq_type,type,
% 0.30/0.46 cartprodfstpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstpairEq,definition,
% 0.30/0.46 ( cartprodfstpairEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndpairEq_type,type,
% 0.30/0.46 cartprodsndpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndpairEq,definition,
% 0.30/0.46 ( cartprodsndpairEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairsurjEq_type,type,
% 0.30/0.46 cartprodpairsurjEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairsurjEq,definition,
% 0.30/0.46 ( cartprodpairsurjEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.46 = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln_type,type,
% 0.30/0.46 breln: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstr_type,type,
% 0.30/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrI_type,type,
% 0.30/0.46 dpsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrI,definition,
% 0.30/0.46 ( dpsetconstrI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrSub_type,type,
% 0.30/0.46 dpsetconstrSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrSub,definition,
% 0.30/0.46 ( dpsetconstrSub
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln_type,type,
% 0.30/0.46 setOfPairsIsBReln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln,definition,
% 0.30/0.46 ( setOfPairsIsBReln
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( breln @ A @ B
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrERa_type,type,
% 0.30/0.46 dpsetconstrERa: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrERa,definition,
% 0.30/0.46 ( dpsetconstrERa
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL1_type,type,
% 0.30/0.46 dpsetconstrEL1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL1,definition,
% 0.30/0.46 ( dpsetconstrEL1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL2_type,type,
% 0.30/0.46 dpsetconstrEL2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL2,definition,
% 0.30/0.46 ( dpsetconstrEL2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrER_type,type,
% 0.30/0.46 dpsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrER,definition,
% 0.30/0.46 ( dpsetconstrER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(func_type,type,
% 0.30/0.46 func: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcSet_type,type,
% 0.30/0.46 funcSet: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(funcImageSingleton_type,type,
% 0.30/0.46 funcImageSingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcImageSingleton,definition,
% 0.30/0.46 ( funcImageSingleton
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(apProp_type,type,
% 0.30/0.46 apProp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(apProp,definition,
% 0.30/0.46 ( apProp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in
% 0.30/0.46 @ ( setunion
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.30/0.46 @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap_type,type,
% 0.30/0.46 ap: $i > $i > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(app_type,type,
% 0.30/0.46 app: $o ).
% 0.30/0.46
% 0.30/0.46 thf(app,definition,
% 0.30/0.46 ( app
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(infuncsetfunc_type,type,
% 0.30/0.46 infuncsetfunc: $o ).
% 0.30/0.46
% 0.30/0.46 thf(infuncsetfunc,definition,
% 0.30/0.46 ( infuncsetfunc
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2p_type,type,
% 0.30/0.46 ap2p: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2p,definition,
% 0.30/0.46 ( ap2p
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcinfuncset_type,type,
% 0.30/0.46 funcinfuncset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcinfuncset,definition,
% 0.30/0.46 ( funcinfuncset
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lamProp_type,type,
% 0.30/0.46 lamProp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lamProp,definition,
% 0.30/0.46 ( lamProp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( func @ A @ B
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam_type,type,
% 0.30/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(lamp_type,type,
% 0.30/0.46 lamp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lamp,definition,
% 0.30/0.46 ( lamp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( func @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam2p_type,type,
% 0.30/0.46 lam2p: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lam2p,definition,
% 0.30/0.46 ( lam2p
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( in
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall1_type,type,
% 0.30/0.46 brelnall1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall1,definition,
% 0.30/0.46 ( brelnall1
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall2_type,type,
% 0.30/0.46 brelnall2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall2,definition,
% 0.30/0.46 ( brelnall2
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E2_type,type,
% 0.30/0.46 ex1E2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E2,definition,
% 0.30/0.46 ( ex1E2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp1_type,type,
% 0.30/0.46 funcGraphProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp1,definition,
% 0.30/0.46 ( funcGraphProp1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp3_type,type,
% 0.30/0.46 funcGraphProp3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp3,definition,
% 0.30/0.46 ( funcGraphProp3
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp2_type,type,
% 0.30/0.46 funcGraphProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp2,definition,
% 0.30/0.46 ( funcGraphProp2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcextLem_type,type,
% 0.30/0.46 funcextLem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcextLem,definition,
% 0.30/0.46 ( funcextLem
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xg )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp4_type,type,
% 0.30/0.46 funcGraphProp4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp4,definition,
% 0.30/0.46 ( funcGraphProp4
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln_type,type,
% 0.30/0.46 subbreln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln,definition,
% 0.30/0.46 ( subbreln
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln_type,type,
% 0.30/0.46 eqbreln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln,definition,
% 0.30/0.46 ( eqbreln
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcext_type,type,
% 0.30/0.46 funcext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcext,definition,
% 0.30/0.46 ( funcext
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xg )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcext2_type,type,
% 0.30/0.46 funcext2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcext2,definition,
% 0.30/0.46 ( funcext2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq1_type,type,
% 0.30/0.46 ap2apEq1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq1,definition,
% 0.30/0.46 ( ap2apEq1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq2_type,type,
% 0.30/0.46 ap2apEq2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq2,definition,
% 0.30/0.46 ( ap2apEq2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(beta1_type,type,
% 0.30/0.46 beta1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(beta1,definition,
% 0.30/0.46 ( beta1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.46 @ Xx )
% 0.30/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eta1_type,type,
% 0.30/0.46 eta1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eta1,definition,
% 0.30/0.46 ( eta1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.46 = Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam2lamEq_type,type,
% 0.30/0.46 lam2lamEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lam2lamEq,definition,
% 0.30/0.46 ( lam2lamEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.46 = ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(beta2_type,type,
% 0.30/0.46 beta2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(beta2,definition,
% 0.30/0.46 ( beta2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.46 @ Xx )
% 0.30/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eta2_type,type,
% 0.30/0.46 eta2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eta2,definition,
% 0.30/0.46 ( eta2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.46 = Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp1_type,type,
% 0.30/0.46 iffalseProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp1,definition,
% 0.30/0.46 ( iffalseProp1
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( in @ Xy
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp2_type,type,
% 0.30/0.46 iffalseProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp2,definition,
% 0.30/0.46 ( iffalseProp2
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) )
% 0.30/0.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp1_type,type,
% 0.30/0.46 iftrueProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp1,definition,
% 0.30/0.46 ( iftrueProp1
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp2_type,type,
% 0.30/0.46 iftrueProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp2,definition,
% 0.30/0.46 ( iftrueProp2
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) )
% 0.30/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ifSingleton_type,type,
% 0.30/0.46 ifSingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ifSingleton,definition,
% 0.30/0.46 ( ifSingleton
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(if_type,type,
% 0.30/0.46 if: $i > $o > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(ifp_type,type,
% 0.30/0.46 ifp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ifp,definition,
% 0.30/0.46 ( ifp
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(theeq_type,type,
% 0.30/0.46 theeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(theeq,definition,
% 0.30/0.46 ( theeq
% 0.30/0.46 = ( ! [X: $i] :
% 0.30/0.46 ( ( singleton @ X )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ X )
% 0.30/0.46 => ( ( setunion @ X )
% 0.30/0.46 = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrue_type,type,
% 0.30/0.46 iftrue: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrue,definition,
% 0.30/0.46 ( iftrue
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.46 = Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalse_type,type,
% 0.30/0.46 iffalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalse,definition,
% 0.30/0.46 ( iffalse
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.46 = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueorfalse_type,type,
% 0.30/0.46 iftrueorfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueorfalse,definition,
% 0.30/0.46 ( iftrueorfalse
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectT_lem_type,type,
% 0.30/0.46 binintersectT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectT_lem,definition,
% 0.30/0.46 ( binintersectT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionT_lem_type,type,
% 0.30/0.46 binunionT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionT_lem,definition,
% 0.30/0.46 ( binunionT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetT_lem_type,type,
% 0.30/0.46 powersetT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetT_lem,definition,
% 0.30/0.46 ( powersetT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusT_lem_type,type,
% 0.30/0.46 setminusT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusT_lem,definition,
% 0.30/0.46 ( setminusT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementT_lem_type,type,
% 0.30/0.46 complementT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementT_lem,definition,
% 0.30/0.46 ( complementT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextT_type,type,
% 0.30/0.46 setextT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextT,definition,
% 0.30/0.46 ( setextT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ( in @ Xx @ X ) ) )
% 0.30/0.46 => ( X = Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTI_type,type,
% 0.30/0.46 subsetTI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTI,definition,
% 0.30/0.46 ( subsetTI
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTI1_type,type,
% 0.30/0.46 powersetTI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTI1,definition,
% 0.30/0.46 ( powersetTI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTE1_type,type,
% 0.30/0.46 powersetTE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTE1,definition,
% 0.30/0.46 ( powersetTE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ X @ ( powerset @ Y ) )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTI1_type,type,
% 0.30/0.46 complementTI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTI1,definition,
% 0.30/0.46 ( complementTI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTE1_type,type,
% 0.30/0.46 complementTE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTE1,definition,
% 0.30/0.46 ( complementTE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTELcontra_type,type,
% 0.30/0.46 binintersectTELcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTELcontra,definition,
% 0.30/0.46 ( binintersectTELcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ X )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTERcontra_type,type,
% 0.30/0.46 binintersectTERcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTERcontra,definition,
% 0.30/0.46 ( binintersectTERcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT_type,type,
% 0.30/0.46 contrasubsetT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT,definition,
% 0.30/0.46 ( contrasubsetT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT1_type,type,
% 0.30/0.46 contrasubsetT1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT1,definition,
% 0.30/0.46 ( contrasubsetT1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT2_type,type,
% 0.30/0.46 contrasubsetT2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT2,definition,
% 0.30/0.46 ( contrasubsetT2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT3_type,type,
% 0.30/0.46 contrasubsetT3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT3,definition,
% 0.30/0.46 ( contrasubsetT3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementI1_type,type,
% 0.30/0.46 doubleComplementI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementI1,definition,
% 0.30/0.46 ( doubleComplementI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementE1_type,type,
% 0.30/0.46 doubleComplementE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementE1,definition,
% 0.30/0.46 ( doubleComplementE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.30/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub1_type,type,
% 0.30/0.46 doubleComplementSub1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub1,definition,
% 0.30/0.46 ( doubleComplementSub1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub2_type,type,
% 0.30/0.46 doubleComplementSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub2,definition,
% 0.30/0.46 ( doubleComplementSub2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementEq_type,type,
% 0.30/0.46 doubleComplementEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementEq,definition,
% 0.30/0.46 ( doubleComplementEq
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( X
% 0.30/0.46 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTnotintersectT_type,type,
% 0.30/0.46 complementTnotintersectT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTnotintersectT,definition,
% 0.30/0.46 ( complementTnotintersectT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementImpComplementIntersect_type,type,
% 0.30/0.46 complementImpComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementImpComplementIntersect,definition,
% 0.30/0.46 ( complementImpComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementSubsetComplementIntersect_type,type,
% 0.30/0.46 complementSubsetComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementSubsetComplementIntersect,definition,
% 0.30/0.46 ( complementSubsetComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementInPowersetComplementIntersect_type,type,
% 0.30/0.46 complementInPowersetComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementInPowersetComplementIntersect,definition,
% 0.30/0.46 ( complementInPowersetComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contraSubsetComplement_type,type,
% 0.30/0.46 contraSubsetComplement: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contraSubsetComplement,definition,
% 0.30/0.46 ( contraSubsetComplement
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTcontraSubset_type,type,
% 0.30/0.46 complementTcontraSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTcontraSubset,definition,
% 0.30/0.46 ( complementTcontraSubset
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTILcontra_type,type,
% 0.30/0.46 binunionTILcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTILcontra,definition,
% 0.30/0.46 ( binunionTILcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTIRcontra_type,type,
% 0.30/0.46 binunionTIRcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTIRcontra,definition,
% 0.30/0.46 ( binunionTIRcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion_type,type,
% 0.30/0.46 inIntersectImpInUnion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion,definition,
% 0.30/0.46 ( inIntersectImpInUnion
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion2_type,type,
% 0.30/0.46 inIntersectImpInUnion2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion2,definition,
% 0.30/0.46 ( inIntersectImpInUnion2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInIntersectUnions_type,type,
% 0.30/0.46 inIntersectImpInIntersectUnions: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInIntersectUnions,definition,
% 0.30/0.46 ( inIntersectImpInIntersectUnions
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(intersectInPowersetIntersectUnions_type,type,
% 0.30/0.46 intersectInPowersetIntersectUnions: $o ).
% 0.30/0.46
% 0.30/0.46 thf(intersectInPowersetIntersectUnions,definition,
% 0.30/0.46 ( intersectInPowersetIntersectUnions
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpNotIn1_type,type,
% 0.30/0.46 inComplementUnionImpNotIn1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpNotIn1,definition,
% 0.30/0.46 ( inComplementUnionImpNotIn1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpInComplement1_type,type,
% 0.30/0.46 inComplementUnionImpInComplement1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpInComplement1,definition,
% 0.30/0.46 ( inComplementUnionImpInComplement1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTE_type,type,
% 0.30/0.46 binunionTE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTE,definition,
% 0.30/0.46 ( binunionTE
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ( ( ( in @ Xx @ X )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ Y )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTEcontra_type,type,
% 0.30/0.46 binunionTEcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTEcontra,definition,
% 0.30/0.46 ( binunionTEcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ X )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a1_type,type,
% 0.30/0.46 demorgan2a1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a1,definition,
% 0.30/0.46 ( demorgan2a1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementUnionInPowersetComplement_type,type,
% 0.30/0.46 complementUnionInPowersetComplement: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementUnionInPowersetComplement,definition,
% 0.30/0.46 ( complementUnionInPowersetComplement
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a2_type,type,
% 0.30/0.46 demorgan2a2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a2,definition,
% 0.30/0.46 ( demorgan2a2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1a_type,type,
% 0.30/0.46 demorgan1a: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1a,definition,
% 0.30/0.46 ( demorgan1a
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1b_type,type,
% 0.30/0.46 demorgan1b: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1b,definition,
% 0.30/0.46 ( demorgan1b
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1_type,type,
% 0.30/0.46 demorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1,definition,
% 0.30/0.46 ( demorgan1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.30/0.46 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a_type,type,
% 0.30/0.46 demorgan2a: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a,definition,
% 0.30/0.46 ( demorgan2a
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b2_type,type,
% 0.30/0.46 demorgan2b2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b2,definition,
% 0.30/0.46 ( demorgan2b2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b_type,type,
% 0.30/0.46 demorgan2b: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b,definition,
% 0.30/0.46 ( demorgan2b
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2_type,type,
% 0.30/0.46 demorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2,definition,
% 0.30/0.46 ( demorgan2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.30/0.46 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule0_type,type,
% 0.30/0.46 woz13rule0: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule0,definition,
% 0.30/0.46 ( woz13rule0
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule1_type,type,
% 0.30/0.46 woz13rule1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule1,definition,
% 0.30/0.46 ( woz13rule1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule2_type,type,
% 0.30/0.46 woz13rule2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule2,definition,
% 0.30/0.46 ( woz13rule2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ Y @ Z )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule3_type,type,
% 0.30/0.46 woz13rule3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule3,definition,
% 0.30/0.46 ( woz13rule3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule4_type,type,
% 0.30/0.46 woz13rule4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule4,definition,
% 0.30/0.46 ( woz13rule4
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [W: $i] :
% 0.30/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( ( subset @ Y @ W )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_1_type,type,
% 0.30/0.46 woz1_1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_1,definition,
% 0.30/0.46 ( woz1_1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_2_type,type,
% 0.30/0.46 woz1_2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_2,definition,
% 0.30/0.46 ( woz1_2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [W: $i] :
% 0.30/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.30/0.46 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_3_type,type,
% 0.30/0.46 woz1_3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_3,definition,
% 0.30/0.46 ( woz1_3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_4_type,type,
% 0.30/0.46 woz1_4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_4,definition,
% 0.30/0.46 ( woz1_4
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_5_type,type,
% 0.30/0.46 woz1_5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_5,definition,
% 0.30/0.46 ( woz1_5
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1_type,type,
% 0.30/0.46 breln1: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all2_type,type,
% 0.30/0.46 breln1all2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all2,definition,
% 0.30/0.46 ( breln1all2
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1Set_type,type,
% 0.30/0.46 breln1Set: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1SetBreln1_type,type,
% 0.30/0.46 breln1SetBreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1SetBreln1,definition,
% 0.30/0.46 ( breln1SetBreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( in @ R @ ( breln1Set @ A ) )
% 0.30/0.46 => ( breln1 @ A @ R ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(transitive_type,type,
% 0.30/0.46 transitive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(antisymmetric_type,type,
% 0.30/0.46 antisymmetric: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(reflexive_type,type,
% 0.30/0.46 reflexive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(refltransitive_type,type,
% 0.30/0.46 refltransitive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(refllinearorder_type,type,
% 0.30/0.46 refllinearorder: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(reflwellordering_type,type,
% 0.30/0.46 reflwellordering: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(choice2fnsingleton_type,type,
% 0.30/0.46 choice2fnsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(choice2fnsingleton,definition,
% 0.30/0.46 ( choice2fnsingleton
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 & ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ! [R: $i] :
% 0.30/0.46 ( ( in @ R @ ( breln1Set @ B ) )
% 0.30/0.46 => ( ( reflwellordering @ B @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx @ Xy )
% 0.30/0.46 & ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ B )
% 0.30/0.46 => ( ( Xphi @ Xx @ Xz )
% 0.30/0.46 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln1_type,type,
% 0.30/0.46 setOfPairsIsBReln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln1,definition,
% 0.30/0.46 ( setOfPairsIsBReln1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( breln1 @ A
% 0.30/0.46 @ ( dpsetconstr @ A @ A
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all1_type,type,
% 0.30/0.46 breln1all1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all1,definition,
% 0.30/0.46 ( breln1all1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln1_type,type,
% 0.30/0.46 subbreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln1,definition,
% 0.30/0.46 ( subbreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln1_type,type,
% 0.30/0.46 eqbreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln1,definition,
% 0.30/0.46 ( eqbreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invset_type,type,
% 0.30/0.46 breln1invset: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invprop_type,type,
% 0.30/0.46 breln1invprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invprop,definition,
% 0.30/0.46 ( breln1invprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invI_type,type,
% 0.30/0.46 breln1invI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invI,definition,
% 0.30/0.46 ( breln1invI
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invE_type,type,
% 0.30/0.46 breln1invE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invE,definition,
% 0.30/0.46 ( breln1invE
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compset_type,type,
% 0.30/0.46 breln1compset: $i > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compprop_type,type,
% 0.30/0.46 breln1compprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compprop,definition,
% 0.30/0.46 ( breln1compprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compI_type,type,
% 0.30/0.46 breln1compI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compI,definition,
% 0.30/0.46 ( breln1compI
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compE_type,type,
% 0.30/0.46 breln1compE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compE,definition,
% 0.30/0.46 ( breln1compE
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.46 => ? [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compEex_type,type,
% 0.30/0.46 breln1compEex: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compEex,definition,
% 0.30/0.46 ( breln1compEex
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.46 => Xphi ) ) )
% 0.30/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionprop_type,type,
% 0.30/0.46 breln1unionprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionprop,definition,
% 0.30/0.46 ( breln1unionprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIL_type,type,
% 0.30/0.46 breln1unionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIL,definition,
% 0.30/0.46 ( breln1unionIL
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIR_type,type,
% 0.30/0.46 breln1unionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIR,definition,
% 0.30/0.46 ( breln1unionIR
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionI_type,type,
% 0.30/0.46 breln1unionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionI,definition,
% 0.30/0.46 ( breln1unionI
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionE_type,type,
% 0.30/0.46 breln1unionE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionE,definition,
% 0.30/0.47 ( breln1unionE
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionEcases_type,type,
% 0.30/0.47 breln1unionEcases: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionEcases,definition,
% 0.30/0.47 ( breln1unionEcases
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.30/0.47 => ! [Xphi: $o] :
% 0.30/0.47 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionCommutes_type,type,
% 0.30/0.47 breln1unionCommutes: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionCommutes,definition,
% 0.30/0.47 ( breln1unionCommutes
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ( ( binunion @ R @ S )
% 0.30/0.47 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz2Ex_type,type,
% 0.30/0.47 woz2Ex: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz2Ex,definition,
% 0.30/0.47 ( woz2Ex
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ( R
% 0.30/0.47 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz2W_type,type,
% 0.30/0.47 woz2W: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz2W,definition,
% 0.30/0.47 ( woz2W
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.47 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz2A_type,type,
% 0.30/0.47 woz2A: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz2A,definition,
% 0.30/0.47 ( woz2A
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [T: $i] :
% 0.30/0.47 ( ( breln1 @ A @ T )
% 0.30/0.47 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.30/0.47 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz2B_type,type,
% 0.30/0.47 woz2B: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz2B,definition,
% 0.30/0.47 ( woz2B
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [T: $i] :
% 0.30/0.47 ( ( breln1 @ A @ T )
% 0.30/0.47 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.30/0.47 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1Ex_type,type,
% 0.30/0.47 image1Ex: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1Ex,definition,
% 0.30/0.47 ( image1Ex
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i] :
% 0.30/0.47 ? [B: $i] :
% 0.30/0.47 ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 <=> ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1Ex1_type,type,
% 0.30/0.47 image1Ex1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1Ex1,definition,
% 0.30/0.47 ( image1Ex1
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i] :
% 0.30/0.47 ( exu
% 0.30/0.47 @ ^ [B: $i] :
% 0.30/0.47 ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 <=> ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1_type,type,
% 0.30/0.47 image1: $i > ( $i > $i ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(image1Equiv_type,type,
% 0.30/0.47 image1Equiv: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1Equiv,definition,
% 0.30/0.47 ( image1Equiv
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx
% 0.30/0.47 @ ( image1 @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.30/0.47 <=> ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1E_type,type,
% 0.30/0.47 image1E: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1E,definition,
% 0.30/0.47 ( image1E
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx
% 0.30/0.47 @ ( image1 @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.30/0.47 => ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1I_type,type,
% 0.30/0.47 image1I: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1I,definition,
% 0.30/0.47 ( image1I
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.47 ( ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) )
% 0.30/0.47 => ( in @ Xx
% 0.30/0.47 @ ( image1 @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(injective_type,type,
% 0.30/0.47 injective: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncSet_type,type,
% 0.30/0.47 injFuncSet: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncSet,definition,
% 0.30/0.47 ( injFuncSet
% 0.30/0.47 = ( ^ [A: $i,B: $i] :
% 0.30/0.47 ( dsetconstr @ ( funcSet @ A @ B )
% 0.30/0.47 @ ^ [Xf: $i] : ( injective @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncInInjFuncSet_type,type,
% 0.30/0.47 injFuncInInjFuncSet: $o ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncInInjFuncSet,definition,
% 0.30/0.47 ( injFuncInInjFuncSet
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( ( injective @ A @ B @ Xf )
% 0.30/0.47 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncSetFuncIn,conjecture,
% 0.30/0.47 ( setextAx
% 0.30/0.47 => ( emptysetAx
% 0.30/0.47 => ( setadjoinAx
% 0.30/0.47 => ( powersetAx
% 0.30/0.47 => ( setunionAx
% 0.30/0.47 => ( omega0Ax
% 0.30/0.47 => ( omegaSAx
% 0.30/0.47 => ( omegaIndAx
% 0.30/0.47 => ( replAx
% 0.30/0.47 => ( foundationAx
% 0.30/0.47 => ( wellorderingAx
% 0.30/0.47 => ( descrp
% 0.30/0.47 => ( dsetconstrI
% 0.30/0.47 => ( dsetconstrEL
% 0.30/0.47 => ( dsetconstrER
% 0.30/0.47 => ( exuE1
% 0.30/0.47 => ( prop2setE
% 0.30/0.47 => ( emptysetE
% 0.30/0.47 => ( emptysetimpfalse
% 0.30/0.47 => ( notinemptyset
% 0.30/0.47 => ( exuE3e
% 0.30/0.47 => ( setext
% 0.30/0.47 => ( emptyI
% 0.30/0.47 => ( noeltsimpempty
% 0.30/0.47 => ( setbeta
% 0.30/0.47 => ( nonemptyE1
% 0.30/0.47 => ( nonemptyI
% 0.30/0.47 => ( nonemptyI1
% 0.30/0.47 => ( setadjoinIL
% 0.30/0.47 => ( emptyinunitempty
% 0.30/0.47 => ( setadjoinIR
% 0.30/0.47 => ( setadjoinE
% 0.30/0.47 => ( setadjoinOr
% 0.30/0.47 => ( setoftrueEq
% 0.30/0.47 => ( powersetI
% 0.30/0.47 => ( emptyinPowerset
% 0.30/0.47 => ( emptyInPowerset
% 0.30/0.47 => ( powersetE
% 0.30/0.47 => ( setunionI
% 0.30/0.47 => ( setunionE
% 0.30/0.47 => ( subPowSU
% 0.30/0.47 => ( exuE2
% 0.30/0.47 => ( nonemptyImpWitness
% 0.30/0.47 => ( uniqinunit
% 0.30/0.47 => ( notinsingleton
% 0.30/0.47 => ( eqinunit
% 0.30/0.47 => ( singletonsswitch
% 0.30/0.47 => ( upairsetE
% 0.30/0.47 => ( upairsetIL
% 0.30/0.47 => ( upairsetIR
% 0.30/0.47 => ( emptyE1
% 0.30/0.47 => ( vacuousDall
% 0.30/0.47 => ( quantDeMorgan1
% 0.30/0.47 => ( quantDeMorgan2
% 0.30/0.47 => ( quantDeMorgan3
% 0.30/0.47 => ( quantDeMorgan4
% 0.30/0.47 => ( prop2setI
% 0.30/0.47 => ( prop2set2propI
% 0.30/0.47 => ( notdexE
% 0.30/0.47 => ( notdallE
% 0.30/0.47 => ( exuI1
% 0.30/0.47 => ( exuI3
% 0.30/0.47 => ( exuI2
% 0.30/0.47 => ( inCongP
% 0.30/0.47 => ( in__Cong
% 0.30/0.47 => ( exuE3u
% 0.30/0.47 => ( exu__Cong
% 0.30/0.47 => ( emptyset__Cong
% 0.30/0.47 => ( setadjoin__Cong
% 0.30/0.47 => ( powerset__Cong
% 0.30/0.47 => ( setunion__Cong
% 0.30/0.47 => ( omega__Cong
% 0.30/0.47 => ( exuEu
% 0.30/0.47 => ( descr__Cong
% 0.30/0.47 => ( dsetconstr__Cong
% 0.30/0.47 => ( subsetI1
% 0.30/0.47 => ( eqimpsubset2
% 0.30/0.47 => ( eqimpsubset1
% 0.30/0.47 => ( subsetI2
% 0.30/0.47 => ( emptysetsubset
% 0.30/0.47 => ( subsetE
% 0.30/0.47 => ( subsetE2
% 0.30/0.47 => ( notsubsetI
% 0.30/0.47 => ( notequalI1
% 0.30/0.47 => ( notequalI2
% 0.30/0.47 => ( subsetRefl
% 0.30/0.47 => ( subsetTrans
% 0.30/0.47 => ( setadjoinSub
% 0.30/0.47 => ( setadjoinSub2
% 0.30/0.47 => ( subset2powerset
% 0.30/0.47 => ( setextsub
% 0.30/0.47 => ( subsetemptysetimpeq
% 0.30/0.47 => ( powersetI1
% 0.30/0.47 => ( powersetE1
% 0.30/0.47 => ( inPowerset
% 0.30/0.47 => ( powersetsubset
% 0.30/0.47 => ( sepInPowerset
% 0.30/0.47 => ( sepSubset
% 0.30/0.47 => ( binunionIL
% 0.30/0.47 => ( upairset2IR
% 0.30/0.47 => ( binunionIR
% 0.30/0.47 => ( binunionEcases
% 0.30/0.47 => ( binunionE
% 0.30/0.47 => ( binunionLsub
% 0.30/0.47 => ( binunionRsub
% 0.30/0.47 => ( binintersectI
% 0.30/0.47 => ( binintersectSubset5
% 0.30/0.47 => ( binintersectEL
% 0.30/0.47 => ( binintersectLsub
% 0.30/0.47 => ( binintersectSubset2
% 0.30/0.47 => ( binintersectSubset3
% 0.30/0.47 => ( binintersectER
% 0.30/0.47 => ( disjointsetsI1
% 0.30/0.47 => ( binintersectRsub
% 0.30/0.47 => ( binintersectSubset4
% 0.30/0.47 => ( binintersectSubset1
% 0.30/0.47 => ( bs114d
% 0.30/0.47 => ( setminusI
% 0.30/0.47 => ( setminusEL
% 0.30/0.47 => ( setminusER
% 0.30/0.47 => ( setminusSubset2
% 0.30/0.47 => ( setminusERneg
% 0.30/0.47 => ( setminusELneg
% 0.30/0.47 => ( setminusILneg
% 0.30/0.47 => ( setminusIRneg
% 0.30/0.47 => ( setminusLsub
% 0.30/0.47 => ( setminusSubset1
% 0.30/0.47 => ( symdiffE
% 0.30/0.47 => ( symdiffI1
% 0.30/0.47 => ( symdiffI2
% 0.30/0.47 => ( symdiffIneg1
% 0.30/0.47 => ( symdiffIneg2
% 0.30/0.47 => ( secondinupair
% 0.30/0.47 => ( setukpairIL
% 0.30/0.47 => ( setukpairIR
% 0.30/0.47 => ( kpairiskpair
% 0.30/0.47 => ( kpairp
% 0.30/0.47 => ( singletonsubset
% 0.30/0.47 => ( singletoninpowerset
% 0.30/0.47 => ( singletoninpowunion
% 0.30/0.47 => ( upairset2E
% 0.30/0.47 => ( upairsubunion
% 0.30/0.47 => ( upairinpowunion
% 0.30/0.47 => ( ubforcartprodlem1
% 0.30/0.47 => ( ubforcartprodlem2
% 0.30/0.47 => ( ubforcartprodlem3
% 0.30/0.47 => ( cartprodpairin
% 0.30/0.47 => ( cartprodmempair1
% 0.30/0.47 => ( cartprodmempair
% 0.30/0.47 => ( setunionE2
% 0.30/0.47 => ( setunionsingleton1
% 0.30/0.47 => ( setunionsingleton2
% 0.30/0.47 => ( setunionsingleton
% 0.30/0.47 => ( singletonprop
% 0.30/0.47 => ( ex1E1
% 0.30/0.47 => ( ex1I
% 0.30/0.47 => ( ex1I2
% 0.30/0.47 => ( singletonsuniq
% 0.30/0.47 => ( setukpairinjL1
% 0.30/0.47 => ( kfstsingleton
% 0.30/0.47 => ( theprop
% 0.30/0.47 => ( kfstpairEq
% 0.30/0.47 => ( cartprodfstin
% 0.30/0.47 => ( setukpairinjL2
% 0.30/0.47 => ( setukpairinjL
% 0.30/0.47 => ( setukpairinjR11
% 0.30/0.47 => ( setukpairinjR12
% 0.30/0.47 => ( setukpairinjR1
% 0.30/0.47 => ( upairequniteq
% 0.30/0.47 => ( setukpairinjR2
% 0.30/0.47 => ( setukpairinjR
% 0.30/0.47 => ( ksndsingleton
% 0.30/0.47 => ( ksndpairEq
% 0.30/0.47 => ( kpairsurjEq
% 0.30/0.47 => ( cartprodsndin
% 0.30/0.47 => ( cartprodpairmemEL
% 0.30/0.47 => ( cartprodpairmemER
% 0.30/0.47 => ( cartprodmempaircEq
% 0.30/0.47 => ( cartprodfstpairEq
% 0.30/0.47 => ( cartprodsndpairEq
% 0.30/0.47 => ( cartprodpairsurjEq
% 0.30/0.47 => ( dpsetconstrI
% 0.30/0.47 => ( dpsetconstrSub
% 0.30/0.47 => ( setOfPairsIsBReln
% 0.30/0.47 => ( dpsetconstrERa
% 0.30/0.47 => ( dpsetconstrEL1
% 0.30/0.47 => ( dpsetconstrEL2
% 0.30/0.47 => ( dpsetconstrER
% 0.30/0.47 => ( funcImageSingleton
% 0.30/0.47 => ( apProp
% 0.30/0.47 => ( app
% 0.30/0.47 => ( infuncsetfunc
% 0.30/0.47 => ( ap2p
% 0.30/0.47 => ( funcinfuncset
% 0.30/0.47 => ( lamProp
% 0.30/0.47 => ( lamp
% 0.30/0.47 => ( lam2p
% 0.30/0.47 => ( brelnall1
% 0.30/0.47 => ( brelnall2
% 0.30/0.47 => ( ex1E2
% 0.30/0.47 => ( funcGraphProp1
% 0.30/0.47 => ( funcGraphProp3
% 0.30/0.47 => ( funcGraphProp2
% 0.30/0.47 => ( funcextLem
% 0.30/0.47 => ( funcGraphProp4
% 0.30/0.47 => ( subbreln
% 0.30/0.47 => ( eqbreln
% 0.30/0.47 => ( funcext
% 0.30/0.47 => ( funcext2
% 0.30/0.47 => ( ap2apEq1
% 0.30/0.47 => ( ap2apEq2
% 0.30/0.47 => ( beta1
% 0.30/0.47 => ( eta1
% 0.30/0.47 => ( lam2lamEq
% 0.30/0.47 => ( beta2
% 0.30/0.47 => ( eta2
% 0.30/0.47 => ( iffalseProp1
% 0.30/0.47 => ( iffalseProp2
% 0.30/0.47 => ( iftrueProp1
% 0.30/0.47 => ( iftrueProp2
% 0.30/0.47 => ( ifSingleton
% 0.30/0.47 => ( ifp
% 0.30/0.47 => ( theeq
% 0.30/0.47 => ( iftrue
% 0.30/0.47 => ( iffalse
% 0.30/0.47 => ( iftrueorfalse
% 0.30/0.47 => ( binintersectT_lem
% 0.30/0.47 => ( binunionT_lem
% 0.30/0.47 => ( powersetT_lem
% 0.30/0.47 => ( setminusT_lem
% 0.30/0.47 => ( complementT_lem
% 0.30/0.47 => ( setextT
% 0.30/0.47 => ( subsetTI
% 0.30/0.47 => ( powersetTI1
% 0.30/0.47 => ( powersetTE1
% 0.30/0.47 => ( complementTI1
% 0.30/0.47 => ( complementTE1
% 0.30/0.47 => ( binintersectTELcontra
% 0.30/0.47 => ( binintersectTERcontra
% 0.30/0.47 => ( contrasubsetT
% 0.30/0.47 => ( contrasubsetT1
% 0.30/0.47 => ( contrasubsetT2
% 0.30/0.47 => ( contrasubsetT3
% 0.30/0.47 => ( doubleComplementI1
% 0.30/0.47 => ( doubleComplementE1
% 0.30/0.47 => ( doubleComplementSub1
% 0.30/0.47 => ( doubleComplementSub2
% 0.30/0.47 => ( doubleComplementEq
% 0.30/0.47 => ( complementTnotintersectT
% 0.30/0.47 => ( complementImpComplementIntersect
% 0.30/0.47 => ( complementSubsetComplementIntersect
% 0.30/0.47 => ( complementInPowersetComplementIntersect
% 0.30/0.47 => ( contraSubsetComplement
% 0.30/0.47 => ( complementTcontraSubset
% 0.30/0.47 => ( binunionTILcontra
% 0.30/0.47 => ( binunionTIRcontra
% 0.30/0.47 => ( inIntersectImpInUnion
% 0.30/0.47 => ( inIntersectImpInUnion2
% 0.30/0.47 => ( inIntersectImpInIntersectUnions
% 0.30/0.47 => ( intersectInPowersetIntersectUnions
% 0.30/0.47 => ( inComplementUnionImpNotIn1
% 0.30/0.47 => ( inComplementUnionImpInComplement1
% 0.30/0.47 => ( binunionTE
% 0.30/0.47 => ( binunionTEcontra
% 0.30/0.47 => ( demorgan2a1
% 0.30/0.47 => ( complementUnionInPowersetComplement
% 0.30/0.47 => ( demorgan2a2
% 0.30/0.47 => ( demorgan1a
% 0.30/0.47 => ( demorgan1b
% 0.30/0.47 => ( demorgan1
% 0.30/0.47 => ( demorgan2a
% 0.30/0.47 => ( demorgan2b2
% 0.30/0.47 => ( demorgan2b
% 0.30/0.47 => ( demorgan2
% 0.30/0.47 => ( woz13rule0
% 0.30/0.47 => ( woz13rule1
% 0.30/0.47 => ( woz13rule2
% 0.30/0.47 => ( woz13rule3
% 0.30/0.47 => ( woz13rule4
% 0.30/0.47 => ( woz1_1
% 0.30/0.47 => ( woz1_2
% 0.30/0.47 => ( woz1_3
% 0.30/0.47 => ( woz1_4
% 0.30/0.47 => ( woz1_5
% 0.30/0.47 => ( breln1all2
% 0.30/0.47 => ( breln1SetBreln1
% 0.30/0.47 => ( choice2fnsingleton
% 0.30/0.47 => ( setOfPairsIsBReln1
% 0.30/0.47 => ( breln1all1
% 0.30/0.47 => ( subbreln1
% 0.30/0.47 => ( eqbreln1
% 0.30/0.47 => ( breln1invprop
% 0.30/0.47 => ( breln1invI
% 0.30/0.47 => ( breln1invE
% 0.30/0.47 => ( breln1compprop
% 0.30/0.47 => ( breln1compI
% 0.30/0.47 => ( breln1compE
% 0.30/0.47 => ( breln1compEex
% 0.30/0.47 => ( breln1unionprop
% 0.30/0.47 => ( breln1unionIL
% 0.30/0.47 => ( breln1unionIR
% 0.30/0.47 => ( breln1unionI
% 0.30/0.47 => ( breln1unionE
% 0.30/0.47 => ( breln1unionEcases
% 0.30/0.47 => ( breln1unionCommutes
% 0.30/0.47 => ( woz2Ex
% 0.30/0.47 => ( woz2W
% 0.30/0.47 => ( woz2A
% 0.30/0.47 => ( woz2B
% 0.30/0.47 => ( image1Ex
% 0.30/0.47 => ( image1Ex1
% 0.30/0.47 => ( image1Equiv
% 0.30/0.47 => ( image1E
% 0.30/0.47 => ( image1I
% 0.30/0.47 => ( injFuncInInjFuncSet
% 0.30/0.47 => ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.30/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.59/1.47 ---- Embedded in THF ---
% 2.59/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.59/1.47 %%% Generated on Tue Feb 24 12:34:25 EST 2026
% 2.59/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.59/1.47 %%% Logic specification used:
% 2.59/1.47 %%% thf(spec, logic, $$dhol).
% 2.59/1.47
% 2.59/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.an1PBkiMQH/DTF2DTF18125.p
% See solution above
% 2.59/1.47 ------------------------
% 2.84/1.51 ---- Cleaned THF ---
% 2.84/1.52 thf(in_type,type,
% 2.84/1.52 in: $i > $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(exu_type,type,
% 2.84/1.52 exu: ( $i > $o ) > $o ).
% 2.84/1.52
% 2.84/1.52 thf(exu,definition,
% 2.84/1.52 ( exu
% 2.84/1.52 = ( ^ [Xphi: $i > $o] :
% 2.84/1.52 ? [Xx: $i] :
% 2.84/1.52 ( ( Xphi @ Xx )
% 2.84/1.52 & ! [Xy: $i] :
% 2.84/1.52 ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setextAx_type,type,
% 2.84/1.52 setextAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setextAx,definition,
% 2.84/1.52 ( setextAx
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 <=> ( in @ Xx @ B ) )
% 2.84/1.52 => ( A = B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptyset_type,type,
% 2.84/1.52 emptyset: $i ).
% 2.84/1.52
% 2.84/1.52 thf(emptysetAx_type,type,
% 2.84/1.52 emptysetAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptysetAx,definition,
% 2.84/1.52 ( emptysetAx
% 2.84/1.52 = ( ! [Xx: $i] :
% 2.84/1.52 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoin_type,type,
% 2.84/1.52 setadjoin: $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinAx_type,type,
% 2.84/1.52 setadjoinAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinAx,definition,
% 2.84/1.52 ( setadjoinAx
% 2.84/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.52 <=> ( ( Xy = Xx )
% 2.84/1.52 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powerset_type,type,
% 2.84/1.52 powerset: $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(powersetAx_type,type,
% 2.84/1.52 powersetAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetAx,definition,
% 2.84/1.52 ( powersetAx
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.52 <=> ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setunion_type,type,
% 2.84/1.52 setunion: $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(setunionAx_type,type,
% 2.84/1.52 setunionAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setunionAx,definition,
% 2.84/1.52 ( setunionAx
% 2.84/1.52 = ( ! [A: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.52 <=> ? [B: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 & ( in @ B @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(omega_type,type,
% 2.84/1.52 omega: $i ).
% 2.84/1.52
% 2.84/1.52 thf(omega0Ax_type,type,
% 2.84/1.52 omega0Ax: $o ).
% 2.84/1.52
% 2.84/1.52 thf(omega0Ax,definition,
% 2.84/1.52 ( omega0Ax
% 2.84/1.52 = ( in @ emptyset @ omega ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(omegaSAx_type,type,
% 2.84/1.52 omegaSAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(omegaSAx,definition,
% 2.84/1.52 ( omegaSAx
% 2.84/1.52 = ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ omega )
% 2.84/1.52 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(omegaIndAx_type,type,
% 2.84/1.52 omegaIndAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(omegaIndAx,definition,
% 2.84/1.52 ( omegaIndAx
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ( ( in @ emptyset @ A )
% 2.84/1.52 & ! [Xx: $i] :
% 2.84/1.52 ( ( ( in @ Xx @ omega )
% 2.84/1.52 & ( in @ Xx @ A ) )
% 2.84/1.52 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ omega )
% 2.84/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(replAx_type,type,
% 2.84/1.52 replAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(replAx,definition,
% 2.84/1.52 ( replAx
% 2.84/1.52 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( exu
% 2.84/1.52 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.84/1.52 => ? [B: $i] :
% 2.84/1.52 ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 <=> ? [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(foundationAx_type,type,
% 2.84/1.52 foundationAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(foundationAx,definition,
% 2.84/1.52 ( foundationAx
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.84/1.52 => ? [B: $i] :
% 2.84/1.52 ( ( in @ B @ A )
% 2.84/1.52 & ~ ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(wellorderingAx_type,type,
% 2.84/1.52 wellorderingAx: $o ).
% 2.84/1.52
% 2.84/1.52 thf(wellorderingAx,definition,
% 2.84/1.52 ( wellorderingAx
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ? [B: $i] :
% 2.84/1.52 ( ! [C: $i] :
% 2.84/1.52 ( ( in @ C @ B )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ C )
% 2.84/1.52 => ( in @ Xx @ A ) ) )
% 2.84/1.52 & ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( ( in @ Xx @ A )
% 2.84/1.52 & ( in @ Xy @ A ) )
% 2.84/1.52 => ( ! [C: $i] :
% 2.84/1.52 ( ( in @ C @ B )
% 2.84/1.52 => ( ( in @ Xx @ C )
% 2.84/1.52 <=> ( in @ Xy @ C ) ) )
% 2.84/1.52 => ( Xx = Xy ) ) )
% 2.84/1.52 & ! [C: $i,D: $i] :
% 2.84/1.52 ( ( ( in @ C @ B )
% 2.84/1.52 & ( in @ D @ B ) )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ C )
% 2.84/1.52 => ( in @ Xx @ D ) )
% 2.84/1.52 | ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ D )
% 2.84/1.52 => ( in @ Xx @ C ) ) ) )
% 2.84/1.52 & ! [C: $i] :
% 2.84/1.52 ( ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ C )
% 2.84/1.52 => ( in @ Xx @ A ) )
% 2.84/1.52 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.84/1.52 => ? [D: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ D @ B )
% 2.84/1.52 & ( in @ Xx @ C )
% 2.84/1.52 & ~ ? [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ D )
% 2.84/1.52 & ( in @ Xy @ C ) )
% 2.84/1.52 & ! [E: $i] :
% 2.84/1.52 ( ( in @ E @ B )
% 2.84/1.52 => ( ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ E )
% 2.84/1.52 => ( in @ Xy @ D ) )
% 2.84/1.52 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(descr_type,type,
% 2.84/1.52 descr: ( $i > $o ) > $i ).
% 2.84/1.52
% 2.84/1.52 thf(descrp_type,type,
% 2.84/1.52 descrp: $o ).
% 2.84/1.52
% 2.84/1.52 thf(descrp,definition,
% 2.84/1.52 ( descrp
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ( Xphi
% 2.84/1.52 @ ( descr
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstr_type,type,
% 2.84/1.52 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstrI_type,type,
% 2.84/1.52 dsetconstrI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstrI,definition,
% 2.84/1.52 ( dsetconstrI
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 => ( in @ Xx
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstrEL_type,type,
% 2.84/1.52 dsetconstrEL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstrEL,definition,
% 2.84/1.52 ( dsetconstrEL
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstrER_type,type,
% 2.84/1.52 dsetconstrER: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstrER,definition,
% 2.84/1.52 ( dsetconstrER
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.52 => ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exuE1_type,type,
% 2.84/1.52 exuE1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exuE1,definition,
% 2.84/1.52 ( exuE1
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ? [Xx: $i] :
% 2.84/1.52 ( ( Xphi @ Xx )
% 2.84/1.52 & ! [Xy: $i] :
% 2.84/1.52 ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(prop2set_type,type,
% 2.84/1.52 prop2set: $o > $i ).
% 2.84/1.52
% 2.84/1.52 thf(prop2setE_type,type,
% 2.84/1.52 prop2setE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(prop2setE,definition,
% 2.84/1.52 ( prop2setE
% 2.84/1.52 = ( ! [Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.84/1.52 => Xphi ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptysetE_type,type,
% 2.84/1.52 emptysetE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptysetE,definition,
% 2.84/1.52 ( emptysetE
% 2.84/1.52 = ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ emptyset )
% 2.84/1.52 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptysetimpfalse_type,type,
% 2.84/1.52 emptysetimpfalse: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptysetimpfalse,definition,
% 2.84/1.52 ( emptysetimpfalse
% 2.84/1.52 = ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ emptyset )
% 2.84/1.52 => $false ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(notinemptyset_type,type,
% 2.84/1.52 notinemptyset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(notinemptyset,definition,
% 2.84/1.52 ( notinemptyset
% 2.84/1.52 = ( ! [Xx: $i] :
% 2.84/1.52 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exuE3e_type,type,
% 2.84/1.52 exuE3e: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exuE3e,definition,
% 2.84/1.52 ( exuE3e
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setext_type,type,
% 2.84/1.52 setext: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setext,definition,
% 2.84/1.52 ( setext
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ Xx @ B ) )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ A ) )
% 2.84/1.52 => ( A = B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptyI_type,type,
% 2.84/1.52 emptyI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptyI,definition,
% 2.84/1.52 ( emptyI
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ~ ( in @ Xx @ A )
% 2.84/1.52 => ( A = emptyset ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(noeltsimpempty_type,type,
% 2.84/1.52 noeltsimpempty: $o ).
% 2.84/1.52
% 2.84/1.52 thf(noeltsimpempty,definition,
% 2.84/1.52 ( noeltsimpempty
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ~ ( in @ Xx @ A )
% 2.84/1.52 => ( A = emptyset ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setbeta_type,type,
% 2.84/1.52 setbeta: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setbeta,definition,
% 2.84/1.52 ( setbeta
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.52 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(nonempty_type,type,
% 2.84/1.52 nonempty: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(nonempty,definition,
% 2.84/1.52 ( nonempty
% 2.84/1.52 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(nonemptyE1_type,type,
% 2.84/1.52 nonemptyE1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(nonemptyE1,definition,
% 2.84/1.52 ( nonemptyE1
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ( nonempty @ A )
% 2.84/1.52 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(nonemptyI_type,type,
% 2.84/1.52 nonemptyI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(nonemptyI,definition,
% 2.84/1.52 ( nonemptyI
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 => ( nonempty
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(nonemptyI1_type,type,
% 2.84/1.52 nonemptyI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(nonemptyI1,definition,
% 2.84/1.52 ( nonemptyI1
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.84/1.52 => ( nonempty @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinIL_type,type,
% 2.84/1.52 setadjoinIL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinIL,definition,
% 2.84/1.52 ( setadjoinIL
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptyinunitempty_type,type,
% 2.84/1.52 emptyinunitempty: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptyinunitempty,definition,
% 2.84/1.52 ( emptyinunitempty
% 2.84/1.52 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinIR_type,type,
% 2.84/1.52 setadjoinIR: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinIR,definition,
% 2.84/1.52 ( setadjoinIR
% 2.84/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinE_type,type,
% 2.84/1.52 setadjoinE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinE,definition,
% 2.84/1.52 ( setadjoinE
% 2.84/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.52 => ! [Xphi: $o] :
% 2.84/1.52 ( ( ( Xy = Xx )
% 2.84/1.52 => Xphi )
% 2.84/1.52 => ( ( ( in @ Xy @ A )
% 2.84/1.52 => Xphi )
% 2.84/1.52 => Xphi ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinOr_type,type,
% 2.84/1.52 setadjoinOr: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinOr,definition,
% 2.84/1.52 ( setadjoinOr
% 2.84/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.52 => ( ( Xy = Xx )
% 2.84/1.52 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setoftrueEq_type,type,
% 2.84/1.52 setoftrueEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setoftrueEq,definition,
% 2.84/1.52 ( setoftrueEq
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xx: $i] : $true )
% 2.84/1.52 = A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powersetI_type,type,
% 2.84/1.52 powersetI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetI,definition,
% 2.84/1.52 ( powersetI
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ A ) )
% 2.84/1.52 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptyinPowerset_type,type,
% 2.84/1.52 emptyinPowerset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptyinPowerset,definition,
% 2.84/1.52 ( emptyinPowerset
% 2.84/1.52 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptyInPowerset_type,type,
% 2.84/1.52 emptyInPowerset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptyInPowerset,definition,
% 2.84/1.52 ( emptyInPowerset
% 2.84/1.52 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powersetE_type,type,
% 2.84/1.52 powersetE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetE,definition,
% 2.84/1.52 ( powersetE
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.52 => ( ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setunionI_type,type,
% 2.84/1.52 setunionI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setunionI,definition,
% 2.84/1.52 ( setunionI
% 2.84/1.52 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 => ( ( in @ B @ A )
% 2.84/1.52 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setunionE_type,type,
% 2.84/1.52 setunionE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setunionE,definition,
% 2.84/1.52 ( setunionE
% 2.84/1.52 = ( ! [A: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.52 => ! [Xphi: $o] :
% 2.84/1.52 ( ! [B: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 => ( ( in @ B @ A )
% 2.84/1.52 => Xphi ) )
% 2.84/1.52 => Xphi ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subPowSU_type,type,
% 2.84/1.52 subPowSU: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subPowSU,definition,
% 2.84/1.52 ( subPowSU
% 2.84/1.52 = ( ! [A: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exuE2_type,type,
% 2.84/1.52 exuE2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exuE2,definition,
% 2.84/1.52 ( exuE2
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ? [Xx: $i] :
% 2.84/1.52 ! [Xy: $i] :
% 2.84/1.52 ( ( Xphi @ Xy )
% 2.84/1.52 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(nonemptyImpWitness_type,type,
% 2.84/1.52 nonemptyImpWitness: $o ).
% 2.84/1.52
% 2.84/1.52 thf(nonemptyImpWitness,definition,
% 2.84/1.52 ( nonemptyImpWitness
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ( nonempty @ A )
% 2.84/1.52 => ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & $true ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(uniqinunit_type,type,
% 2.84/1.52 uniqinunit: $o ).
% 2.84/1.52
% 2.84/1.52 thf(uniqinunit,definition,
% 2.84/1.52 ( uniqinunit
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(notinsingleton_type,type,
% 2.84/1.52 notinsingleton: $o ).
% 2.84/1.52
% 2.84/1.52 thf(notinsingleton,definition,
% 2.84/1.52 ( notinsingleton
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx != Xy )
% 2.84/1.52 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(eqinunit_type,type,
% 2.84/1.52 eqinunit: $o ).
% 2.84/1.52
% 2.84/1.52 thf(eqinunit,definition,
% 2.84/1.52 ( eqinunit
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx = Xy )
% 2.84/1.52 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(singletonsswitch_type,type,
% 2.84/1.52 singletonsswitch: $o ).
% 2.84/1.52
% 2.84/1.52 thf(singletonsswitch,definition,
% 2.84/1.52 ( singletonsswitch
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.52 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(upairsetE_type,type,
% 2.84/1.52 upairsetE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairsetE,definition,
% 2.84/1.52 ( upairsetE
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.52 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.84/1.52 => ( ( Xz = Xx )
% 2.84/1.52 | ( Xz = Xy ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(upairsetIL_type,type,
% 2.84/1.52 upairsetIL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairsetIL,definition,
% 2.84/1.52 ( upairsetIL
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(upairsetIR_type,type,
% 2.84/1.52 upairsetIR: $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairsetIR,definition,
% 2.84/1.52 ( upairsetIR
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptyE1_type,type,
% 2.84/1.52 emptyE1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptyE1,definition,
% 2.84/1.52 ( emptyE1
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ( Xphi @ Xx ) )
% 2.84/1.52 => ( ( ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 = emptyset )
% 2.84/1.52 => $false ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(vacuousDall_type,type,
% 2.84/1.52 vacuousDall: $o ).
% 2.84/1.52
% 2.84/1.52 thf(vacuousDall,definition,
% 2.84/1.52 ( vacuousDall
% 2.84/1.52 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ emptyset )
% 2.84/1.52 => ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(quantDeMorgan1_type,type,
% 2.84/1.52 quantDeMorgan1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(quantDeMorgan1,definition,
% 2.84/1.52 ( quantDeMorgan1
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ~ ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( Xphi @ Xx ) )
% 2.84/1.52 => ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(quantDeMorgan2_type,type,
% 2.84/1.52 quantDeMorgan2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(quantDeMorgan2,definition,
% 2.84/1.52 ( quantDeMorgan2
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ~ ( Xphi @ Xx ) )
% 2.84/1.52 => ~ ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(quantDeMorgan3_type,type,
% 2.84/1.52 quantDeMorgan3: $o ).
% 2.84/1.52
% 2.84/1.52 thf(quantDeMorgan3,definition,
% 2.84/1.52 ( quantDeMorgan3
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ~ ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ( Xphi @ Xx ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(quantDeMorgan4_type,type,
% 2.84/1.52 quantDeMorgan4: $o ).
% 2.84/1.52
% 2.84/1.52 thf(quantDeMorgan4,definition,
% 2.84/1.52 ( quantDeMorgan4
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ~ ( Xphi @ Xx ) )
% 2.84/1.52 => ~ ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(prop2setI_type,type,
% 2.84/1.52 prop2setI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(prop2setI,definition,
% 2.84/1.52 ( prop2setI
% 2.84/1.52 = ( ! [Xphi: $o] :
% 2.84/1.52 ( Xphi
% 2.84/1.52 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(set2prop_type,type,
% 2.84/1.52 set2prop: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(prop2set2propI_type,type,
% 2.84/1.52 prop2set2propI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(prop2set2propI,definition,
% 2.84/1.52 ( prop2set2propI
% 2.84/1.52 = ( ! [Xphi: $o] :
% 2.84/1.52 ( Xphi
% 2.84/1.52 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(notdexE_type,type,
% 2.84/1.52 notdexE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(notdexE,definition,
% 2.84/1.52 ( notdexE
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ~ ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ( Xphi @ Xx ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(notdallE_type,type,
% 2.84/1.52 notdallE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(notdallE,definition,
% 2.84/1.52 ( notdallE
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ~ ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( Xphi @ Xx ) )
% 2.84/1.52 => ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exuI1_type,type,
% 2.84/1.52 exuI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exuI1,definition,
% 2.84/1.52 ( exuI1
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ? [Xx: $i] :
% 2.84/1.52 ( ( Xphi @ Xx )
% 2.84/1.52 & ! [Xy: $i] :
% 2.84/1.52 ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) )
% 2.84/1.52 => ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exuI3_type,type,
% 2.84/1.52 exuI3: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exuI3,definition,
% 2.84/1.52 ( exuI3
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.84/1.52 => ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xphi @ Xx )
% 2.84/1.52 => ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) )
% 2.84/1.52 => ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exuI2_type,type,
% 2.84/1.52 exuI2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exuI2,definition,
% 2.84/1.52 ( exuI2
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ? [Xx: $i] :
% 2.84/1.52 ! [Xy: $i] :
% 2.84/1.52 ( ( Xphi @ Xy )
% 2.84/1.52 <=> ( Xy = Xx ) )
% 2.84/1.52 => ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(inCongP_type,type,
% 2.84/1.52 inCongP: $o ).
% 2.84/1.52
% 2.84/1.52 thf(inCongP,definition,
% 2.84/1.52 ( inCongP
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( A = B )
% 2.84/1.52 => ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx = Xy )
% 2.84/1.52 => ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(in__Cong_type,type,
% 2.84/1.52 in__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(in__Cong,definition,
% 2.84/1.52 ( in__Cong
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( A = B )
% 2.84/1.52 => ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx = Xy )
% 2.84/1.52 => ( ( in @ Xx @ A )
% 2.84/1.52 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exuE3u_type,type,
% 2.84/1.52 exuE3u: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exuE3u,definition,
% 2.84/1.52 ( exuE3u
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xphi @ Xx )
% 2.84/1.52 => ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exu__Cong_type,type,
% 2.84/1.52 exu__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exu__Cong,definition,
% 2.84/1.52 ( exu__Cong
% 2.84/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.52 ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx = Xy )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 <=> ( Xpsi @ Xy ) ) )
% 2.84/1.52 => ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 <=> ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptyset__Cong_type,type,
% 2.84/1.52 emptyset__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptyset__Cong,definition,
% 2.84/1.52 ( emptyset__Cong
% 2.84/1.52 = ( emptyset = emptyset ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoin__Cong_type,type,
% 2.84/1.52 setadjoin__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoin__Cong,definition,
% 2.84/1.52 ( setadjoin__Cong
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx = Xy )
% 2.84/1.52 => ! [Xz: $i,Xu: $i] :
% 2.84/1.52 ( ( Xz = Xu )
% 2.84/1.52 => ( ( setadjoin @ Xx @ Xz )
% 2.84/1.52 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powerset__Cong_type,type,
% 2.84/1.52 powerset__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powerset__Cong,definition,
% 2.84/1.52 ( powerset__Cong
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( A = B )
% 2.84/1.52 => ( ( powerset @ A )
% 2.84/1.52 = ( powerset @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setunion__Cong_type,type,
% 2.84/1.52 setunion__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setunion__Cong,definition,
% 2.84/1.52 ( setunion__Cong
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( A = B )
% 2.84/1.52 => ( ( setunion @ A )
% 2.84/1.52 = ( setunion @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(omega__Cong_type,type,
% 2.84/1.52 omega__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(omega__Cong,definition,
% 2.84/1.52 ( omega__Cong
% 2.84/1.52 = ( omega = omega ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(exuEu_type,type,
% 2.84/1.52 exuEu: $o ).
% 2.84/1.52
% 2.84/1.52 thf(exuEu,definition,
% 2.84/1.52 ( exuEu
% 2.84/1.52 = ( ! [Xphi: $i > $o] :
% 2.84/1.52 ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xphi @ Xx )
% 2.84/1.52 => ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(descr__Cong_type,type,
% 2.84/1.52 descr__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(descr__Cong,definition,
% 2.84/1.52 ( descr__Cong
% 2.84/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.52 ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx = Xy )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 <=> ( Xpsi @ Xy ) ) )
% 2.84/1.52 => ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ( ( exu
% 2.84/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.84/1.52 => ( ( descr
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 = ( descr
% 2.84/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstr__Cong_type,type,
% 2.84/1.52 dsetconstr__Cong: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dsetconstr__Cong,definition,
% 2.84/1.52 ( dsetconstr__Cong
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( A = B )
% 2.84/1.52 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( Xx = Xy )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.84/1.52 => ( ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 = ( dsetconstr @ B
% 2.84/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subset_type,type,
% 2.84/1.52 subset: $i > $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(disjoint_type,type,
% 2.84/1.52 disjoint: $i > $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(setsmeet_type,type,
% 2.84/1.52 setsmeet: $i > $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetI1_type,type,
% 2.84/1.52 subsetI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetI1,definition,
% 2.84/1.52 ( subsetI1
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ Xx @ B ) )
% 2.84/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(eqimpsubset2_type,type,
% 2.84/1.52 eqimpsubset2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(eqimpsubset2,definition,
% 2.84/1.52 ( eqimpsubset2
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( A = B )
% 2.84/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(eqimpsubset1_type,type,
% 2.84/1.52 eqimpsubset1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(eqimpsubset1,definition,
% 2.84/1.52 ( eqimpsubset1
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( A = B )
% 2.84/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subsetI2_type,type,
% 2.84/1.52 subsetI2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetI2,definition,
% 2.84/1.52 ( subsetI2
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ Xx @ B ) )
% 2.84/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(emptysetsubset_type,type,
% 2.84/1.52 emptysetsubset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(emptysetsubset,definition,
% 2.84/1.52 ( emptysetsubset
% 2.84/1.52 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subsetE_type,type,
% 2.84/1.52 subsetE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetE,definition,
% 2.84/1.52 ( subsetE
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subsetE2_type,type,
% 2.84/1.52 subsetE2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetE2,definition,
% 2.84/1.52 ( subsetE2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( ~ ( in @ Xx @ B )
% 2.84/1.52 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(notsubsetI_type,type,
% 2.84/1.52 notsubsetI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(notsubsetI,definition,
% 2.84/1.52 ( notsubsetI
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ B )
% 2.84/1.52 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(notequalI1_type,type,
% 2.84/1.52 notequalI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(notequalI1,definition,
% 2.84/1.52 ( notequalI1
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ~ ( subset @ A @ B )
% 2.84/1.52 => ( A != B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(notequalI2_type,type,
% 2.84/1.52 notequalI2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(notequalI2,definition,
% 2.84/1.52 ( notequalI2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ B )
% 2.84/1.52 => ( A != B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subsetRefl_type,type,
% 2.84/1.52 subsetRefl: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetRefl,definition,
% 2.84/1.52 ( subsetRefl
% 2.84/1.52 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subsetTrans_type,type,
% 2.84/1.52 subsetTrans: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetTrans,definition,
% 2.84/1.52 ( subsetTrans
% 2.84/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( ( subset @ B @ C )
% 2.84/1.52 => ( subset @ A @ C ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinSub_type,type,
% 2.84/1.52 setadjoinSub: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinSub,definition,
% 2.84/1.52 ( setadjoinSub
% 2.84/1.52 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinSub2_type,type,
% 2.84/1.52 setadjoinSub2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setadjoinSub2,definition,
% 2.84/1.52 ( setadjoinSub2
% 2.84/1.52 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subset2powerset_type,type,
% 2.84/1.52 subset2powerset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subset2powerset,definition,
% 2.84/1.52 ( subset2powerset
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setextsub_type,type,
% 2.84/1.52 setextsub: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setextsub,definition,
% 2.84/1.52 ( setextsub
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( ( subset @ B @ A )
% 2.84/1.52 => ( A = B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subsetemptysetimpeq_type,type,
% 2.84/1.52 subsetemptysetimpeq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetemptysetimpeq,definition,
% 2.84/1.52 ( subsetemptysetimpeq
% 2.84/1.52 = ( ! [A: $i] :
% 2.84/1.52 ( ( subset @ A @ emptyset )
% 2.84/1.52 => ( A = emptyset ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powersetI1_type,type,
% 2.84/1.52 powersetI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetI1,definition,
% 2.84/1.52 ( powersetI1
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( subset @ B @ A )
% 2.84/1.52 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powersetE1_type,type,
% 2.84/1.52 powersetE1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetE1,definition,
% 2.84/1.52 ( powersetE1
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(inPowerset_type,type,
% 2.84/1.52 inPowerset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(inPowerset,definition,
% 2.84/1.52 ( inPowerset
% 2.84/1.52 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powersetsubset_type,type,
% 2.84/1.52 powersetsubset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetsubset,definition,
% 2.84/1.52 ( powersetsubset
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(sepInPowerset_type,type,
% 2.84/1.52 sepInPowerset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(sepInPowerset,definition,
% 2.84/1.52 ( sepInPowerset
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( in
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 @ ( powerset @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(sepSubset_type,type,
% 2.84/1.52 sepSubset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(sepSubset,definition,
% 2.84/1.52 ( sepSubset
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( subset
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunion_type,type,
% 2.84/1.52 binunion: $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(binunionIL_type,type,
% 2.84/1.52 binunionIL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionIL,definition,
% 2.84/1.52 ( binunionIL
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(upairset2IR_type,type,
% 2.84/1.52 upairset2IR: $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairset2IR,definition,
% 2.84/1.52 ( upairset2IR
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionIR_type,type,
% 2.84/1.52 binunionIR: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionIR,definition,
% 2.84/1.52 ( binunionIR
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionEcases_type,type,
% 2.84/1.52 binunionEcases: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionEcases,definition,
% 2.84/1.52 ( binunionEcases
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.84/1.52 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.84/1.52 => ( ( ( in @ Xx @ A )
% 2.84/1.52 => Xphi )
% 2.84/1.52 => ( ( ( in @ Xx @ B )
% 2.84/1.52 => Xphi )
% 2.84/1.52 => Xphi ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionE_type,type,
% 2.84/1.52 binunionE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionE,definition,
% 2.84/1.52 ( binunionE
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.84/1.52 => ( ( in @ Xx @ A )
% 2.84/1.52 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionLsub_type,type,
% 2.84/1.52 binunionLsub: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionLsub,definition,
% 2.84/1.52 ( binunionLsub
% 2.84/1.52 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionRsub_type,type,
% 2.84/1.52 binunionRsub: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionRsub,definition,
% 2.84/1.52 ( binunionRsub
% 2.84/1.52 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersect_type,type,
% 2.84/1.52 binintersect: $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectI_type,type,
% 2.84/1.52 binintersectI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectI,definition,
% 2.84/1.52 ( binintersectI
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset5_type,type,
% 2.84/1.52 binintersectSubset5: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset5,definition,
% 2.84/1.52 ( binintersectSubset5
% 2.84/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.52 ( ( subset @ C @ A )
% 2.84/1.52 => ( ( subset @ C @ B )
% 2.84/1.52 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectEL_type,type,
% 2.84/1.52 binintersectEL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectEL,definition,
% 2.84/1.52 ( binintersectEL
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.84/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectLsub_type,type,
% 2.84/1.52 binintersectLsub: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectLsub,definition,
% 2.84/1.52 ( binintersectLsub
% 2.84/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset2_type,type,
% 2.84/1.52 binintersectSubset2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset2,definition,
% 2.84/1.52 ( binintersectSubset2
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( ( binintersect @ A @ B )
% 2.84/1.52 = A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset3_type,type,
% 2.84/1.52 binintersectSubset3: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset3,definition,
% 2.84/1.52 ( binintersectSubset3
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( ( binintersect @ A @ B )
% 2.84/1.52 = B )
% 2.84/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectER_type,type,
% 2.84/1.52 binintersectER: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectER,definition,
% 2.84/1.52 ( binintersectER
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.84/1.52 => ( in @ Xx @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(disjointsetsI1_type,type,
% 2.84/1.52 disjointsetsI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(disjointsetsI1,definition,
% 2.84/1.52 ( disjointsetsI1
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ~ ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ( in @ Xx @ B ) )
% 2.84/1.52 => ( ( binintersect @ A @ B )
% 2.84/1.52 = emptyset ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectRsub_type,type,
% 2.84/1.52 binintersectRsub: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectRsub,definition,
% 2.84/1.52 ( binintersectRsub
% 2.84/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset4_type,type,
% 2.84/1.52 binintersectSubset4: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset4,definition,
% 2.84/1.52 ( binintersectSubset4
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( subset @ B @ A )
% 2.84/1.52 => ( ( binintersect @ A @ B )
% 2.84/1.52 = B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset1_type,type,
% 2.84/1.52 binintersectSubset1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectSubset1,definition,
% 2.84/1.52 ( binintersectSubset1
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( ( binintersect @ A @ B )
% 2.84/1.52 = A )
% 2.84/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(bs114d_type,type,
% 2.84/1.52 bs114d: $o ).
% 2.84/1.52
% 2.84/1.52 thf(bs114d,definition,
% 2.84/1.52 ( bs114d
% 2.84/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.52 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.84/1.52 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(regular_type,type,
% 2.84/1.52 regular: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminus_type,type,
% 2.84/1.52 setminus: $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(setminusI_type,type,
% 2.84/1.52 setminusI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusI,definition,
% 2.84/1.52 ( setminusI
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusEL_type,type,
% 2.84/1.52 setminusEL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusEL,definition,
% 2.84/1.52 ( setminusEL
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusER_type,type,
% 2.84/1.52 setminusER: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusER,definition,
% 2.84/1.52 ( setminusER
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.52 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusSubset2_type,type,
% 2.84/1.52 setminusSubset2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusSubset2,definition,
% 2.84/1.52 ( setminusSubset2
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( subset @ A @ B )
% 2.84/1.52 => ( ( setminus @ A @ B )
% 2.84/1.52 = emptyset ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusERneg_type,type,
% 2.84/1.52 setminusERneg: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusERneg,definition,
% 2.84/1.52 ( setminusERneg
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.52 => ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusELneg_type,type,
% 2.84/1.52 setminusELneg: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusELneg,definition,
% 2.84/1.52 ( setminusELneg
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.52 => ( ~ ( in @ Xx @ B )
% 2.84/1.52 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusILneg_type,type,
% 2.84/1.52 setminusILneg: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusILneg,definition,
% 2.84/1.52 ( setminusILneg
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ~ ( in @ Xx @ A )
% 2.84/1.52 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusIRneg_type,type,
% 2.84/1.52 setminusIRneg: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusIRneg,definition,
% 2.84/1.52 ( setminusIRneg
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ B )
% 2.84/1.52 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusLsub_type,type,
% 2.84/1.52 setminusLsub: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusLsub,definition,
% 2.84/1.52 ( setminusLsub
% 2.84/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusSubset1_type,type,
% 2.84/1.52 setminusSubset1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusSubset1,definition,
% 2.84/1.52 ( setminusSubset1
% 2.84/1.52 = ( ! [A: $i,B: $i] :
% 2.84/1.52 ( ( ( setminus @ A @ B )
% 2.84/1.52 = emptyset )
% 2.84/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(symdiff_type,type,
% 2.84/1.52 symdiff: $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffE_type,type,
% 2.84/1.52 symdiffE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffE,definition,
% 2.84/1.52 ( symdiffE
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.84/1.52 => ! [Xphi: $o] :
% 2.84/1.52 ( ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ B )
% 2.84/1.52 => Xphi ) )
% 2.84/1.52 => ( ( ~ ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ B )
% 2.84/1.52 => Xphi ) )
% 2.84/1.52 => Xphi ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffI1_type,type,
% 2.84/1.52 symdiffI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffI1,definition,
% 2.84/1.52 ( symdiffI1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffI2_type,type,
% 2.84/1.52 symdiffI2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffI2,definition,
% 2.84/1.52 ( symdiffI2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ~ ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ B )
% 2.84/1.52 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffIneg1_type,type,
% 2.84/1.52 symdiffIneg1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffIneg1,definition,
% 2.84/1.52 ( symdiffIneg1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ B )
% 2.84/1.52 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffIneg2_type,type,
% 2.84/1.52 symdiffIneg2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(symdiffIneg2,definition,
% 2.84/1.52 ( symdiffIneg2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ~ ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ B )
% 2.84/1.52 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(iskpair_type,type,
% 2.84/1.52 iskpair: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(secondinupair_type,type,
% 2.84/1.52 secondinupair: $o ).
% 2.84/1.52
% 2.84/1.52 thf(secondinupair,definition,
% 2.84/1.52 ( secondinupair
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairIL_type,type,
% 2.84/1.52 setukpairIL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairIL,definition,
% 2.84/1.52 ( setukpairIL
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairIR_type,type,
% 2.84/1.52 setukpairIR: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairIR,definition,
% 2.84/1.52 ( setukpairIR
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(kpairiskpair_type,type,
% 2.84/1.52 kpairiskpair: $o ).
% 2.84/1.52
% 2.84/1.52 thf(kpairiskpair,definition,
% 2.84/1.52 ( kpairiskpair
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(kpair_type,type,
% 2.84/1.52 kpair: $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(kpairp_type,type,
% 2.84/1.52 kpairp: $o ).
% 2.84/1.52
% 2.84/1.52 thf(kpairp,definition,
% 2.84/1.52 ( kpairp
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprod_type,type,
% 2.84/1.52 cartprod: $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(singletonsubset_type,type,
% 2.84/1.52 singletonsubset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(singletonsubset,definition,
% 2.84/1.52 ( singletonsubset
% 2.84/1.52 = ( ! [A: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(singletoninpowerset_type,type,
% 2.84/1.52 singletoninpowerset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(singletoninpowerset,definition,
% 2.84/1.52 ( singletoninpowerset
% 2.84/1.52 = ( ! [A: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(singletoninpowunion_type,type,
% 2.84/1.52 singletoninpowunion: $o ).
% 2.84/1.52
% 2.84/1.52 thf(singletoninpowunion,definition,
% 2.84/1.52 ( singletoninpowunion
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(upairset2E_type,type,
% 2.84/1.52 upairset2E: $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairset2E,definition,
% 2.84/1.52 ( upairset2E
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.52 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.84/1.52 => ( ( Xz = Xx )
% 2.84/1.52 | ( Xz = Xy ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(upairsubunion_type,type,
% 2.84/1.52 upairsubunion: $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairsubunion,definition,
% 2.84/1.52 ( upairsubunion
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(upairinpowunion_type,type,
% 2.84/1.52 upairinpowunion: $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairinpowunion,definition,
% 2.84/1.52 ( upairinpowunion
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ubforcartprodlem1_type,type,
% 2.84/1.52 ubforcartprodlem1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ubforcartprodlem1,definition,
% 2.84/1.52 ( ubforcartprodlem1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ubforcartprodlem2_type,type,
% 2.84/1.52 ubforcartprodlem2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ubforcartprodlem2,definition,
% 2.84/1.52 ( ubforcartprodlem2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ubforcartprodlem3_type,type,
% 2.84/1.52 ubforcartprodlem3: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ubforcartprodlem3,definition,
% 2.84/1.52 ( ubforcartprodlem3
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodpairin_type,type,
% 2.84/1.52 cartprodpairin: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodpairin,definition,
% 2.84/1.52 ( cartprodpairin
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodmempair1_type,type,
% 2.84/1.52 cartprodmempair1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodmempair1,definition,
% 2.84/1.52 ( cartprodmempair1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.52 => ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ? [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 & ( Xu
% 2.84/1.52 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodmempair_type,type,
% 2.84/1.52 cartprodmempair: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodmempair,definition,
% 2.84/1.52 ( cartprodmempair
% 2.84/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.52 => ( iskpair @ Xu ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setunionE2_type,type,
% 2.84/1.52 setunionE2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setunionE2,definition,
% 2.84/1.52 ( setunionE2
% 2.84/1.52 = ( ! [A: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.52 => ? [X: $i] :
% 2.84/1.52 ( ( in @ X @ A )
% 2.84/1.52 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setunionsingleton1_type,type,
% 2.84/1.52 setunionsingleton1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setunionsingleton1,definition,
% 2.84/1.52 ( setunionsingleton1
% 2.84/1.52 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setunionsingleton2_type,type,
% 2.84/1.52 setunionsingleton2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setunionsingleton2,definition,
% 2.84/1.52 ( setunionsingleton2
% 2.84/1.52 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setunionsingleton_type,type,
% 2.84/1.52 setunionsingleton: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setunionsingleton,definition,
% 2.84/1.52 ( setunionsingleton
% 2.84/1.52 = ( ! [Xx: $i] :
% 2.84/1.52 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.84/1.52 = Xx ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(singleton_type,type,
% 2.84/1.52 singleton: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(singletonprop_type,type,
% 2.84/1.52 singletonprop: $o ).
% 2.84/1.52
% 2.84/1.52 thf(singletonprop,definition,
% 2.84/1.52 ( singletonprop
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 => ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) )
% 2.84/1.52 => ( ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ( Xphi @ Xx ) )
% 2.84/1.52 => ( singleton
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ex1_type,type,
% 2.84/1.52 ex1: $i > ( $i > $o ) > $o ).
% 2.84/1.52
% 2.84/1.52 thf(ex1E1_type,type,
% 2.84/1.52 ex1E1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ex1E1,definition,
% 2.84/1.52 ( ex1E1
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ( ex1 @ A
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ex1I_type,type,
% 2.84/1.52 ex1I: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ex1I,definition,
% 2.84/1.52 ( ex1I
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 => ( ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xy = Xx ) ) )
% 2.84/1.52 => ( ex1 @ A
% 2.84/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ex1I2_type,type,
% 2.84/1.52 ex1I2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ex1I2,definition,
% 2.84/1.52 ( ex1I2
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 => ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) )
% 2.84/1.52 => ( ? [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 & ( Xphi @ Xx ) )
% 2.84/1.52 => ( ex1 @ A
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(singletonsuniq_type,type,
% 2.84/1.52 singletonsuniq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(singletonsuniq,definition,
% 2.84/1.52 ( singletonsuniq
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( ( setadjoin @ Xx @ emptyset )
% 2.84/1.52 = ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(atmost1p_type,type,
% 2.84/1.52 atmost1p: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(atleast2p_type,type,
% 2.84/1.52 atleast2p: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(atmost2p_type,type,
% 2.84/1.52 atmost2p: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairsetp_type,type,
% 2.84/1.52 upairsetp: $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjL1_type,type,
% 2.84/1.52 setukpairinjL1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjL1,definition,
% 2.84/1.52 ( setukpairinjL1
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.52 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(kfstsingleton_type,type,
% 2.84/1.52 kfstsingleton: $o ).
% 2.84/1.52
% 2.84/1.52 thf(kfstsingleton,definition,
% 2.84/1.52 ( kfstsingleton
% 2.84/1.52 = ( ! [Xu: $i] :
% 2.84/1.52 ( ( iskpair @ Xu )
% 2.84/1.52 => ( singleton
% 2.84/1.52 @ ( dsetconstr @ ( setunion @ Xu )
% 2.84/1.52 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(theprop_type,type,
% 2.84/1.52 theprop: $o ).
% 2.84/1.52
% 2.84/1.52 thf(theprop,definition,
% 2.84/1.52 ( theprop
% 2.84/1.52 = ( ! [X: $i] :
% 2.84/1.52 ( ( singleton @ X )
% 2.84/1.52 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(kfst_type,type,
% 2.84/1.52 kfst: $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(kfstpairEq_type,type,
% 2.84/1.52 kfstpairEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(kfstpairEq,definition,
% 2.84/1.52 ( kfstpairEq
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.84/1.52 = Xx ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodfstin_type,type,
% 2.84/1.52 cartprodfstin: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodfstin,definition,
% 2.84/1.52 ( cartprodfstin
% 2.84/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.52 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjL2_type,type,
% 2.84/1.52 setukpairinjL2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjL2,definition,
% 2.84/1.52 ( setukpairinjL2
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjL_type,type,
% 2.84/1.52 setukpairinjL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjL,definition,
% 2.84/1.52 ( setukpairinjL
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.52 ( ( ( kpair @ Xx @ Xy )
% 2.84/1.52 = ( kpair @ Xz @ Xu ) )
% 2.84/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR11_type,type,
% 2.84/1.52 setukpairinjR11: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR11,definition,
% 2.84/1.52 ( setukpairinjR11
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx = Xy )
% 2.84/1.52 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.52 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR12_type,type,
% 2.84/1.52 setukpairinjR12: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR12,definition,
% 2.84/1.52 ( setukpairinjR12
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xx = Xy )
% 2.84/1.52 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.52 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR1_type,type,
% 2.84/1.52 setukpairinjR1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR1,definition,
% 2.84/1.52 ( setukpairinjR1
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.52 => ( ( Xz = Xu )
% 2.84/1.52 => ( Xy = Xu ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(upairequniteq_type,type,
% 2.84/1.52 upairequniteq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(upairequniteq,definition,
% 2.84/1.52 ( upairequniteq
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.52 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.52 = ( setadjoin @ Xz @ emptyset ) )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR2_type,type,
% 2.84/1.52 setukpairinjR2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR2,definition,
% 2.84/1.52 ( setukpairinjR2
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.52 => ( Xy = Xu ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR_type,type,
% 2.84/1.52 setukpairinjR: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setukpairinjR,definition,
% 2.84/1.52 ( setukpairinjR
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.52 ( ( ( kpair @ Xx @ Xy )
% 2.84/1.52 = ( kpair @ Xz @ Xu ) )
% 2.84/1.52 => ( Xy = Xu ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ksndsingleton_type,type,
% 2.84/1.52 ksndsingleton: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ksndsingleton,definition,
% 2.84/1.52 ( ksndsingleton
% 2.84/1.52 = ( ! [Xu: $i] :
% 2.84/1.52 ( ( iskpair @ Xu )
% 2.84/1.52 => ( singleton
% 2.84/1.52 @ ( dsetconstr @ ( setunion @ Xu )
% 2.84/1.52 @ ^ [Xx: $i] :
% 2.84/1.52 ( Xu
% 2.84/1.52 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ksnd_type,type,
% 2.84/1.52 ksnd: $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(ksndpairEq_type,type,
% 2.84/1.52 ksndpairEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ksndpairEq,definition,
% 2.84/1.52 ( ksndpairEq
% 2.84/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.84/1.52 = Xy ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(kpairsurjEq_type,type,
% 2.84/1.52 kpairsurjEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(kpairsurjEq,definition,
% 2.84/1.52 ( kpairsurjEq
% 2.84/1.52 = ( ! [Xu: $i] :
% 2.84/1.52 ( ( iskpair @ Xu )
% 2.84/1.52 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.84/1.52 = Xu ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodsndin_type,type,
% 2.84/1.52 cartprodsndin: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodsndin,definition,
% 2.84/1.52 ( cartprodsndin
% 2.84/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.52 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodpairmemEL_type,type,
% 2.84/1.52 cartprodpairmemEL: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodpairmemEL,definition,
% 2.84/1.52 ( cartprodpairmemEL
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.84/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodpairmemER_type,type,
% 2.84/1.52 cartprodpairmemER: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodpairmemER,definition,
% 2.84/1.52 ( cartprodpairmemER
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.84/1.52 => ( in @ Xy @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodmempaircEq_type,type,
% 2.84/1.52 cartprodmempaircEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodmempaircEq,definition,
% 2.84/1.52 ( cartprodmempaircEq
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( kpair @ Xx @ Xy )
% 2.84/1.52 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodfstpairEq_type,type,
% 2.84/1.52 cartprodfstpairEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodfstpairEq,definition,
% 2.84/1.52 ( cartprodfstpairEq
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.84/1.52 = Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodsndpairEq_type,type,
% 2.84/1.52 cartprodsndpairEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodsndpairEq,definition,
% 2.84/1.52 ( cartprodsndpairEq
% 2.84/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.84/1.52 = Xy ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodpairsurjEq_type,type,
% 2.84/1.52 cartprodpairsurjEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(cartprodpairsurjEq,definition,
% 2.84/1.52 ( cartprodpairsurjEq
% 2.84/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.52 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.84/1.52 = Xu ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(breln_type,type,
% 2.84/1.52 breln: $i > $i > $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstr_type,type,
% 2.84/1.52 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrI_type,type,
% 2.84/1.52 dpsetconstrI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrI,definition,
% 2.84/1.52 ( dpsetconstrI
% 2.84/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( Xphi @ Xx @ Xy )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.52 @ ( dpsetconstr @ A @ B
% 2.84/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrSub_type,type,
% 2.84/1.52 dpsetconstrSub: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrSub,definition,
% 2.84/1.52 ( dpsetconstrSub
% 2.84/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.84/1.52 ( subset
% 2.84/1.52 @ ( dpsetconstr @ A @ B
% 2.84/1.52 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.84/1.52 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setOfPairsIsBReln_type,type,
% 2.84/1.52 setOfPairsIsBReln: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setOfPairsIsBReln,definition,
% 2.84/1.52 ( setOfPairsIsBReln
% 2.84/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.84/1.52 ( breln @ A @ B
% 2.84/1.52 @ ( dpsetconstr @ A @ B
% 2.84/1.52 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrERa_type,type,
% 2.84/1.52 dpsetconstrERa: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrERa,definition,
% 2.84/1.52 ( dpsetconstrERa
% 2.84/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.52 @ ( dpsetconstr @ A @ B
% 2.84/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.52 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrEL1_type,type,
% 2.84/1.52 dpsetconstrEL1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrEL1,definition,
% 2.84/1.52 ( dpsetconstrEL1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.52 @ ( dpsetconstr @ A @ B
% 2.84/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrEL2_type,type,
% 2.84/1.52 dpsetconstrEL2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrEL2,definition,
% 2.84/1.52 ( dpsetconstrEL2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.52 @ ( dpsetconstr @ A @ B
% 2.84/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.52 => ( in @ Xy @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrER_type,type,
% 2.84/1.52 dpsetconstrER: $o ).
% 2.84/1.52
% 2.84/1.52 thf(dpsetconstrER,definition,
% 2.84/1.52 ( dpsetconstrER
% 2.84/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.52 @ ( dpsetconstr @ A @ B
% 2.84/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.52 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(func_type,type,
% 2.84/1.52 func: $i > $i > $i > $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcSet_type,type,
% 2.84/1.52 funcSet: $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(funcImageSingleton_type,type,
% 2.84/1.52 funcImageSingleton: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcImageSingleton,definition,
% 2.84/1.52 ( funcImageSingleton
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( singleton
% 2.84/1.52 @ ( dsetconstr @ B
% 2.84/1.52 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(apProp_type,type,
% 2.84/1.52 apProp: $o ).
% 2.84/1.52
% 2.84/1.52 thf(apProp,definition,
% 2.84/1.52 ( apProp
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in
% 2.84/1.52 @ ( setunion
% 2.84/1.52 @ ( dsetconstr @ B
% 2.84/1.52 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.84/1.52 @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ap_type,type,
% 2.84/1.52 ap: $i > $i > $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(app_type,type,
% 2.84/1.52 app: $o ).
% 2.84/1.52
% 2.84/1.52 thf(app,definition,
% 2.84/1.52 ( app
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(infuncsetfunc_type,type,
% 2.84/1.52 infuncsetfunc: $o ).
% 2.84/1.52
% 2.84/1.52 thf(infuncsetfunc,definition,
% 2.84/1.52 ( infuncsetfunc
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.52 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ap2p_type,type,
% 2.84/1.52 ap2p: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ap2p,definition,
% 2.84/1.52 ( ap2p
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(funcinfuncset_type,type,
% 2.84/1.52 funcinfuncset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcinfuncset,definition,
% 2.84/1.52 ( funcinfuncset
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(lamProp_type,type,
% 2.84/1.52 lamProp: $o ).
% 2.84/1.52
% 2.84/1.52 thf(lamProp,definition,
% 2.84/1.52 ( lamProp
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.52 => ( func @ A @ B
% 2.84/1.52 @ ( dpsetconstr @ A @ B
% 2.84/1.52 @ ^ [Xx: $i,Xy: $i] :
% 2.84/1.52 ( ( Xf @ Xx )
% 2.84/1.52 = Xy ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(lam_type,type,
% 2.84/1.52 lam: $i > $i > ( $i > $i ) > $i ).
% 2.84/1.52
% 2.84/1.52 thf(lamp_type,type,
% 2.84/1.52 lamp: $o ).
% 2.84/1.52
% 2.84/1.52 thf(lamp,definition,
% 2.84/1.52 ( lamp
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.52 => ( func @ A @ B
% 2.84/1.52 @ ( lam @ A @ B
% 2.84/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(lam2p_type,type,
% 2.84/1.52 lam2p: $o ).
% 2.84/1.52
% 2.84/1.52 thf(lam2p,definition,
% 2.84/1.52 ( lam2p
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.52 => ( in
% 2.84/1.52 @ ( lam @ A @ B
% 2.84/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.84/1.52 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(brelnall1_type,type,
% 2.84/1.52 brelnall1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(brelnall1,definition,
% 2.84/1.52 ( brelnall1
% 2.84/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.84/1.52 ( ( breln @ A @ B @ R )
% 2.84/1.52 => ! [Xphi: $i > $o] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ R )
% 2.84/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(brelnall2_type,type,
% 2.84/1.52 brelnall2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(brelnall2,definition,
% 2.84/1.52 ( brelnall2
% 2.84/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.84/1.52 ( ( breln @ A @ B @ R )
% 2.84/1.52 => ! [Xphi: $i > $o] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ R )
% 2.84/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ex1E2_type,type,
% 2.84/1.52 ex1E2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ex1E2,definition,
% 2.84/1.52 ( ex1E2
% 2.84/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.52 ( ( ex1 @ A
% 2.84/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( ( Xphi @ Xx )
% 2.84/1.52 => ( ( Xphi @ Xy )
% 2.84/1.52 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(funcGraphProp1_type,type,
% 2.84/1.52 funcGraphProp1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcGraphProp1,definition,
% 2.84/1.52 ( funcGraphProp1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(funcGraphProp3_type,type,
% 2.84/1.52 funcGraphProp3: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcGraphProp3,definition,
% 2.84/1.52 ( funcGraphProp3
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(funcGraphProp2_type,type,
% 2.84/1.52 funcGraphProp2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcGraphProp2,definition,
% 2.84/1.52 ( funcGraphProp2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.84/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.52 = Xy ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(funcextLem_type,type,
% 2.84/1.52 funcextLem: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcextLem,definition,
% 2.84/1.52 ( funcextLem
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ! [Xg: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xg )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(funcGraphProp4_type,type,
% 2.84/1.52 funcGraphProp4: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcGraphProp4,definition,
% 2.84/1.52 ( funcGraphProp4
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.84/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.52 = Xy ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subbreln_type,type,
% 2.84/1.52 subbreln: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subbreln,definition,
% 2.84/1.52 ( subbreln
% 2.84/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.84/1.52 ( ( breln @ A @ B @ R )
% 2.84/1.52 => ! [S: $i] :
% 2.84/1.52 ( ( breln @ A @ B @ S )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.84/1.52 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(eqbreln_type,type,
% 2.84/1.52 eqbreln: $o ).
% 2.84/1.52
% 2.84/1.52 thf(eqbreln,definition,
% 2.84/1.52 ( eqbreln
% 2.84/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.84/1.52 ( ( breln @ A @ B @ R )
% 2.84/1.52 => ! [S: $i] :
% 2.84/1.52 ( ( breln @ A @ B @ S )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ B )
% 2.84/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.84/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.84/1.52 => ( R = S ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(funcext_type,type,
% 2.84/1.52 funcext: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcext,definition,
% 2.84/1.52 ( funcext
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ! [Xg: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xg )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.84/1.52 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(funcext2_type,type,
% 2.84/1.52 funcext2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(funcext2,definition,
% 2.84/1.52 ( funcext2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.52 => ! [Xg: $i] :
% 2.84/1.52 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.84/1.52 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ap2apEq1_type,type,
% 2.84/1.52 ap2apEq1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ap2apEq1,definition,
% 2.84/1.52 ( ap2apEq1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.52 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ap2apEq2_type,type,
% 2.84/1.52 ap2apEq2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ap2apEq2,definition,
% 2.84/1.52 ( ap2apEq2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.52 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(beta1_type,type,
% 2.84/1.52 beta1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(beta1,definition,
% 2.84/1.52 ( beta1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( ap @ A @ B
% 2.84/1.52 @ ( lam @ A @ B
% 2.84/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.84/1.52 @ Xx )
% 2.84/1.52 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(eta1_type,type,
% 2.84/1.52 eta1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(eta1,definition,
% 2.84/1.52 ( eta1
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( func @ A @ B @ Xf )
% 2.84/1.52 => ( ( lam @ A @ B
% 2.84/1.52 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.84/1.52 = Xf ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(lam2lamEq_type,type,
% 2.84/1.52 lam2lamEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(lam2lamEq,definition,
% 2.84/1.52 ( lam2lamEq
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.52 => ( ( lam @ A @ B
% 2.84/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.84/1.52 = ( lam @ A @ B
% 2.84/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(beta2_type,type,
% 2.84/1.52 beta2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(beta2,definition,
% 2.84/1.52 ( beta2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.52 ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( ap @ A @ B
% 2.84/1.52 @ ( lam @ A @ B
% 2.84/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.84/1.52 @ Xx )
% 2.84/1.52 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(eta2_type,type,
% 2.84/1.52 eta2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(eta2,definition,
% 2.84/1.52 ( eta2
% 2.84/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.52 => ( ( lam @ A @ B
% 2.84/1.52 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.84/1.52 = Xf ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(iffalseProp1_type,type,
% 2.84/1.52 iffalseProp1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(iffalseProp1,definition,
% 2.84/1.52 ( iffalseProp1
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( ~ Xphi
% 2.84/1.52 => ( in @ Xy
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xz: $i] :
% 2.84/1.52 ( ( Xphi
% 2.84/1.52 & ( Xz = Xx ) )
% 2.84/1.52 | ( ~ Xphi
% 2.84/1.52 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(iffalseProp2_type,type,
% 2.84/1.52 iffalseProp2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(iffalseProp2,definition,
% 2.84/1.52 ( iffalseProp2
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( ~ Xphi
% 2.84/1.52 => ( ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xz: $i] :
% 2.84/1.52 ( ( Xphi
% 2.84/1.52 & ( Xz = Xx ) )
% 2.84/1.52 | ( ~ Xphi
% 2.84/1.52 & ( Xz = Xy ) ) ) )
% 2.84/1.52 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(iftrueProp1_type,type,
% 2.84/1.52 iftrueProp1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(iftrueProp1,definition,
% 2.84/1.52 ( iftrueProp1
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( Xphi
% 2.84/1.52 => ( in @ Xx
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xz: $i] :
% 2.84/1.52 ( ( Xphi
% 2.84/1.52 & ( Xz = Xx ) )
% 2.84/1.52 | ( ~ Xphi
% 2.84/1.52 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(iftrueProp2_type,type,
% 2.84/1.52 iftrueProp2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(iftrueProp2,definition,
% 2.84/1.52 ( iftrueProp2
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( Xphi
% 2.84/1.52 => ( ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xz: $i] :
% 2.84/1.52 ( ( Xphi
% 2.84/1.52 & ( Xz = Xx ) )
% 2.84/1.52 | ( ~ Xphi
% 2.84/1.52 & ( Xz = Xy ) ) ) )
% 2.84/1.52 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(ifSingleton_type,type,
% 2.84/1.52 ifSingleton: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ifSingleton,definition,
% 2.84/1.52 ( ifSingleton
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( singleton
% 2.84/1.52 @ ( dsetconstr @ A
% 2.84/1.52 @ ^ [Xz: $i] :
% 2.84/1.52 ( ( Xphi
% 2.84/1.52 & ( Xz = Xx ) )
% 2.84/1.52 | ( ~ Xphi
% 2.84/1.52 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(if_type,type,
% 2.84/1.52 if: $i > $o > $i > $i > $i ).
% 2.84/1.52
% 2.84/1.52 thf(ifp_type,type,
% 2.84/1.52 ifp: $o ).
% 2.84/1.52
% 2.84/1.52 thf(ifp,definition,
% 2.84/1.52 ( ifp
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(theeq_type,type,
% 2.84/1.52 theeq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(theeq,definition,
% 2.84/1.52 ( theeq
% 2.84/1.52 = ( ! [X: $i] :
% 2.84/1.52 ( ( singleton @ X )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ X )
% 2.84/1.52 => ( ( setunion @ X )
% 2.84/1.52 = Xx ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(iftrue_type,type,
% 2.84/1.52 iftrue: $o ).
% 2.84/1.52
% 2.84/1.52 thf(iftrue,definition,
% 2.84/1.52 ( iftrue
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( Xphi
% 2.84/1.52 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.84/1.52 = Xx ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(iffalse_type,type,
% 2.84/1.52 iffalse: $o ).
% 2.84/1.52
% 2.84/1.52 thf(iffalse,definition,
% 2.84/1.52 ( iffalse
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( ~ Xphi
% 2.84/1.52 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.84/1.52 = Xy ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(iftrueorfalse_type,type,
% 2.84/1.52 iftrueorfalse: $o ).
% 2.84/1.52
% 2.84/1.52 thf(iftrueorfalse,definition,
% 2.84/1.52 ( iftrueorfalse
% 2.84/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ! [Xy: $i] :
% 2.84/1.52 ( ( in @ Xy @ A )
% 2.84/1.52 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectT_lem_type,type,
% 2.84/1.52 binintersectT_lem: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectT_lem,definition,
% 2.84/1.52 ( binintersectT_lem
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionT_lem_type,type,
% 2.84/1.52 binunionT_lem: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionT_lem,definition,
% 2.84/1.52 ( binunionT_lem
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powersetT_lem_type,type,
% 2.84/1.52 powersetT_lem: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetT_lem,definition,
% 2.84/1.52 ( powersetT_lem
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setminusT_lem_type,type,
% 2.84/1.52 setminusT_lem: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setminusT_lem,definition,
% 2.84/1.52 ( setminusT_lem
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(complementT_lem_type,type,
% 2.84/1.52 complementT_lem: $o ).
% 2.84/1.52
% 2.84/1.52 thf(complementT_lem,definition,
% 2.84/1.52 ( complementT_lem
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(setextT_type,type,
% 2.84/1.52 setextT: $o ).
% 2.84/1.52
% 2.84/1.52 thf(setextT,definition,
% 2.84/1.52 ( setextT
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ X )
% 2.84/1.52 => ( in @ Xx @ Y ) ) )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ Y )
% 2.84/1.52 => ( in @ Xx @ X ) ) )
% 2.84/1.52 => ( X = Y ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subsetTI_type,type,
% 2.84/1.52 subsetTI: $o ).
% 2.84/1.52
% 2.84/1.52 thf(subsetTI,definition,
% 2.84/1.52 ( subsetTI
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ X )
% 2.84/1.52 => ( in @ Xx @ Y ) ) )
% 2.84/1.52 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powersetTI1_type,type,
% 2.84/1.52 powersetTI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetTI1,definition,
% 2.84/1.52 ( powersetTI1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ X )
% 2.84/1.52 => ( in @ Xx @ Y ) ) )
% 2.84/1.52 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(powersetTE1_type,type,
% 2.84/1.52 powersetTE1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(powersetTE1,definition,
% 2.84/1.52 ( powersetTE1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ X @ ( powerset @ Y ) )
% 2.84/1.52 => ( ( in @ Xx @ X )
% 2.84/1.52 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(complementTI1_type,type,
% 2.84/1.52 complementTI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(complementTI1,definition,
% 2.84/1.52 ( complementTI1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ X )
% 2.84/1.52 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(complementTE1_type,type,
% 2.84/1.52 complementTE1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(complementTE1,definition,
% 2.84/1.52 ( complementTE1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.84/1.52 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectTELcontra_type,type,
% 2.84/1.52 binintersectTELcontra: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectTELcontra,definition,
% 2.84/1.52 ( binintersectTELcontra
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ X )
% 2.84/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectTERcontra_type,type,
% 2.84/1.52 binintersectTERcontra: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binintersectTERcontra,definition,
% 2.84/1.52 ( binintersectTERcontra
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ Y )
% 2.84/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(contrasubsetT_type,type,
% 2.84/1.52 contrasubsetT: $o ).
% 2.84/1.52
% 2.84/1.52 thf(contrasubsetT,definition,
% 2.84/1.52 ( contrasubsetT
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.84/1.52 => ( ( in @ Xx @ Y )
% 2.84/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(contrasubsetT1_type,type,
% 2.84/1.52 contrasubsetT1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(contrasubsetT1,definition,
% 2.84/1.52 ( contrasubsetT1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( subset @ X @ Y )
% 2.84/1.52 => ( ~ ( in @ Xx @ Y )
% 2.84/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(contrasubsetT2_type,type,
% 2.84/1.52 contrasubsetT2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(contrasubsetT2,definition,
% 2.84/1.52 ( contrasubsetT2
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( ( subset @ X @ Y )
% 2.84/1.52 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(contrasubsetT3_type,type,
% 2.84/1.52 contrasubsetT3: $o ).
% 2.84/1.52
% 2.84/1.52 thf(contrasubsetT3,definition,
% 2.84/1.52 ( contrasubsetT3
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.84/1.52 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementI1_type,type,
% 2.84/1.52 doubleComplementI1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementI1,definition,
% 2.84/1.52 ( doubleComplementI1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ X )
% 2.84/1.52 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementE1_type,type,
% 2.84/1.52 doubleComplementE1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementE1,definition,
% 2.84/1.52 ( doubleComplementE1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.84/1.52 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementSub1_type,type,
% 2.84/1.52 doubleComplementSub1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementSub1,definition,
% 2.84/1.52 ( doubleComplementSub1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementSub2_type,type,
% 2.84/1.52 doubleComplementSub2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementSub2,definition,
% 2.84/1.52 ( doubleComplementSub2
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementEq_type,type,
% 2.84/1.52 doubleComplementEq: $o ).
% 2.84/1.52
% 2.84/1.52 thf(doubleComplementEq,definition,
% 2.84/1.52 ( doubleComplementEq
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ( X
% 2.84/1.52 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(complementTnotintersectT_type,type,
% 2.84/1.52 complementTnotintersectT: $o ).
% 2.84/1.52
% 2.84/1.52 thf(complementTnotintersectT,definition,
% 2.84/1.52 ( complementTnotintersectT
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.84/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(complementImpComplementIntersect_type,type,
% 2.84/1.52 complementImpComplementIntersect: $o ).
% 2.84/1.52
% 2.84/1.52 thf(complementImpComplementIntersect,definition,
% 2.84/1.52 ( complementImpComplementIntersect
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.84/1.52 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(complementSubsetComplementIntersect_type,type,
% 2.84/1.52 complementSubsetComplementIntersect: $o ).
% 2.84/1.52
% 2.84/1.52 thf(complementSubsetComplementIntersect,definition,
% 2.84/1.52 ( complementSubsetComplementIntersect
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(complementInPowersetComplementIntersect_type,type,
% 2.84/1.52 complementInPowersetComplementIntersect: $o ).
% 2.84/1.52
% 2.84/1.52 thf(complementInPowersetComplementIntersect,definition,
% 2.84/1.52 ( complementInPowersetComplementIntersect
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(contraSubsetComplement_type,type,
% 2.84/1.52 contraSubsetComplement: $o ).
% 2.84/1.52
% 2.84/1.52 thf(contraSubsetComplement,definition,
% 2.84/1.52 ( contraSubsetComplement
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ Y )
% 2.84/1.52 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(complementTcontraSubset_type,type,
% 2.84/1.52 complementTcontraSubset: $o ).
% 2.84/1.52
% 2.84/1.52 thf(complementTcontraSubset,definition,
% 2.84/1.52 ( complementTcontraSubset
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.84/1.52 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionTILcontra_type,type,
% 2.84/1.52 binunionTILcontra: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionTILcontra,definition,
% 2.84/1.52 ( binunionTILcontra
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.84/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionTIRcontra_type,type,
% 2.84/1.52 binunionTIRcontra: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionTIRcontra,definition,
% 2.84/1.52 ( binunionTIRcontra
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.84/1.52 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(inIntersectImpInUnion_type,type,
% 2.84/1.52 inIntersectImpInUnion: $o ).
% 2.84/1.52
% 2.84/1.52 thf(inIntersectImpInUnion,definition,
% 2.84/1.52 ( inIntersectImpInUnion
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Z: $i] :
% 2.84/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.84/1.52 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(inIntersectImpInUnion2_type,type,
% 2.84/1.52 inIntersectImpInUnion2: $o ).
% 2.84/1.52
% 2.84/1.52 thf(inIntersectImpInUnion2,definition,
% 2.84/1.52 ( inIntersectImpInUnion2
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Z: $i] :
% 2.84/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.84/1.52 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(inIntersectImpInIntersectUnions_type,type,
% 2.84/1.52 inIntersectImpInIntersectUnions: $o ).
% 2.84/1.52
% 2.84/1.52 thf(inIntersectImpInIntersectUnions,definition,
% 2.84/1.52 ( inIntersectImpInIntersectUnions
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Z: $i] :
% 2.84/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.84/1.52 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(intersectInPowersetIntersectUnions_type,type,
% 2.84/1.52 intersectInPowersetIntersectUnions: $o ).
% 2.84/1.52
% 2.84/1.52 thf(intersectInPowersetIntersectUnions,definition,
% 2.84/1.52 ( intersectInPowersetIntersectUnions
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Z: $i] :
% 2.84/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.52 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(inComplementUnionImpNotIn1_type,type,
% 2.84/1.52 inComplementUnionImpNotIn1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(inComplementUnionImpNotIn1,definition,
% 2.84/1.52 ( inComplementUnionImpNotIn1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(inComplementUnionImpInComplement1_type,type,
% 2.84/1.52 inComplementUnionImpInComplement1: $o ).
% 2.84/1.52
% 2.84/1.52 thf(inComplementUnionImpInComplement1,definition,
% 2.84/1.52 ( inComplementUnionImpInComplement1
% 2.84/1.52 = ( ! [A: $i,X: $i] :
% 2.84/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.52 => ! [Y: $i] :
% 2.84/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.52 => ! [Xx: $i] :
% 2.84/1.52 ( ( in @ Xx @ A )
% 2.84/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.52 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(binunionTE_type,type,
% 2.84/1.52 binunionTE: $o ).
% 2.84/1.52
% 2.84/1.52 thf(binunionTE,definition,
% 2.84/1.52 ( binunionTE
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xphi: $o,Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.84/1.53 => ( ( ( in @ Xx @ X )
% 2.84/1.53 => Xphi )
% 2.84/1.53 => ( ( ( in @ Xx @ Y )
% 2.84/1.53 => Xphi )
% 2.84/1.53 => Xphi ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(binunionTEcontra_type,type,
% 2.84/1.53 binunionTEcontra: $o ).
% 2.84/1.53
% 2.84/1.53 thf(binunionTEcontra,definition,
% 2.84/1.53 ( binunionTEcontra
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ~ ( in @ Xx @ X )
% 2.84/1.53 => ( ~ ( in @ Xx @ Y )
% 2.84/1.53 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2a1_type,type,
% 2.84/1.53 demorgan2a1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2a1,definition,
% 2.84/1.53 ( demorgan2a1
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.53 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(complementUnionInPowersetComplement_type,type,
% 2.84/1.53 complementUnionInPowersetComplement: $o ).
% 2.84/1.53
% 2.84/1.53 thf(complementUnionInPowersetComplement,definition,
% 2.84/1.53 ( complementUnionInPowersetComplement
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2a2_type,type,
% 2.84/1.53 demorgan2a2: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2a2,definition,
% 2.84/1.53 ( demorgan2a2
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.53 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan1a_type,type,
% 2.84/1.53 demorgan1a: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan1a,definition,
% 2.84/1.53 ( demorgan1a
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.84/1.53 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan1b_type,type,
% 2.84/1.53 demorgan1b: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan1b,definition,
% 2.84/1.53 ( demorgan1b
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.84/1.53 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan1_type,type,
% 2.84/1.53 demorgan1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan1,definition,
% 2.84/1.53 ( demorgan1
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.84/1.53 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2a_type,type,
% 2.84/1.53 demorgan2a: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2a,definition,
% 2.84/1.53 ( demorgan2a
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.53 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2b2_type,type,
% 2.84/1.53 demorgan2b2: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2b2,definition,
% 2.84/1.53 ( demorgan2b2
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.84/1.53 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.84/1.53 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2b_type,type,
% 2.84/1.53 demorgan2b: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2b,definition,
% 2.84/1.53 ( demorgan2b
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.84/1.53 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2_type,type,
% 2.84/1.53 demorgan2: $o ).
% 2.84/1.53
% 2.84/1.53 thf(demorgan2,definition,
% 2.84/1.53 ( demorgan2
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.84/1.53 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule0_type,type,
% 2.84/1.53 woz13rule0: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule0,definition,
% 2.84/1.53 ( woz13rule0
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.84/1.53 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule1_type,type,
% 2.84/1.53 woz13rule1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule1,definition,
% 2.84/1.53 ( woz13rule1
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Z: $i] :
% 2.84/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.53 => ( ( subset @ X @ Z )
% 2.84/1.53 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule2_type,type,
% 2.84/1.53 woz13rule2: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule2,definition,
% 2.84/1.53 ( woz13rule2
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Z: $i] :
% 2.84/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.53 => ( ( subset @ Y @ Z )
% 2.84/1.53 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule3_type,type,
% 2.84/1.53 woz13rule3: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule3,definition,
% 2.84/1.53 ( woz13rule3
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Z: $i] :
% 2.84/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.53 => ( ( subset @ X @ Y )
% 2.84/1.53 => ( ( subset @ X @ Z )
% 2.84/1.53 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule4_type,type,
% 2.84/1.53 woz13rule4: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz13rule4,definition,
% 2.84/1.53 ( woz13rule4
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Z: $i] :
% 2.84/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.53 => ! [W: $i] :
% 2.84/1.53 ( ( in @ W @ ( powerset @ A ) )
% 2.84/1.53 => ( ( subset @ X @ Z )
% 2.84/1.53 => ( ( subset @ Y @ W )
% 2.84/1.53 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_1_type,type,
% 2.84/1.53 woz1_1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_1,definition,
% 2.84/1.53 ( woz1_1
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_2_type,type,
% 2.84/1.53 woz1_2: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_2,definition,
% 2.84/1.53 ( woz1_2
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Z: $i] :
% 2.84/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.53 => ! [W: $i] :
% 2.84/1.53 ( ( in @ W @ ( powerset @ A ) )
% 2.84/1.53 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.84/1.53 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_3_type,type,
% 2.84/1.53 woz1_3: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_3,definition,
% 2.84/1.53 ( woz1_3
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ! [Z: $i] :
% 2.84/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.53 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_4_type,type,
% 2.84/1.53 woz1_4: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_4,definition,
% 2.84/1.53 ( woz1_4
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.84/1.53 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_5_type,type,
% 2.84/1.53 woz1_5: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz1_5,definition,
% 2.84/1.53 ( woz1_5
% 2.84/1.53 = ( ! [A: $i,X: $i] :
% 2.84/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.53 => ! [Y: $i] :
% 2.84/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.53 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1_type,type,
% 2.84/1.53 breln1: $i > $i > $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1all2_type,type,
% 2.84/1.53 breln1all2: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1all2,definition,
% 2.84/1.53 ( breln1all2
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [Xphi: $i > $o] :
% 2.84/1.53 ( ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ R )
% 2.84/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1Set_type,type,
% 2.84/1.53 breln1Set: $i > $i ).
% 2.84/1.53
% 2.84/1.53 thf(breln1SetBreln1_type,type,
% 2.84/1.53 breln1SetBreln1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1SetBreln1,definition,
% 2.84/1.53 ( breln1SetBreln1
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( in @ R @ ( breln1Set @ A ) )
% 2.84/1.53 => ( breln1 @ A @ R ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(transitive_type,type,
% 2.84/1.53 transitive: $i > $i > $o ).
% 2.84/1.53
% 2.84/1.53 thf(antisymmetric_type,type,
% 2.84/1.53 antisymmetric: $i > $i > $o ).
% 2.84/1.53
% 2.84/1.53 thf(reflexive_type,type,
% 2.84/1.53 reflexive: $i > $i > $o ).
% 2.84/1.53
% 2.84/1.53 thf(refltransitive_type,type,
% 2.84/1.53 refltransitive: $i > $i > $o ).
% 2.84/1.53
% 2.84/1.53 thf(refllinearorder_type,type,
% 2.84/1.53 refllinearorder: $i > $i > $o ).
% 2.84/1.53
% 2.84/1.53 thf(reflwellordering_type,type,
% 2.84/1.53 reflwellordering: $i > $i > $o ).
% 2.84/1.53
% 2.84/1.53 thf(choice2fnsingleton_type,type,
% 2.84/1.53 choice2fnsingleton: $o ).
% 2.84/1.53
% 2.84/1.53 thf(choice2fnsingleton,definition,
% 2.84/1.53 ( choice2fnsingleton
% 2.84/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.84/1.53 ( ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ? [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ B )
% 2.84/1.53 & ( Xphi @ Xx @ Xy ) ) )
% 2.84/1.53 => ! [R: $i] :
% 2.84/1.53 ( ( in @ R @ ( breln1Set @ B ) )
% 2.84/1.53 => ( ( reflwellordering @ B @ R )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ( singleton
% 2.84/1.53 @ ( dsetconstr @ B
% 2.84/1.53 @ ^ [Xy: $i] :
% 2.84/1.53 ( ( Xphi @ Xx @ Xy )
% 2.84/1.53 & ! [Xz: $i] :
% 2.84/1.53 ( ( in @ Xz @ B )
% 2.84/1.53 => ( ( Xphi @ Xx @ Xz )
% 2.84/1.53 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(setOfPairsIsBReln1_type,type,
% 2.84/1.53 setOfPairsIsBReln1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(setOfPairsIsBReln1,definition,
% 2.84/1.53 ( setOfPairsIsBReln1
% 2.84/1.53 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 2.84/1.53 ( breln1 @ A
% 2.84/1.53 @ ( dpsetconstr @ A @ A
% 2.84/1.53 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1all1_type,type,
% 2.84/1.53 breln1all1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1all1,definition,
% 2.84/1.53 ( breln1all1
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [Xphi: $i > $o] :
% 2.84/1.53 ( ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ R )
% 2.84/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(subbreln1_type,type,
% 2.84/1.53 subbreln1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(subbreln1,definition,
% 2.84/1.53 ( subbreln1
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ( ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.84/1.53 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(eqbreln1_type,type,
% 2.84/1.53 eqbreln1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(eqbreln1,definition,
% 2.84/1.53 ( eqbreln1
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ( ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.84/1.53 => ( ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.84/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.84/1.53 => ( R = S ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1invset_type,type,
% 2.84/1.53 breln1invset: $i > $i > $i ).
% 2.84/1.53
% 2.84/1.53 thf(breln1invprop_type,type,
% 2.84/1.53 breln1invprop: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1invprop,definition,
% 2.84/1.53 ( breln1invprop
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1invI_type,type,
% 2.84/1.53 breln1invI: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1invI,definition,
% 2.84/1.53 ( breln1invI
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1invE_type,type,
% 2.84/1.53 breln1invE: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1invE,definition,
% 2.84/1.53 ( breln1invE
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 2.84/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compset_type,type,
% 2.84/1.53 breln1compset: $i > $i > $i > $i ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compprop_type,type,
% 2.84/1.53 breln1compprop: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compprop,definition,
% 2.84/1.53 ( breln1compprop
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compI_type,type,
% 2.84/1.53 breln1compI: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compI,definition,
% 2.84/1.53 ( breln1compI
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ! [Xz: $i] :
% 2.84/1.53 ( ( in @ Xz @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.84/1.53 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.84/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compE_type,type,
% 2.84/1.53 breln1compE: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compE,definition,
% 2.84/1.53 ( breln1compE
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.84/1.53 => ? [Xz: $i] :
% 2.84/1.53 ( ( in @ Xz @ A )
% 2.84/1.53 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.84/1.53 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compEex_type,type,
% 2.84/1.53 breln1compEex: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1compEex,definition,
% 2.84/1.53 ( breln1compEex
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.84/1.53 => ! [Xphi: $o] :
% 2.84/1.53 ( ! [Xz: $i] :
% 2.84/1.53 ( ( in @ Xz @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.84/1.53 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.84/1.53 => Xphi ) ) )
% 2.84/1.53 => Xphi ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionprop_type,type,
% 2.84/1.53 breln1unionprop: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionprop,definition,
% 2.84/1.53 ( breln1unionprop
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionIL_type,type,
% 2.84/1.53 breln1unionIL: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionIL,definition,
% 2.84/1.53 ( breln1unionIL
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionIR_type,type,
% 2.84/1.53 breln1unionIR: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionIR,definition,
% 2.84/1.53 ( breln1unionIR
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.84/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionI_type,type,
% 2.84/1.53 breln1unionI: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionI,definition,
% 2.84/1.53 ( breln1unionI
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 2.84/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionE_type,type,
% 2.84/1.53 breln1unionE: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionE,definition,
% 2.84/1.53 ( breln1unionE
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionEcases_type,type,
% 2.84/1.53 breln1unionEcases: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionEcases,definition,
% 2.84/1.53 ( breln1unionEcases
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ A )
% 2.84/1.53 => ! [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.84/1.53 => ! [Xphi: $o] :
% 2.84/1.53 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.53 => Xphi )
% 2.84/1.53 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.84/1.53 => Xphi )
% 2.84/1.53 => Xphi ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionCommutes_type,type,
% 2.84/1.53 breln1unionCommutes: $o ).
% 2.84/1.53
% 2.84/1.53 thf(breln1unionCommutes,definition,
% 2.84/1.53 ( breln1unionCommutes
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ( ( binunion @ R @ S )
% 2.84/1.53 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz2Ex_type,type,
% 2.84/1.53 woz2Ex: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz2Ex,definition,
% 2.84/1.53 ( woz2Ex
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ( R
% 2.84/1.53 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz2W_type,type,
% 2.84/1.53 woz2W: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz2W,definition,
% 2.84/1.53 ( woz2W
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 2.84/1.53 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz2A_type,type,
% 2.84/1.53 woz2A: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz2A,definition,
% 2.84/1.53 ( woz2A
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [T: $i] :
% 2.84/1.53 ( ( breln1 @ A @ T )
% 2.84/1.53 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 2.84/1.53 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(woz2B_type,type,
% 2.84/1.53 woz2B: $o ).
% 2.84/1.53
% 2.84/1.53 thf(woz2B,definition,
% 2.84/1.53 ( woz2B
% 2.84/1.53 = ( ! [A: $i,R: $i] :
% 2.84/1.53 ( ( breln1 @ A @ R )
% 2.84/1.53 => ! [S: $i] :
% 2.84/1.53 ( ( breln1 @ A @ S )
% 2.84/1.53 => ! [T: $i] :
% 2.84/1.53 ( ( breln1 @ A @ T )
% 2.84/1.53 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 2.84/1.53 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(image1Ex_type,type,
% 2.84/1.53 image1Ex: $o ).
% 2.84/1.53
% 2.84/1.53 thf(image1Ex,definition,
% 2.84/1.53 ( image1Ex
% 2.84/1.53 = ( ! [A: $i,Xf: $i > $i] :
% 2.84/1.53 ? [B: $i] :
% 2.84/1.53 ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ B )
% 2.84/1.53 <=> ? [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 & ( Xx
% 2.84/1.53 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(image1Ex1_type,type,
% 2.84/1.53 image1Ex1: $o ).
% 2.84/1.53
% 2.84/1.53 thf(image1Ex1,definition,
% 2.84/1.53 ( image1Ex1
% 2.84/1.53 = ( ! [A: $i,Xf: $i > $i] :
% 2.84/1.53 ( exu
% 2.84/1.53 @ ^ [B: $i] :
% 2.84/1.53 ! [Xx: $i] :
% 2.84/1.53 ( ( in @ Xx @ B )
% 2.84/1.53 <=> ? [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 & ( Xx
% 2.84/1.53 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(image1_type,type,
% 2.84/1.53 image1: $i > ( $i > $i ) > $i ).
% 2.84/1.53
% 2.84/1.53 thf(image1Equiv_type,type,
% 2.84/1.53 image1Equiv: $o ).
% 2.84/1.53
% 2.84/1.53 thf(image1Equiv,definition,
% 2.84/1.53 ( image1Equiv
% 2.84/1.53 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.84/1.53 ( ( in @ Xx
% 2.84/1.53 @ ( image1 @ A
% 2.84/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 2.84/1.53 <=> ? [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 & ( Xx
% 2.84/1.53 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(image1E_type,type,
% 2.84/1.53 image1E: $o ).
% 2.84/1.53
% 2.84/1.53 thf(image1E,definition,
% 2.84/1.53 ( image1E
% 2.84/1.53 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.84/1.53 ( ( in @ Xx
% 2.84/1.53 @ ( image1 @ A
% 2.84/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 2.84/1.53 => ? [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 & ( Xx
% 2.84/1.53 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(image1I_type,type,
% 2.84/1.53 image1I: $o ).
% 2.84/1.53
% 2.84/1.53 thf(image1I,definition,
% 2.84/1.53 ( image1I
% 2.84/1.53 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.84/1.53 ( ? [Xy: $i] :
% 2.84/1.53 ( ( in @ Xy @ A )
% 2.84/1.53 & ( Xx
% 2.84/1.53 = ( Xf @ Xy ) ) )
% 2.84/1.53 => ( in @ Xx
% 2.84/1.53 @ ( image1 @ A
% 2.84/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(injective_type,type,
% 2.84/1.53 injective: $i > $i > $i > $o ).
% 2.84/1.53
% 2.84/1.53 thf(injFuncSet_type,type,
% 2.84/1.53 injFuncSet: $i > $i > $i ).
% 2.84/1.53
% 2.84/1.53 thf(injFuncSet,definition,
% 2.84/1.53 ( injFuncSet
% 2.84/1.53 = ( ^ [A: $i,B: $i] :
% 2.84/1.53 ( dsetconstr @ ( funcSet @ A @ B )
% 2.84/1.53 @ ^ [Xf: $i] : ( injective @ A @ B @ Xf ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(injFuncInInjFuncSet_type,type,
% 2.84/1.53 injFuncInInjFuncSet: $o ).
% 2.84/1.53
% 2.84/1.53 thf(injFuncInInjFuncSet,definition,
% 2.84/1.53 ( injFuncInInjFuncSet
% 2.84/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.53 => ( ( injective @ A @ B @ Xf )
% 2.84/1.53 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 2.84/1.53
% 2.84/1.53 thf(injFuncSetFuncIn,conjecture,
% 2.84/1.53 ( setextAx
% 2.84/1.53 => ( emptysetAx
% 2.84/1.53 => ( setadjoinAx
% 2.84/1.53 => ( powersetAx
% 2.84/1.53 => ( setunionAx
% 2.84/1.53 => ( omega0Ax
% 2.84/1.53 => ( omegaSAx
% 2.84/1.53 => ( omegaIndAx
% 2.84/1.53 => ( replAx
% 2.84/1.53 => ( foundationAx
% 2.84/1.53 => ( wellorderingAx
% 2.84/1.53 => ( descrp
% 2.84/1.53 => ( dsetconstrI
% 2.84/1.53 => ( dsetconstrEL
% 2.84/1.53 => ( dsetconstrER
% 2.84/1.53 => ( exuE1
% 2.84/1.53 => ( prop2setE
% 2.84/1.53 => ( emptysetE
% 2.84/1.53 => ( emptysetimpfalse
% 2.84/1.53 => ( notinemptyset
% 2.84/1.53 => ( exuE3e
% 2.84/1.53 => ( setext
% 2.84/1.53 => ( emptyI
% 2.84/1.53 => ( noeltsimpempty
% 2.84/1.53 => ( setbeta
% 2.84/1.53 => ( nonemptyE1
% 2.84/1.53 => ( nonemptyI
% 2.84/1.53 => ( nonemptyI1
% 2.84/1.53 => ( setadjoinIL
% 2.84/1.53 => ( emptyinunitempty
% 2.84/1.53 => ( setadjoinIR
% 2.84/1.53 => ( setadjoinE
% 2.84/1.53 => ( setadjoinOr
% 2.84/1.53 => ( setoftrueEq
% 2.84/1.53 => ( powersetI
% 2.84/1.53 => ( emptyinPowerset
% 2.84/1.53 => ( emptyInPowerset
% 2.84/1.53 => ( powersetE
% 2.84/1.53 => ( setunionI
% 2.84/1.53 => ( setunionE
% 2.84/1.53 => ( subPowSU
% 2.84/1.53 => ( exuE2
% 2.84/1.53 => ( nonemptyImpWitness
% 2.84/1.53 => ( uniqinunit
% 2.84/1.53 => ( notinsingleton
% 2.84/1.53 => ( eqinunit
% 2.84/1.53 => ( singletonsswitch
% 2.84/1.53 => ( upairsetE
% 2.84/1.53 => ( upairsetIL
% 2.84/1.53 => ( upairsetIR
% 2.84/1.53 => ( emptyE1
% 2.84/1.53 => ( vacuousDall
% 2.84/1.53 => ( quantDeMorgan1
% 2.84/1.53 => ( quantDeMorgan2
% 2.84/1.53 => ( quantDeMorgan3
% 2.84/1.53 => ( quantDeMorgan4
% 2.84/1.53 => ( prop2setI
% 2.84/1.53 => ( prop2set2propI
% 2.84/1.53 => ( notdexE
% 2.84/1.53 => ( notdallE
% 2.84/1.53 => ( exuI1
% 2.84/1.53 => ( exuI3
% 2.84/1.53 => ( exuI2
% 2.84/1.53 => ( inCongP
% 2.84/1.53 => ( in__Cong
% 2.84/1.53 => ( exuE3u
% 2.84/1.53 => ( exu__Cong
% 2.84/1.53 => ( emptyset__Cong
% 2.84/1.53 => ( setadjoin__Cong
% 2.84/1.53 => ( powerset__Cong
% 2.84/1.53 => ( setunion__Cong
% 2.84/1.53 => ( omega__Cong
% 2.84/1.53 => ( exuEu
% 2.84/1.53 => ( descr__Cong
% 2.84/1.53 => ( dsetconstr__Cong
% 2.84/1.53 => ( subsetI1
% 2.84/1.53 => ( eqimpsubset2
% 2.84/1.53 => ( eqimpsubset1
% 2.84/1.53 => ( subsetI2
% 2.84/1.53 => ( emptysetsubset
% 2.84/1.53 => ( subsetE
% 2.84/1.53 => ( subsetE2
% 2.84/1.53 => ( notsubsetI
% 2.84/1.53 => ( notequalI1
% 2.84/1.53 => ( notequalI2
% 2.84/1.53 => ( subsetRefl
% 2.84/1.53 => ( subsetTrans
% 2.84/1.53 => ( setadjoinSub
% 2.84/1.53 => ( setadjoinSub2
% 2.84/1.53 => ( subset2powerset
% 2.84/1.53 => ( setextsub
% 2.84/1.53 => ( subsetemptysetimpeq
% 2.84/1.53 => ( powersetI1
% 2.84/1.53 => ( powersetE1
% 2.84/1.53 => ( inPowerset
% 2.84/1.53 => ( powersetsubset
% 2.84/1.53 => ( sepInPowerset
% 2.84/1.53 => ( sepSubset
% 2.84/1.53 => ( binunionIL
% 2.84/1.53 => ( upairset2IR
% 2.84/1.53 => ( binunionIR
% 2.84/1.53 => ( binunionEcases
% 2.84/1.53 => ( binunionE
% 2.84/1.53 => ( binunionLsub
% 2.84/1.53 => ( binunionRsub
% 2.84/1.53 => ( binintersectI
% 2.84/1.53 => ( binintersectSubset5
% 2.84/1.53 => ( binintersectEL
% 2.84/1.53 => ( binintersectLsub
% 2.84/1.53 => ( binintersectSubset2
% 2.84/1.53 => ( binintersectSubset3
% 2.84/1.53 => ( binintersectER
% 2.84/1.53 => ( disjointsetsI1
% 2.84/1.53 => ( binintersectRsub
% 2.84/1.53 => ( binintersectSubset4
% 2.84/1.53 => ( binintersectSubset1
% 2.84/1.53 => ( bs114d
% 2.84/1.53 => ( setminusI
% 2.84/1.53 => ( setminusEL
% 2.84/1.53 => ( setminusER
% 2.84/1.53 => ( setminusSubset2
% 2.84/1.53 => ( setminusERneg
% 2.84/1.53 => ( setminusELneg
% 2.84/1.53 => ( setminusILneg
% 2.84/1.53 => ( setminusIRneg
% 2.84/1.53 => ( setminusLsub
% 2.84/1.53 => ( setminusSubset1
% 2.84/1.53 => ( symdiffE
% 2.84/1.53 => ( symdiffI1
% 2.84/1.53 => ( symdiffI2
% 2.84/1.53 => ( symdiffIneg1
% 2.84/1.53 => ( symdiffIneg2
% 2.84/1.53 => ( secondinupair
% 2.84/1.53 => ( setukpairIL
% 2.84/1.53 => ( setukpairIR
% 2.84/1.53 => ( kpairiskpair
% 2.84/1.53 => ( kpairp
% 2.84/1.53 => ( singletonsubset
% 2.84/1.53 => ( singletoninpowerset
% 2.84/1.53 => ( singletoninpowunion
% 2.84/1.53 => ( upairset2E
% 2.84/1.53 => ( upairsubunion
% 2.84/1.53 => ( upairinpowunion
% 2.84/1.53 => ( ubforcartprodlem1
% 2.84/1.53 => ( ubforcartprodlem2
% 2.84/1.53 => ( ubforcartprodlem3
% 2.84/1.53 => ( cartprodpairin
% 2.84/1.53 => ( cartprodmempair1
% 2.84/1.53 => ( cartprodmempair
% 2.84/1.53 => ( setunionE2
% 2.84/1.53 => ( setunionsingleton1
% 2.84/1.53 => ( setunionsingleton2
% 2.84/1.53 => ( setunionsingleton
% 2.84/1.53 => ( singletonprop
% 2.84/1.53 => ( ex1E1
% 2.84/1.53 => ( ex1I
% 2.84/1.53 => ( ex1I2
% 2.84/1.53 => ( singletonsuniq
% 2.84/1.53 => ( setukpairinjL1
% 2.84/1.53 => ( kfstsingleton
% 2.84/1.53 => ( theprop
% 2.84/1.53 => ( kfstpairEq
% 2.84/1.53 => ( cartprodfstin
% 2.84/1.53 => ( setukpairinjL2
% 2.84/1.53 => ( setukpairinjL
% 2.84/1.53 => ( setukpairinjR11
% 2.84/1.53 => ( setukpairinjR12
% 2.84/1.53 => ( setukpairinjR1
% 2.84/1.53 => ( upairequniteq
% 2.84/1.53 => ( setukpairinjR2
% 2.84/1.53 => ( setukpairinjR
% 2.84/1.53 => ( ksndsingleton
% 2.84/1.53 => ( ksndpairEq
% 2.84/1.53 => ( kpairsurjEq
% 2.84/1.53 => ( cartprodsndin
% 2.84/1.53 => ( cartprodpairmemEL
% 2.84/1.53 => ( cartprodpairmemER
% 2.84/1.53 => ( cartprodmempaircEq
% 2.84/1.53 => ( cartprodfstpairEq
% 2.84/1.53 => ( cartprodsndpairEq
% 2.84/1.53 => ( cartprodpairsurjEq
% 2.84/1.53 => ( dpsetconstrI
% 2.84/1.53 => ( dpsetconstrSub
% 2.84/1.53 => ( setOfPairsIsBReln
% 2.84/1.53 => ( dpsetconstrERa
% 2.84/1.53 => ( dpsetconstrEL1
% 2.84/1.53 => ( dpsetconstrEL2
% 2.84/1.53 => ( dpsetconstrER
% 2.84/1.53 => ( funcImageSingleton
% 2.84/1.53 => ( apProp
% 2.84/1.53 => ( app
% 2.84/1.53 => ( infuncsetfunc
% 2.84/1.53 => ( ap2p
% 2.84/1.53 => ( funcinfuncset
% 2.84/1.53 => ( lamProp
% 2.84/1.53 => ( lamp
% 2.84/1.53 => ( lam2p
% 2.84/1.53 => ( brelnall1
% 2.84/1.53 => ( brelnall2
% 2.84/1.53 => ( ex1E2
% 2.84/1.53 => ( funcGraphProp1
% 2.84/1.53 => ( funcGraphProp3
% 2.84/1.53 => ( funcGraphProp2
% 2.84/1.53 => ( funcextLem
% 2.84/1.53 => ( funcGraphProp4
% 2.84/1.53 => ( subbreln
% 2.84/1.53 => ( eqbreln
% 2.84/1.53 => ( funcext
% 2.84/1.53 => ( funcext2
% 2.84/1.53 => ( ap2apEq1
% 2.84/1.53 => ( ap2apEq2
% 2.84/1.53 => ( beta1
% 2.84/1.53 => ( eta1
% 2.84/1.53 => ( lam2lamEq
% 2.84/1.53 => ( beta2
% 2.84/1.53 => ( eta2
% 2.84/1.53 => ( iffalseProp1
% 2.84/1.53 => ( iffalseProp2
% 2.84/1.53 => ( iftrueProp1
% 2.84/1.53 => ( iftrueProp2
% 2.84/1.53 => ( ifSingleton
% 2.84/1.53 => ( ifp
% 2.84/1.53 => ( theeq
% 2.84/1.53 => ( iftrue
% 2.84/1.53 => ( iffalse
% 2.84/1.53 => ( iftrueorfalse
% 2.84/1.53 => ( binintersectT_lem
% 2.84/1.53 => ( binunionT_lem
% 2.84/1.53 => ( powersetT_lem
% 2.84/1.53 => ( setminusT_lem
% 2.84/1.53 => ( complementT_lem
% 2.84/1.53 => ( setextT
% 2.84/1.53 => ( subsetTI
% 2.84/1.53 => ( powersetTI1
% 2.84/1.53 => ( powersetTE1
% 2.84/1.53 => ( complementTI1
% 2.84/1.53 => ( complementTE1
% 2.84/1.53 => ( binintersectTELcontra
% 2.84/1.53 => ( binintersectTERcontra
% 2.84/1.53 => ( contrasubsetT
% 2.84/1.53 => ( contrasubsetT1
% 2.84/1.53 => ( contrasubsetT2
% 2.84/1.53 => ( contrasubsetT3
% 2.84/1.53 => ( doubleComplementI1
% 2.84/1.53 => ( doubleComplementE1
% 2.84/1.53 => ( doubleComplementSub1
% 2.84/1.53 => ( doubleComplementSub2
% 2.84/1.53 => ( doubleComplementEq
% 2.84/1.53 => ( complementTnotintersectT
% 2.84/1.53 => ( complementImpComplementIntersect
% 2.84/1.53 => ( complementSubsetComplementIntersect
% 2.84/1.53 => ( complementInPowersetComplementIntersect
% 2.84/1.53 => ( contraSubsetComplement
% 2.84/1.53 => ( complementTcontraSubset
% 2.84/1.53 => ( binunionTILcontra
% 2.84/1.53 => ( binunionTIRcontra
% 2.84/1.53 => ( inIntersectImpInUnion
% 2.84/1.53 => ( inIntersectImpInUnion2
% 2.84/1.53 => ( inIntersectImpInIntersectUnions
% 2.84/1.53 => ( intersectInPowersetIntersectUnions
% 2.84/1.53 => ( inComplementUnionImpNotIn1
% 2.84/1.53 => ( inComplementUnionImpInComplement1
% 2.84/1.53 => ( binunionTE
% 2.84/1.53 => ( binunionTEcontra
% 2.84/1.53 => ( demorgan2a1
% 2.84/1.53 => ( complementUnionInPowersetComplement
% 2.84/1.53 => ( demorgan2a2
% 2.84/1.53 => ( demorgan1a
% 2.84/1.53 => ( demorgan1b
% 2.84/1.53 => ( demorgan1
% 2.84/1.53 => ( demorgan2a
% 2.84/1.53 => ( demorgan2b2
% 2.84/1.53 => ( demorgan2b
% 2.84/1.53 => ( demorgan2
% 2.84/1.53 => ( woz13rule0
% 2.84/1.53 => ( woz13rule1
% 2.84/1.53 => ( woz13rule2
% 2.84/1.53 => ( woz13rule3
% 2.84/1.53 => ( woz13rule4
% 2.84/1.53 => ( woz1_1
% 2.84/1.53 => ( woz1_2
% 2.84/1.53 => ( woz1_3
% 2.84/1.53 => ( woz1_4
% 2.84/1.53 => ( woz1_5
% 2.84/1.53 => ( breln1all2
% 2.84/1.53 => ( breln1SetBreln1
% 2.84/1.53 => ( choice2fnsingleton
% 2.84/1.53 => ( setOfPairsIsBReln1
% 2.84/1.53 => ( breln1all1
% 2.84/1.53 => ( subbreln1
% 2.84/1.53 => ( eqbreln1
% 2.84/1.53 => ( breln1invprop
% 2.84/1.53 => ( breln1invI
% 2.84/1.53 => ( breln1invE
% 2.84/1.53 => ( breln1compprop
% 2.84/1.53 => ( breln1compI
% 2.84/1.53 => ( breln1compE
% 2.84/1.53 => ( breln1compEex
% 2.84/1.53 => ( breln1unionprop
% 2.84/1.53 => ( breln1unionIL
% 2.84/1.53 => ( breln1unionIR
% 2.84/1.53 => ( breln1unionI
% 2.84/1.53 => ( breln1unionE
% 2.84/1.53 => ( breln1unionEcases
% 2.84/1.53 => ( breln1unionCommutes
% 2.84/1.53 => ( woz2Ex
% 2.84/1.53 => ( woz2W
% 2.84/1.53 => ( woz2A
% 2.84/1.53 => ( woz2B
% 2.84/1.53 => ( image1Ex
% 2.84/1.53 => ( image1Ex1
% 2.84/1.53 => ( image1Equiv
% 2.84/1.53 => ( image1E
% 2.84/1.53 => ( image1I
% 2.84/1.53 => ( injFuncInInjFuncSet
% 2.84/1.53 => ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.53 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 2.84/1.53 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.53 ------------------------
% 2.84/1.56 % (18317)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/183Mi)
% 2.84/1.56 % (18319)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_18125 for (2999ds/27Mi)
% 2.84/1.56 % (18318)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_18125 for (2999ds/4Mi)
% 2.84/1.56 % (18322)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_18125 for (2999ds/275Mi)
% 2.84/1.56 % (18318)Instruction limit reached!
% 2.84/1.56 % (18318)------------------------------
% 2.84/1.56 % (18318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.56 % (18318)Termination reason: Unknown
% 2.84/1.56 % (18318)Termination phase: shuffling
% 2.84/1.56
% 2.84/1.56 % (18318)Memory used [KB]: 1663
% 2.84/1.56 % (18318)Time elapsed: 0.003 s
% 2.84/1.56 % (18318)Instructions burned: 4 (million)
% 2.84/1.56 % (18318)------------------------------
% 2.84/1.56 % (18318)------------------------------
% 2.84/1.56 % (18323)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/18Mi)
% 2.84/1.56 % (18320)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/2Mi)
% 2.84/1.56 % (18320)Instruction limit reached!
% 2.84/1.56 % (18320)------------------------------
% 2.84/1.56 % (18320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.56 % (18320)Termination reason: Unknown
% 2.84/1.56 % (18320)Termination phase: shuffling
% 2.84/1.56
% 2.84/1.56 % (18320)Memory used [KB]: 1535
% 2.84/1.56 % (18320)Time elapsed: 0.003 s
% 2.84/1.56 % (18320)Instructions burned: 2 (million)
% 2.84/1.56 % (18320)------------------------------
% 2.84/1.56 % (18320)------------------------------
% 2.84/1.57 % (18321)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_18125 for (2999ds/2Mi)
% 2.84/1.57 % (18319)Instruction limit reached!
% 2.84/1.57 % (18319)------------------------------
% 2.84/1.57 % (18319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.57 % (18319)Termination reason: Unknown
% 2.84/1.57 % (18319)Termination phase: shuffling
% 2.84/1.57
% 2.84/1.57 % (18319)Memory used [KB]: 2174
% 2.84/1.57 % (18319)Time elapsed: 0.010 s
% 2.84/1.57 % (18319)Instructions burned: 29 (million)
% 2.84/1.57 % (18319)------------------------------
% 2.84/1.57 % (18319)------------------------------
% 2.84/1.57 % (18321)Instruction limit reached!
% 2.84/1.57 % (18321)------------------------------
% 2.84/1.57 % (18321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.57 % (18321)Termination reason: Unknown
% 2.84/1.57 % (18321)Termination phase: shuffling
% 2.84/1.57
% 2.84/1.57 % (18321)Memory used [KB]: 1663
% 2.84/1.57 % (18321)Time elapsed: 0.003 s
% 2.84/1.57 % (18321)Instructions burned: 3 (million)
% 2.84/1.57 % (18321)------------------------------
% 2.84/1.57 % (18321)------------------------------
% 2.84/1.57 % (18323)Instruction limit reached!
% 2.84/1.57 % (18323)------------------------------
% 2.84/1.57 % (18323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.57 % (18323)Termination reason: Unknown
% 2.84/1.57 % (18323)Termination phase: shuffling
% 2.84/1.57
% 2.84/1.57 % (18323)Memory used [KB]: 1918
% 2.84/1.57 % (18323)Time elapsed: 0.009 s
% 2.84/1.57 % (18323)Instructions burned: 20 (million)
% 2.84/1.57 % (18323)------------------------------
% 2.84/1.57 % (18323)------------------------------
% 2.84/1.57 % (18324)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/3Mi)
% 2.84/1.57 % (18324)Instruction limit reached!
% 2.84/1.57 % (18324)------------------------------
% 2.84/1.57 % (18324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.57 % (18324)Termination reason: Unknown
% 2.84/1.57 % (18324)Termination phase: shuffling
% 2.84/1.57
% 2.84/1.57 % (18324)Memory used [KB]: 1663
% 2.84/1.57 % (18324)Time elapsed: 0.002 s
% 2.84/1.57 % (18324)Instructions burned: 3 (million)
% 2.84/1.57 % (18324)------------------------------
% 2.84/1.57 % (18324)------------------------------
% 2.84/1.58 % (18326)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_18125 for (2999ds/15Mi)
% 2.84/1.58 % (18328)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_18125 for (2999ds/1041Mi)
% 2.84/1.58 % (18325)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_18125 for (2999ds/37Mi)
% 2.84/1.58 % (18326)Instruction limit reached!
% 2.84/1.58 % (18326)------------------------------
% 2.84/1.58 % (18326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.58 % (18326)Termination reason: Unknown
% 2.84/1.58 % (18326)Termination phase: shuffling
% 2.84/1.58
% 2.84/1.58 % (18326)Memory used [KB]: 1918
% 2.84/1.58 % (18326)Time elapsed: 0.007 s
% 2.84/1.58 % (18326)Instructions burned: 17 (million)
% 2.84/1.58 % (18326)------------------------------
% 2.84/1.58 % (18326)------------------------------
% 2.84/1.59 % (18329)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/7Mi)
% 2.84/1.59 % (18327)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_18125 for (2999ds/3Mi)
% 2.84/1.59 % (18327)Instruction limit reached!
% 2.84/1.59 % (18327)------------------------------
% 2.84/1.59 % (18327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.59 % (18327)Termination reason: Unknown
% 2.84/1.59 % (18327)Termination phase: shuffling
% 2.84/1.59
% 2.84/1.59 % (18327)Memory used [KB]: 1663
% 2.84/1.59 % (18327)Time elapsed: 0.003 s
% 2.84/1.59 % (18327)Instructions burned: 3 (million)
% 2.84/1.59 % (18327)------------------------------
% 2.84/1.59 % (18327)------------------------------
% 2.84/1.59 % (18329)Instruction limit reached!
% 2.84/1.59 % (18329)------------------------------
% 2.84/1.59 % (18329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.59 % (18329)Termination reason: Unknown
% 2.84/1.59 % (18329)Termination phase: shuffling
% 2.84/1.59
% 2.84/1.59 % (18329)Memory used [KB]: 1791
% 2.84/1.59 % (18329)Time elapsed: 0.006 s
% 2.84/1.59 % (18329)Instructions burned: 8 (million)
% 2.84/1.59 % (18329)------------------------------
% 2.84/1.59 % (18329)------------------------------
% 2.84/1.60 % (18330)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/16Mi)
% 2.88/1.60 % (18325)Instruction limit reached!
% 2.88/1.60 % (18325)------------------------------
% 2.88/1.60 % (18325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (18325)Termination reason: Unknown
% 2.88/1.60 % (18325)Termination phase: shuffling
% 2.88/1.60
% 2.88/1.60 % (18325)Memory used [KB]: 2430
% 2.88/1.60 % (18325)Time elapsed: 0.018 s
% 2.88/1.60 % (18325)Instructions burned: 38 (million)
% 2.88/1.60 % (18325)------------------------------
% 2.88/1.60 % (18325)------------------------------
% 2.88/1.60 % (18332)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_18125 for (2999ds/3Mi)
% 2.88/1.60 % (18330)Instruction limit reached!
% 2.88/1.60 % (18330)------------------------------
% 2.88/1.60 % (18330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (18330)Termination reason: Unknown
% 2.88/1.60 % (18330)Termination phase: shuffling
% 2.88/1.60
% 2.88/1.60 % (18330)Memory used [KB]: 1918
% 2.88/1.60 % (18330)Time elapsed: 0.006 s
% 2.88/1.60 % (18330)Instructions burned: 16 (million)
% 2.88/1.60 % (18330)------------------------------
% 2.88/1.60 % (18330)------------------------------
% 2.88/1.60 % (18332)Instruction limit reached!
% 2.88/1.60 % (18332)------------------------------
% 2.88/1.60 % (18332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (18332)Termination reason: Unknown
% 2.88/1.60 % (18332)Termination phase: shuffling
% 2.88/1.60
% 2.88/1.60 % (18332)Memory used [KB]: 1663
% 2.88/1.60 % (18332)Time elapsed: 0.003 s
% 2.88/1.60 % (18332)Instructions burned: 6 (million)
% 2.88/1.60 % (18332)------------------------------
% 2.88/1.60 % (18332)------------------------------
% 2.88/1.61 % (18317)Instruction limit reached!
% 2.88/1.61 % (18317)------------------------------
% 2.88/1.61 % (18317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (18317)Termination reason: Unknown
% 2.88/1.61 % (18317)Termination phase: Property scanning
% 2.88/1.61
% 2.88/1.61 % (18317)Memory used [KB]: 3070
% 2.88/1.61 % (18317)Time elapsed: 0.048 s
% 2.88/1.61 % (18317)Instructions burned: 183 (million)
% 2.88/1.61 % (18317)------------------------------
% 2.88/1.61 % (18317)------------------------------
% 2.88/1.61 % (18331)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_18125 for (2999ds/3Mi)
% 2.88/1.61 % (18331)Instruction limit reached!
% 2.88/1.61 % (18331)------------------------------
% 2.88/1.61 % (18331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (18331)Termination reason: Unknown
% 2.88/1.61 % (18331)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (18331)Memory used [KB]: 1663
% 2.88/1.61 % (18331)Time elapsed: 0.004 s
% 2.88/1.61 % (18331)Instructions burned: 4 (million)
% 2.88/1.61 % (18331)------------------------------
% 2.88/1.61 % (18331)------------------------------
% 2.88/1.61 % (18334)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/3Mi)
% 2.88/1.61 % (18334)Instruction limit reached!
% 2.88/1.61 % (18334)------------------------------
% 2.88/1.61 % (18334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (18334)Termination reason: Unknown
% 2.88/1.61 % (18334)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (18334)Memory used [KB]: 1663
% 2.88/1.61 % (18334)Time elapsed: 0.002 s
% 2.88/1.61 % (18334)Instructions burned: 3 (million)
% 2.88/1.61 % (18334)------------------------------
% 2.88/1.61 % (18334)------------------------------
% 2.88/1.61 % (18335)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/4Mi)
% 2.88/1.61 % (18335)Instruction limit reached!
% 2.88/1.61 % (18335)------------------------------
% 2.88/1.61 % (18335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (18335)Termination reason: Unknown
% 2.88/1.61 % (18335)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (18335)Memory used [KB]: 1663
% 2.88/1.61 % (18335)Time elapsed: 0.003 s
% 2.88/1.61 % (18335)Instructions burned: 4 (million)
% 2.88/1.61 % (18335)------------------------------
% 2.88/1.61 % (18335)------------------------------
% 2.88/1.61 % (18333)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/7Mi)
% 2.88/1.62 % (18336)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/18Mi)
% 2.88/1.62 % (18333)Instruction limit reached!
% 2.88/1.62 % (18333)------------------------------
% 2.88/1.62 % (18333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (18333)Termination reason: Unknown
% 2.88/1.62 % (18333)Termination phase: shuffling
% 2.88/1.62
% 2.88/1.62 % (18333)Memory used [KB]: 1663
% 2.88/1.62 % (18333)Time elapsed: 0.005 s
% 2.88/1.62 % (18333)Instructions burned: 7 (million)
% 2.88/1.62 % (18333)------------------------------
% 2.88/1.62 % (18333)------------------------------
% 2.88/1.62 % (18336)Instruction limit reached!
% 2.88/1.62 % (18336)------------------------------
% 2.88/1.62 % (18336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (18336)Termination reason: Unknown
% 2.88/1.62 % (18336)Termination phase: shuffling
% 2.88/1.62
% 2.88/1.62 % (18336)Memory used [KB]: 1918
% 2.88/1.62 % (18336)Time elapsed: 0.006 s
% 2.88/1.62 % (18336)Instructions burned: 18 (million)
% 2.88/1.62 % (18336)------------------------------
% 2.88/1.62 % (18336)------------------------------
% 2.88/1.62 % (18338)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/6Mi)
% 2.88/1.62 % (18338)Instruction limit reached!
% 2.88/1.62 % (18338)------------------------------
% 2.88/1.62 % (18338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (18338)Termination reason: Unknown
% 2.88/1.62 % (18338)Termination phase: shuffling
% 2.88/1.62
% 2.88/1.62 % (18338)Memory used [KB]: 1663
% 2.88/1.62 % (18338)Time elapsed: 0.003 s
% 2.88/1.62 % (18338)Instructions burned: 7 (million)
% 2.88/1.62 % (18338)------------------------------
% 2.88/1.62 % (18338)------------------------------
% 2.88/1.62 % (18339)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_18125 for (2999ds/902Mi)
% 2.88/1.62 % (18337)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/710Mi)
% 2.88/1.63 % (18341)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/5Mi)
% 2.88/1.63 % (18341)Instruction limit reached!
% 2.88/1.63 % (18341)------------------------------
% 2.88/1.63 % (18341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.63 % (18341)Termination reason: Unknown
% 2.88/1.63 % (18341)Termination phase: shuffling
% 2.88/1.63
% 2.88/1.63 % (18341)Memory used [KB]: 1663
% 2.88/1.63 % (18341)Time elapsed: 0.003 s
% 2.88/1.63 % (18341)Instructions burned: 7 (million)
% 2.88/1.63 % (18341)------------------------------
% 2.88/1.63 % (18341)------------------------------
% 2.88/1.63 % (18340)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_18125 for (2999ds/21Mi)
% 2.88/1.63 % (18342)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_18125 for (2999ds/6Mi)
% 2.88/1.64 % (18342)Instruction limit reached!
% 2.88/1.64 % (18342)------------------------------
% 2.88/1.64 % (18342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.64 % (18342)Termination reason: Unknown
% 2.88/1.64 % (18342)Termination phase: shuffling
% 2.88/1.64
% 2.88/1.64 % (18342)Memory used [KB]: 1663
% 2.88/1.64 % (18342)Time elapsed: 0.003 s
% 2.88/1.64 % (18342)Instructions burned: 7 (million)
% 2.88/1.64 % (18342)------------------------------
% 2.88/1.64 % (18342)------------------------------
% 2.88/1.64 % (18322)Instruction limit reached!
% 2.88/1.64 % (18322)------------------------------
% 2.88/1.64 % (18322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.64 % (18322)Termination reason: Unknown
% 2.88/1.64 % (18322)Termination phase: Saturation
% 2.88/1.64
% 2.88/1.64 % (18322)Memory used [KB]: 5884
% 2.88/1.64 % (18322)Time elapsed: 0.079 s
% 2.88/1.64 % (18322)Instructions burned: 276 (million)
% 2.88/1.64 % (18322)------------------------------
% 2.88/1.64 % (18322)------------------------------
% 2.88/1.64 % (18343)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/377Mi)
% 2.88/1.64 % (18340)Instruction limit reached!
% 2.88/1.64 % (18340)------------------------------
% 2.88/1.64 % (18340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.64 % (18340)Termination reason: Unknown
% 2.88/1.64 % (18340)Termination phase: shuffling
% 2.88/1.64
% 2.88/1.64 % (18340)Memory used [KB]: 2046
% 2.88/1.64 % (18340)Time elapsed: 0.012 s
% 2.88/1.64 % (18340)Instructions burned: 22 (million)
% 2.88/1.64 % (18340)------------------------------
% 2.88/1.64 % (18340)------------------------------
% 2.88/1.65 % (18344)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_18125 for (2999ds/779Mi)
% 2.88/1.65 % (18345)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_18125 for (2999ds/19Mi)
% 2.88/1.65 % (18345)Instruction limit reached!
% 2.88/1.65 % (18345)------------------------------
% 2.88/1.65 % (18345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.65 % (18345)Termination reason: Unknown
% 2.88/1.65 % (18345)Termination phase: shuffling
% 2.88/1.65
% 2.88/1.65 % (18345)Memory used [KB]: 1918
% 2.88/1.65 % (18345)Time elapsed: 0.007 s
% 2.88/1.65 % (18345)Instructions burned: 21 (million)
% 2.88/1.65 % (18345)------------------------------
% 2.88/1.65 % (18345)------------------------------
% 2.88/1.66 % (18346)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_18125 for (2999ds/879Mi)
% 2.88/1.66 % (18347)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_18125 for (2998ds/17Mi)
% 2.88/1.67 % (18347)Instruction limit reached!
% 2.88/1.67 % (18347)------------------------------
% 2.88/1.67 % (18347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.67 % (18347)Termination reason: Unknown
% 2.88/1.67 % (18347)Termination phase: shuffling
% 2.88/1.67
% 2.88/1.67 % (18347)Memory used [KB]: 1918
% 2.88/1.67 % (18347)Time elapsed: 0.007 s
% 2.88/1.67 % (18347)Instructions burned: 20 (million)
% 2.88/1.67 % (18347)------------------------------
% 2.88/1.67 % (18347)------------------------------
% 2.88/1.68 % (18348)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18125 for (2998ds/3Mi)
% 2.88/1.68 % (18348)Instruction limit reached!
% 2.88/1.68 % (18348)------------------------------
% 2.88/1.68 % (18348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.68 % (18348)Termination reason: Unknown
% 2.88/1.68 % (18348)Termination phase: shuffling
% 2.88/1.68
% 2.88/1.68 % (18348)Memory used [KB]: 1663
% 2.88/1.68 % (18348)Time elapsed: 0.003 s
% 2.88/1.68 % (18348)Instructions burned: 5 (million)
% 2.88/1.68 % (18348)------------------------------
% 2.88/1.68 % (18348)------------------------------
% 2.88/1.69 % (18349)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_18125 for (2998ds/30Mi)
% 2.88/1.70 % (18349)Instruction limit reached!
% 2.88/1.70 % (18349)------------------------------
% 2.88/1.70 % (18349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.70 % (18349)Termination reason: Unknown
% 2.88/1.70 % (18349)Termination phase: shuffling
% 2.88/1.70
% 2.88/1.70 % (18349)Memory used [KB]: 2174
% 2.88/1.70 % (18349)Time elapsed: 0.010 s
% 2.88/1.70 % (18349)Instructions burned: 30 (million)
% 2.88/1.70 % (18349)------------------------------
% 2.88/1.70 % (18349)------------------------------
% 2.88/1.71 % (18350)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_18125 for (2998ds/127Mi)
% 2.88/1.74 % (18343)Instruction limit reached!
% 2.88/1.74 % (18343)------------------------------
% 2.88/1.74 % (18343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.74 % (18343)Termination reason: Unknown
% 2.88/1.74 % (18343)Termination phase: Saturation
% 2.88/1.74
% 2.88/1.74 % (18343)Memory used [KB]: 9466
% 2.88/1.74 % (18343)Time elapsed: 0.102 s
% 2.88/1.74 % (18343)Instructions burned: 378 (million)
% 2.88/1.74 % (18343)------------------------------
% 2.88/1.74 % (18343)------------------------------
% 2.88/1.75 % (18350)Instruction limit reached!
% 2.88/1.75 % (18350)------------------------------
% 2.88/1.75 % (18350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.75 % (18350)Termination reason: Unknown
% 2.88/1.75 % (18350)Termination phase: Property scanning
% 2.88/1.75
% 2.88/1.75 % (18350)Memory used [KB]: 2942
% 2.88/1.75 % (18350)Time elapsed: 0.037 s
% 2.88/1.75 % (18350)Instructions burned: 127 (million)
% 2.88/1.75 % (18350)------------------------------
% 2.88/1.75 % (18350)------------------------------
% 2.88/1.75 % (18356)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_18125 for (2997ds/100Mi)
% 2.88/1.75 % (18357)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_18125 for (2997ds/3Mi)
% 2.88/1.76 % (18357)Instruction limit reached!
% 2.88/1.76 % (18357)------------------------------
% 2.88/1.76 % (18357)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.76 % (18357)Termination reason: Unknown
% 2.88/1.76 % (18357)Termination phase: shuffling
% 2.88/1.76
% 2.88/1.76 % (18357)Memory used [KB]: 1663
% 2.88/1.76 % (18357)Time elapsed: 0.002 s
% 2.88/1.76 % (18357)Instructions burned: 3 (million)
% 2.88/1.76 % (18357)------------------------------
% 2.88/1.76 % (18357)------------------------------
% 2.88/1.77 % (18358)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_18125 for (2997ds/20Mi)
% 2.88/1.77 % (18358)Instruction limit reached!
% 2.88/1.77 % (18358)------------------------------
% 2.88/1.77 % (18358)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.77 % (18358)Termination reason: Unknown
% 2.88/1.77 % (18358)Termination phase: shuffling
% 2.88/1.77
% 2.88/1.77 % (18358)Memory used [KB]: 1918
% 2.88/1.77 % (18358)Time elapsed: 0.008 s
% 2.88/1.77 % (18358)Instructions burned: 22 (million)
% 2.88/1.77 % (18358)------------------------------
% 2.88/1.77 % (18358)------------------------------
% 2.88/1.78 % (18356)Instruction limit reached!
% 2.88/1.78 % (18356)------------------------------
% 2.88/1.78 % (18356)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.78 % (18356)Termination reason: Unknown
% 2.88/1.78 % (18356)Termination phase: Property scanning
% 2.88/1.78
% 2.88/1.78 % (18356)Memory used [KB]: 2942
% 2.88/1.78 % (18356)Time elapsed: 0.027 s
% 2.88/1.78 % (18356)Instructions burned: 101 (million)
% 2.88/1.78 % (18356)------------------------------
% 2.88/1.78 % (18356)------------------------------
% 2.88/1.78 % (18360)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_18125 for (2997ds/86Mi)
% 2.88/1.79 % (18362)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_18125 for (2997ds/107Mi)
% 2.88/1.81 % (18360)Instruction limit reached!
% 2.88/1.81 % (18360)------------------------------
% 2.88/1.81 % (18360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.81 % (18360)Termination reason: Unknown
% 2.88/1.81 % (18360)Termination phase: Property scanning
% 2.88/1.81
% 2.88/1.81 % (18360)Memory used [KB]: 2686
% 2.88/1.81 % (18360)Time elapsed: 0.032 s
% 2.88/1.81 % (18360)Instructions burned: 86 (million)
% 2.88/1.81 % (18360)------------------------------
% 2.88/1.81 % (18360)------------------------------
% 2.88/1.83 % (18362)Instruction limit reached!
% 2.88/1.83 % (18362)------------------------------
% 2.88/1.83 % (18362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.83 % (18362)Termination reason: Unknown
% 2.88/1.83 % (18373)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_18125 for (2997ds/39Mi)
% 2.88/1.83 % (18362)Termination phase: SInE selection
% 2.88/1.83
% 2.88/1.83 % (18362)Memory used [KB]: 2814
% 2.88/1.83 % (18362)Time elapsed: 0.045 s
% 2.88/1.83 % (18362)Instructions burned: 109 (million)
% 2.88/1.83 % (18362)------------------------------
% 2.88/1.83 % (18362)------------------------------
% 2.88/1.85 % (18377)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_18125 for (2997ds/448Mi)
% 2.88/1.85 % (18373)Instruction limit reached!
% 2.88/1.85 % (18373)------------------------------
% 2.88/1.85 % (18373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.85 % (18373)Termination reason: Unknown
% 2.88/1.85 % (18373)Termination phase: shuffling
% 2.88/1.85
% 2.88/1.85 % (18373)Memory used [KB]: 2430
% 2.88/1.85 % (18373)Time elapsed: 0.019 s
% 2.88/1.85 % (18373)Instructions burned: 40 (million)
% 2.88/1.85 % (18373)------------------------------
% 2.88/1.85 % (18373)------------------------------
% 2.88/1.86 % (18344)Instruction limit reached!
% 2.88/1.86 % (18344)------------------------------
% 2.88/1.86 % (18344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.86 % (18344)Termination reason: Unknown
% 2.88/1.86 % (18344)Termination phase: Saturation
% 2.88/1.86
% 2.88/1.86 % (18344)Memory used [KB]: 7931
% 2.88/1.86 % (18344)Time elapsed: 0.214 s
% 2.88/1.86 % (18344)Instructions burned: 779 (million)
% 2.88/1.86 % (18344)------------------------------
% 2.88/1.86 % (18344)------------------------------
% 2.88/1.87 % (18381)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_18125 for (2996ds/46Mi)
% 2.88/1.88 % (18384)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_18125 for (2996ds/98Mi)
% 2.88/1.88 % (18328)Instruction limit reached!
% 2.88/1.88 % (18328)------------------------------
% 2.88/1.88 % (18328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.88 % (18328)Termination reason: Unknown
% 2.88/1.88 % (18328)Termination phase: Saturation
% 2.88/1.88
% 2.88/1.88 % (18328)Memory used [KB]: 18166
% 2.88/1.88 % (18328)Time elapsed: 0.304 s
% 2.88/1.88 % (18328)Instructions burned: 1043 (million)
% 2.88/1.88 % (18328)------------------------------
% 2.88/1.88 % (18328)------------------------------
% 2.88/1.89 % (18381)Instruction limit reached!
% 2.88/1.89 % (18381)------------------------------
% 2.88/1.89 % (18381)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.89 % (18381)Termination reason: Unknown
% 2.88/1.89 % (18381)Termination phase: shuffling
% 2.88/1.89
% 2.88/1.89 % (18381)Memory used [KB]: 2558
% 2.88/1.89 % (18381)Time elapsed: 0.023 s
% 2.88/1.89 % (18381)Instructions burned: 47 (million)
% 2.88/1.89 % (18381)------------------------------
% 2.88/1.89 % (18381)------------------------------
% 2.88/1.89 % (18391)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_18125 for (2996ds/507Mi)
% 2.88/1.90 % (18392)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_18125 for (2996ds/149Mi)
% 2.88/1.91 % (18337)Instruction limit reached!
% 2.88/1.91 % (18337)------------------------------
% 2.88/1.91 % (18337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.91 % (18337)Termination reason: Unknown
% 2.88/1.91 % (18337)Termination phase: Saturation
% 2.88/1.91
% 2.88/1.91 % (18337)Memory used [KB]: 11769
% 2.88/1.91 % (18337)Time elapsed: 0.290 s
% 2.88/1.91 % (18337)Instructions burned: 710 (million)
% 2.88/1.91 % (18337)------------------------------
% 2.88/1.91 % (18337)------------------------------
% 2.88/1.92 % (18384)Instruction limit reached!
% 2.88/1.92 % (18384)------------------------------
% 2.88/1.92 % (18384)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.92 % (18384)Termination reason: Unknown
% 2.88/1.92 % (18384)Termination phase: Property scanning
% 2.88/1.92
% 2.88/1.92 % (18384)Memory used [KB]: 2814
% 2.88/1.92 % (18384)Time elapsed: 0.044 s
% 2.88/1.92 % (18384)Instructions burned: 99 (million)
% 2.88/1.92 % (18384)------------------------------
% 2.88/1.92 % (18384)------------------------------
% 2.88/1.93 % (18399)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_18125 for (2996ds/39Mi)
% 2.88/1.94 % (18400)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_18125 for (2996ds/6Mi)
% 2.88/1.94 % (18399)Instruction limit reached!
% 2.88/1.94 % (18399)------------------------------
% 2.88/1.94 % (18399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.94 % (18399)Termination reason: Unknown
% 2.88/1.94 % (18399)Termination phase: shuffling
% 2.88/1.94
% 2.88/1.94 % (18399)Memory used [KB]: 2430
% 2.88/1.94 % (18399)Time elapsed: 0.012 s
% 2.88/1.94 % (18399)Instructions burned: 40 (million)
% 2.88/1.94 % (18399)------------------------------
% 2.88/1.94 % (18399)------------------------------
% 2.88/1.94 % (18400)Instruction limit reached!
% 2.88/1.94 % (18400)------------------------------
% 2.88/1.94 % (18400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.94 % (18400)Termination reason: Unknown
% 2.88/1.94 % (18400)Termination phase: shuffling
% 2.88/1.94
% 2.88/1.94 % (18400)Memory used [KB]: 1663
% 2.88/1.94 % (18400)Time elapsed: 0.005 s
% 2.88/1.94 % (18400)Instructions burned: 6 (million)
% 2.88/1.94 % (18400)------------------------------
% 2.88/1.94 % (18400)------------------------------
% 2.88/1.95 % (18392)Instruction limit reached!
% 2.88/1.95 % (18392)------------------------------
% 2.88/1.95 % (18392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.95 % (18392)Termination reason: Unknown
% 2.88/1.95 % (18392)Termination phase: Function definition elimination
% 2.88/1.95
% 2.88/1.95 % (18392)Memory used [KB]: 2942
% 2.88/1.95 % (18392)Time elapsed: 0.049 s
% 2.88/1.95 % (18392)Instructions burned: 150 (million)
% 2.88/1.95 % (18392)------------------------------
% 2.88/1.95 % (18392)------------------------------
% 2.88/1.95 % (18402)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_18125 for (2996ds/1087Mi)
% 2.88/1.96 % (18403)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_18125 for (2995ds/47Mi)
% 2.88/1.96 % (18404)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_18125 for (2995ds/44Mi)
% 2.88/1.98 % (18339)Instruction limit reached!
% 2.88/1.98 % (18339)------------------------------
% 2.88/1.98 % (18339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.98 % (18339)Termination reason: Unknown
% 2.88/1.98 % (18339)Termination phase: Saturation
% 2.88/1.98
% 2.88/1.98 % (18339)Memory used [KB]: 13176
% 2.88/1.98 % (18339)Time elapsed: 0.354 s
% 2.88/1.98 % (18339)Instructions burned: 903 (million)
% 2.88/1.98 % (18339)------------------------------
% 2.88/1.98 % (18339)------------------------------
% 2.88/1.98 % (18403)Instruction limit reached!
% 2.88/1.98 % (18403)------------------------------
% 2.88/1.98 % (18403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.98 % (18403)Termination reason: Unknown
% 2.88/1.98 % (18403)Termination phase: shuffling
% 2.88/1.98
% 2.88/1.98 % (18403)Memory used [KB]: 2558
% 2.88/1.98 % (18403)Time elapsed: 0.023 s
% 2.88/1.98 % (18403)Instructions burned: 47 (million)
% 2.88/1.98 % (18403)------------------------------
% 2.88/1.98 % (18403)------------------------------
% 2.88/1.98 % (18404)Instruction limit reached!
% 2.88/1.98 % (18404)------------------------------
% 2.88/1.98 % (18404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.98 % (18404)Termination reason: Unknown
% 2.88/1.98 % (18404)Termination phase: shuffling
% 2.88/1.98
% 2.88/1.98 % (18404)Memory used [KB]: 2430
% 2.88/1.98 % (18404)Time elapsed: 0.022 s
% 2.88/1.98 % (18404)Instructions burned: 45 (million)
% 2.88/1.98 % (18404)------------------------------
% 2.88/1.98 % (18404)------------------------------
% 2.88/1.99 % (18408)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_18125 for (2995ds/62Mi)
% 2.88/1.99 % (18407)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_18125 for (2995ds/432Mi)
% 2.88/2.00 % (18410)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_18125 for (2995ds/31Mi)
% 2.88/2.01 % (18346)Instruction limit reached!
% 2.88/2.01 % (18346)------------------------------
% 2.88/2.01 % (18346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/2.01 % (18346)Termination reason: Unknown
% 2.88/2.01 % (18346)Termination phase: Saturation
% 2.88/2.01
% 2.88/2.01 % (18346)Memory used [KB]: 10746
% 2.88/2.01 % (18346)Time elapsed: 0.368 s
% 2.88/2.01 % (18346)Instructions burned: 880 (million)
% 2.88/2.01 % (18346)------------------------------
% 2.88/2.01 % (18346)------------------------------
% 2.88/2.01 % (18410)Instruction limit reached!
% 2.88/2.01 % (18410)------------------------------
% 2.88/2.01 % (18410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/2.01 % (18410)Termination reason: Unknown
% 2.88/2.01 % (18410)Termination phase: shuffling
% 2.88/2.01
% 2.88/2.01 % (18410)Memory used [KB]: 2302
% 2.88/2.01 % (18410)Time elapsed: 0.016 s
% 2.88/2.01 % (18410)Instructions burned: 32 (million)
% 2.88/2.01 % (18410)------------------------------
% 2.88/2.01 % (18410)------------------------------
% 2.88/2.02 % (18377)Instruction limit reached!
% 2.88/2.02 % (18377)------------------------------
% 2.88/2.02 % (18377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/2.02 % (18377)Termination reason: Unknown
% 2.88/2.02 % (18377)Termination phase: Saturation
% 2.88/2.02
% 2.88/2.02 % (18377)Memory used [KB]: 7803
% 2.88/2.02 % (18377)Time elapsed: 0.175 s
% 2.88/2.02 % (18377)Instructions burned: 448 (million)
% 2.88/2.02 % (18377)------------------------------
% 2.88/2.02 % (18377)------------------------------
% 2.88/2.02 % (18408)Instruction limit reached!
% 2.88/2.02 % (18408)------------------------------
% 2.88/2.02 % (18408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/2.02 % (18408)Termination reason: Unknown
% 2.88/2.02 % (18408)Termination phase: shuffling
% 2.88/2.02
% 2.88/2.02 % (18408)Memory used [KB]: 2686
% 2.88/2.02 % (18408)Time elapsed: 0.029 s
% 2.88/2.02 % (18408)Instructions burned: 62 (million)
% 2.88/2.02 % (18408)------------------------------
% 2.88/2.02 % (18408)------------------------------
% 6.09/2.03 % (18412)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_18125 for (2995ds/453Mi)
% 6.09/2.03 % (18411)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_18125 for (2995ds/96Mi)
% 6.09/2.04 % (18413)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_18125 for (2995ds/71Mi)
% 6.09/2.04 % (18414)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_18125 for (2995ds/83Mi)
% 6.09/2.07 % (18413)Instruction limit reached!
% 6.09/2.07 % (18413)------------------------------
% 6.09/2.07 % (18413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.07 % (18413)Termination reason: Unknown
% 6.09/2.07 % (18413)Termination phase: Property scanning
% 6.09/2.07
% 6.09/2.07 % (18413)Memory used [KB]: 2814
% 6.09/2.07 % (18413)Time elapsed: 0.034 s
% 6.09/2.07 % (18413)Instructions burned: 72 (million)
% 6.09/2.07 % (18413)------------------------------
% 6.09/2.07 % (18413)------------------------------
% 6.09/2.07 % (18411)Instruction limit reached!
% 6.09/2.07 % (18411)------------------------------
% 6.09/2.07 % (18411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.07 % (18411)Termination reason: Unknown
% 6.09/2.07 % (18411)Termination phase: Property scanning
% 6.09/2.07
% 6.09/2.07 % (18411)Memory used [KB]: 2814
% 6.09/2.08 % (18411)Time elapsed: 0.044 s
% 6.09/2.08 % (18411)Instructions burned: 98 (million)
% 6.09/2.08 % (18411)------------------------------
% 6.09/2.08 % (18411)------------------------------
% 6.09/2.08 % (18391)Instruction limit reached!
% 6.09/2.08 % (18391)------------------------------
% 6.09/2.08 % (18391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.08 % (18391)Termination reason: Unknown
% 6.09/2.08 % (18391)Termination phase: Saturation
% 6.09/2.08
% 6.09/2.08 % (18391)Memory used [KB]: 8315
% 6.09/2.08 % (18391)Time elapsed: 0.179 s
% 6.09/2.08 % (18391)Instructions burned: 508 (million)
% 6.09/2.08 % (18391)------------------------------
% 6.09/2.08 % (18391)------------------------------
% 6.09/2.08 % (18414)Instruction limit reached!
% 6.09/2.08 % (18414)------------------------------
% 6.09/2.08 % (18414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.08 % (18414)Termination reason: Unknown
% 6.09/2.08 % (18414)Termination phase: Property scanning
% 6.09/2.08
% 6.09/2.08 % (18414)Memory used [KB]: 2814
% 6.09/2.08 % (18414)Time elapsed: 0.037 s
% 6.09/2.08 % (18414)Instructions burned: 83 (million)
% 6.09/2.08 % (18414)------------------------------
% 6.09/2.08 % (18414)------------------------------
% 6.09/2.09 % (18416)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_18125 for (2994ds/11Mi)
% 6.09/2.09 % (18416)Instruction limit reached!
% 6.09/2.09 % (18416)------------------------------
% 6.09/2.09 % (18416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.09 % (18416)Termination reason: Unknown
% 6.09/2.09 % (18416)Termination phase: shuffling
% 6.09/2.09
% 6.09/2.09 % (18416)Memory used [KB]: 1791
% 6.09/2.09 % (18416)Time elapsed: 0.007 s
% 6.09/2.09 % (18416)Instructions burned: 11 (million)
% 6.09/2.09 % (18416)------------------------------
% 6.09/2.09 % (18416)------------------------------
% 6.09/2.09 % (18417)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_18125 for (2994ds/41Mi)
% 6.09/2.10 % (18419)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18125 for (2994ds/6Mi)
% 6.09/2.10 % (18420)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_18125 for (2994ds/150Mi)
% 6.09/2.10 % (18419)Instruction limit reached!
% 6.09/2.10 % (18419)------------------------------
% 6.09/2.10 % (18419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.10 % (18419)Termination reason: Unknown
% 6.09/2.10 % (18419)Termination phase: shuffling
% 6.09/2.10
% 6.09/2.10 % (18419)Memory used [KB]: 1663
% 6.09/2.10 % (18419)Time elapsed: 0.005 s
% 6.09/2.10 % (18419)Instructions burned: 7 (million)
% 6.09/2.10 % (18419)------------------------------
% 6.09/2.10 % (18419)------------------------------
% 6.09/2.11 % (18417)Instruction limit reached!
% 6.09/2.11 % (18417)------------------------------
% 6.09/2.11 % (18417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.11 % (18417)Termination reason: Unknown
% 6.09/2.11 % (18417)Termination phase: shuffling
% 6.09/2.11
% 6.09/2.11 % (18417)Memory used [KB]: 2430
% 6.09/2.11 % (18417)Time elapsed: 0.014 s
% 6.09/2.11 % (18417)Instructions burned: 44 (million)
% 6.09/2.11 % (18417)------------------------------
% 6.09/2.11 % (18417)------------------------------
% 6.09/2.11 % (18421)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_18125 for (2994ds/343Mi)
% 6.09/2.12 % (18423)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_18125 for (2994ds/109Mi)
% 6.09/2.12 % (18425)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_18125 for (2994ds/253Mi)
% 6.61/2.15 % (18423)Instruction limit reached!
% 6.61/2.15 % (18423)------------------------------
% 6.61/2.15 % (18423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.15 % (18423)Termination reason: Unknown
% 6.61/2.15 % (18423)Termination phase: SInE selection
% 6.61/2.15
% 6.61/2.15 % (18423)Memory used [KB]: 2814
% 6.61/2.15 % (18423)Time elapsed: 0.030 s
% 6.61/2.15 % (18423)Instructions burned: 111 (million)
% 6.61/2.15 % (18423)------------------------------
% 6.61/2.15 % (18423)------------------------------
% 6.61/2.16 % (18428)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_18125 for (2994ds/140Mi)
% 6.61/2.17 % (18420)Instruction limit reached!
% 6.61/2.17 % (18420)------------------------------
% 6.61/2.17 % (18420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.17 % (18420)Termination reason: Unknown
% 6.61/2.17 % (18420)Termination phase: Preprocessing 3
% 6.61/2.17
% 6.61/2.17 % (18420)Memory used [KB]: 5245
% 6.61/2.17 % (18420)Time elapsed: 0.074 s
% 6.61/2.17 % (18420)Instructions burned: 151 (million)
% 6.61/2.17 % (18420)------------------------------
% 6.61/2.17 % (18420)------------------------------
% 6.61/2.17 % (18407)Instruction limit reached!
% 6.61/2.17 % (18407)------------------------------
% 6.61/2.17 % (18407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.17 % (18407)Termination reason: Unknown
% 6.61/2.17 % (18407)Termination phase: Saturation
% 6.61/2.17
% 6.61/2.17 % (18407)Memory used [KB]: 3326
% 6.61/2.17 % (18407)Time elapsed: 0.180 s
% 6.61/2.17 % (18407)Instructions burned: 433 (million)
% 6.61/2.17 % (18407)------------------------------
% 6.61/2.17 % (18407)------------------------------
% 6.61/2.19 % (18431)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_18125 for (2993ds/614Mi)
% 6.61/2.19 % (18432)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_18125 for (2993ds/202Mi)
% 6.61/2.20 % (18428)Instruction limit reached!
% 6.61/2.20 % (18428)------------------------------
% 6.61/2.20 % (18428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.20 % (18428)Termination reason: Unknown
% 6.61/2.20 % (18428)Termination phase: Property scanning
% 6.61/2.20
% 6.61/2.20 % (18428)Memory used [KB]: 3070
% 6.61/2.20 % (18428)Time elapsed: 0.042 s
% 6.61/2.20 % (18428)Instructions burned: 142 (million)
% 6.61/2.20 % (18428)------------------------------
% 6.61/2.20 % (18428)------------------------------
% 6.87/2.21 % (18435)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_18125 for (2993ds/339Mi)
% 6.87/2.22 % (18412)Instruction limit reached!
% 6.87/2.22 % (18412)------------------------------
% 6.87/2.22 % (18412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.22 % (18412)Termination reason: Unknown
% 6.87/2.22 % (18412)Termination phase: Saturation
% 6.87/2.22
% 6.87/2.22 % (18412)Memory used [KB]: 7803
% 6.87/2.22 % (18412)Time elapsed: 0.188 s
% 6.87/2.22 % (18412)Instructions burned: 453 (million)
% 6.87/2.22 % (18412)------------------------------
% 6.87/2.22 % (18412)------------------------------
% 6.87/2.23 % (18421)Instruction limit reached!
% 6.87/2.23 % (18421)------------------------------
% 6.87/2.23 % (18421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.23 % (18421)Termination reason: Unknown
% 6.87/2.23 % (18421)Termination phase: Saturation
% 6.87/2.23
% 6.87/2.23 % (18421)Memory used [KB]: 8443
% 6.87/2.23 % (18421)Time elapsed: 0.126 s
% 6.87/2.23 % (18421)Instructions burned: 343 (million)
% 6.87/2.23 % (18421)------------------------------
% 6.87/2.23 % (18421)------------------------------
% 6.87/2.24 % (18439)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_18125 for (2993ds/283Mi)
% 6.87/2.24 % (18425)Instruction limit reached!
% 6.87/2.24 % (18425)------------------------------
% 6.87/2.24 % (18425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.24 % (18425)Termination reason: Unknown
% 6.87/2.24 % (18425)Termination phase: Property scanning
% 6.87/2.24
% 6.87/2.24 % (18425)Memory used [KB]: 5884
% 6.87/2.24 % (18425)Time elapsed: 0.120 s
% 6.87/2.24 % (18425)Instructions burned: 254 (million)
% 6.87/2.24 % (18425)------------------------------
% 6.87/2.24 % (18425)------------------------------
% 6.87/2.24 % (18441)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_18125 for (2993ds/134Mi)
% 6.87/2.25 % (18444)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_18125 for (2992ds/298Mi)
% 6.87/2.28 % (18432)Instruction limit reached!
% 6.87/2.28 % (18432)------------------------------
% 6.87/2.28 % (18432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.28 % (18432)Termination reason: Unknown
% 6.87/2.28 % (18432)Termination phase: Property scanning
% 6.87/2.28
% 6.87/2.28 % (18432)Memory used [KB]: 5628
% 6.87/2.28 % (18432)Time elapsed: 0.090 s
% 6.87/2.28 % (18432)Instructions burned: 202 (million)
% 6.87/2.28 % (18432)------------------------------
% 6.87/2.28 % (18432)------------------------------
% 6.87/2.28 % (18441)Instruction limit reached!
% 6.87/2.28 % (18441)------------------------------
% 6.87/2.28 % (18441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.28 % (18441)Termination reason: Unknown
% 6.87/2.28 % (18441)Termination phase: Property scanning
% 6.87/2.28
% 6.87/2.28 % (18441)Memory used [KB]: 2942
% 6.87/2.28 % (18441)Time elapsed: 0.041 s
% 6.87/2.28 % (18441)Instructions burned: 135 (million)
% 6.87/2.28 % (18441)------------------------------
% 6.87/2.28 % (18441)------------------------------
% 6.87/2.29 % (18448)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_18125 for (2992ds/192Mi)
% 7.65/2.30 % (18449)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_18125 for (2992ds/258Mi)
% 7.65/2.35 % (18444)Instruction limit reached!
% 7.65/2.35 % (18444)------------------------------
% 7.65/2.35 % (18444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.65/2.35 % (18444)Termination reason: Unknown
% 7.65/2.35 % (18444)Termination phase: Saturation
% 7.65/2.35
% 7.65/2.35 % (18444)Memory used [KB]: 7675
% 7.65/2.35 % (18444)Time elapsed: 0.101 s
% 7.65/2.35 % (18444)Instructions burned: 298 (million)
% 7.65/2.35 % (18444)------------------------------
% 7.65/2.35 % (18444)------------------------------
% 7.65/2.35 % (18448)Instruction limit reached!
% 7.65/2.35 % (18448)------------------------------
% 7.65/2.35 % (18448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.65/2.35 % (18448)Termination reason: Unknown
% 7.65/2.35 % (18448)Termination phase: Saturation
% 7.65/2.35
% 7.65/2.35 % (18448)Memory used [KB]: 7675
% 7.65/2.35 % (18448)Time elapsed: 0.066 s
% 7.65/2.35 % (18448)Instructions burned: 192 (million)
% 7.65/2.35 % (18448)------------------------------
% 7.65/2.35 % (18448)------------------------------
% 7.65/2.35 % (18439)Instruction limit reached!
% 7.65/2.35 % (18439)------------------------------
% 7.65/2.35 % (18439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.65/2.35 % (18439)Termination reason: Unknown
% 7.65/2.35 % (18439)Termination phase: Saturation
% 7.65/2.35
% 7.65/2.35 % (18439)Memory used [KB]: 7675
% 7.65/2.35 % (18439)Time elapsed: 0.121 s
% 7.65/2.35 % (18439)Instructions burned: 283 (million)
% 7.65/2.35 % (18439)------------------------------
% 7.65/2.35 % (18439)------------------------------
% 7.65/2.36 % (18435)Instruction limit reached!
% 7.65/2.36 % (18435)------------------------------
% 7.65/2.36 % (18435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.65/2.36 % (18435)Termination reason: Unknown
% 7.65/2.36 % (18435)Termination phase: Saturation
% 7.65/2.36
% 7.65/2.36 % (18435)Memory used [KB]: 7931
% 7.65/2.36 % (18435)Time elapsed: 0.152 s
% 7.65/2.36 % (18435)Instructions burned: 339 (million)
% 7.65/2.36 % (18435)------------------------------
% 7.65/2.36 % (18435)------------------------------
% 7.65/2.37 % (18454)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_18125 for (2991ds/90Mi)
% 7.65/2.37 % (18455)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_18125 for (2991ds/231Mi)
% 7.65/2.37 % (18402)First to succeed.
% 7.65/2.37 % (18456)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_18125 for (2991ds/458Mi)
% 7.65/2.38 % (18402)Refutation found. Thanks to Tanya!
% 7.65/2.38 % SZS status Theorem for DTF2THF_18125
% 7.65/2.38 % SZS output start Proof for DTF2THF_18125
% 7.65/2.38 thf(func_def_0, type, in: $i > $i > $o).
% 7.65/2.38 thf(func_def_1, type, exu: ($i > $o) > $o).
% 7.65/2.38 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 7.65/2.38 thf(func_def_8, type, powerset: $i > $i).
% 7.65/2.38 thf(func_def_10, type, setunion: $i > $i).
% 7.65/2.38 thf(func_def_19, type, descr: ($i > $o) > $i).
% 7.65/2.38 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_26, type, prop2set: $o > $i).
% 7.65/2.38 thf(func_def_36, type, nonempty: $i > $o).
% 7.65/2.38 thf(func_def_69, type, set2prop: $i > $o).
% 7.65/2.38 thf(func_def_88, type, subset: $i > $i > $o).
% 7.65/2.38 thf(func_def_89, type, disjoint: $i > $i > $o).
% 7.65/2.38 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 7.65/2.38 thf(func_def_114, type, binunion: $i > $i > $i).
% 7.65/2.38 thf(func_def_122, type, binintersect: $i > $i > $i).
% 7.65/2.38 thf(func_def_135, type, regular: $i > $o).
% 7.65/2.38 thf(func_def_136, type, setminus: $i > $i > $i).
% 7.65/2.38 thf(func_def_147, type, symdiff: $i > $i > $i).
% 7.65/2.38 thf(func_def_153, type, iskpair: $i > $o).
% 7.65/2.38 thf(func_def_158, type, kpair: $i > $i > $i).
% 7.65/2.38 thf(func_def_160, type, cartprod: $i > $i > $i).
% 7.65/2.38 thf(func_def_177, type, singleton: $i > $o).
% 7.65/2.38 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_184, type, atmost1p: $i > $o).
% 7.65/2.38 thf(func_def_185, type, atleast2p: $i > $o).
% 7.65/2.38 thf(func_def_186, type, atmost2p: $i > $o).
% 7.65/2.38 thf(func_def_187, type, upairsetp: $i > $o).
% 7.65/2.38 thf(func_def_191, type, kfst: $i > $i).
% 7.65/2.38 thf(func_def_203, type, ksnd: $i > $i).
% 7.65/2.38 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 7.65/2.38 thf(func_def_222, type, func: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_223, type, funcSet: $i > $i > $i).
% 7.65/2.38 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 7.65/2.38 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 7.65/2.38 thf(func_def_323, type, breln1: $i > $i > $o).
% 7.65/2.38 thf(func_def_325, type, breln1Set: $i > $i).
% 7.65/2.38 thf(func_def_327, type, transitive: $i > $i > $o).
% 7.65/2.38 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 7.65/2.38 thf(func_def_329, type, reflexive: $i > $i > $o).
% 7.65/2.38 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 7.65/2.38 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 7.65/2.38 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 7.65/2.38 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 7.65/2.38 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 7.65/2.38 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 7.65/2.38 thf(func_def_382, type, sP1: $i > $o > $i > $i > $o).
% 7.65/2.38 thf(func_def_383, type, sP2: $i > $i > $i > $o > $o).
% 7.65/2.38 thf(func_def_384, type, sP3: $i > $o).
% 7.65/2.38 thf(func_def_385, type, sP4: $i > ($i > $i) > $i > $o).
% 7.65/2.38 thf(func_def_387, type, sP6: ($i > $o) > $o).
% 7.65/2.38 thf(func_def_388, type, sP7: $i > ($i > $i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_389, type, sP8: $i > ($i > $i > $o) > $o).
% 7.65/2.38 thf(func_def_391, type, sP10: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_393, type, sP12: ($i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_395, type, sP14: ($i > $o) > ($i > $o) > $i > $i > $o).
% 7.65/2.38 thf(func_def_396, type, sP15: ($i > $o) > $o).
% 7.65/2.38 thf(func_def_397, type, sP16: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_398, type, sP17: $i > $i > ($i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_399, type, sP18: ($i > $o) > $o).
% 7.65/2.38 thf(func_def_402, type, sP21: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_403, type, sP22: ($i > $o) > $o).
% 7.65/2.38 thf(func_def_404, type, sP23: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_406, type, sP25: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_409, type, sP28: $i > $i > $o).
% 7.65/2.38 thf(func_def_411, type, sP30: $i > $i > $o > $o).
% 7.65/2.38 thf(func_def_412, type, sP31: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_416, type, sP35: $i > $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_417, type, sP36: $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_418, type, sP37: $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_419, type, sP38: $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_420, type, sP39: $i > $i > $o).
% 7.65/2.38 thf(func_def_422, type, sP41: $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_423, type, sP42: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_425, type, sP44: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_426, type, sP45: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_427, type, sP46: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_429, type, sP48: ($i > $o) > $o).
% 7.65/2.38 thf(func_def_430, type, sP49: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_432, type, sP51: ($i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_434, type, sP53: $o > $i > $i > $o).
% 7.65/2.38 thf(func_def_436, type, sP55: $i > $i > ($i > $i) > $o).
% 7.65/2.38 thf(func_def_438, type, sP57: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_439, type, sP58: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_440, type, sP59: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_441, type, sP60: $i > ($i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_442, type, sP61: ($i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_443, type, sP62: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_444, type, sP63: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_446, type, sP65: ($i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_447, type, sP66: ($i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_448, type, sP67: $i > $i > $i > $o > $o).
% 7.65/2.38 thf(func_def_449, type, sP68: $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_451, type, sP70: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_452, type, sP71: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_453, type, sP72: $i > ($i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_454, type, sP73: ($i > $i > $o) > $i > $i > $o).
% 7.65/2.38 thf(func_def_455, type, sP74: $i > ($i > $i > $o) > $i > $o).
% 7.65/2.38 thf(func_def_456, type, sP75: $i > $i > $o).
% 7.65/2.38 thf(func_def_459, type, sP78: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_461, type, sP80: $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_462, type, sP81: $i > $i > $i > ($i > $o) > $o).
% 7.65/2.38 thf(func_def_463, type, sP82: $o > $i > $i > $o).
% 7.65/2.38 thf(func_def_464, type, sP83: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_465, type, sP84: ($i > $i) > $i > $i > $o).
% 7.65/2.38 thf(func_def_467, type, sP86: $i > $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_468, type, sP87: ($i > $i) > $i > $i > $o).
% 7.65/2.38 thf(func_def_470, type, sP89: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_471, type, sP90: $i > $i > $o).
% 7.65/2.38 thf(func_def_472, type, sP91: $i > $o).
% 7.65/2.38 thf(func_def_473, type, sP92: $i > $i > $o).
% 7.65/2.38 thf(func_def_474, type, sP93: $i > $i > $o).
% 7.65/2.38 thf(func_def_475, type, sP94: $i > $i > $o).
% 7.65/2.38 thf(func_def_476, type, sP95: $i > $o > $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_477, type, sP96: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_478, type, sP97: $i > $o).
% 7.65/2.38 thf(func_def_479, type, sP98: $i > $o).
% 7.65/2.38 thf(func_def_480, type, sP99: $o > $i > $i > $o).
% 7.65/2.38 thf(func_def_481, type, sP100: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_483, type, sP102: $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_493, type, sK112: $i > $i > $i).
% 7.65/2.38 thf(func_def_522, type, sK141: $i > $o).
% 7.65/2.38 thf(func_def_530, type, sK149: $i > $o).
% 7.65/2.38 thf(func_def_537, type, sK156: $i > $i > $i).
% 7.65/2.38 thf(func_def_538, type, sK157: $i > $i).
% 7.65/2.38 thf(func_def_541, type, sK160: $i > $o).
% 7.65/2.38 thf(func_def_543, type, sK162: $i > $i > $o).
% 7.65/2.38 thf(func_def_551, type, sK170: $i > $o).
% 7.65/2.38 thf(func_def_563, type, sK182: $i > $o).
% 7.65/2.38 thf(func_def_566, type, sK185: $i > $o).
% 7.65/2.38 thf(func_def_569, type, sK188: $i > $o).
% 7.65/2.38 thf(func_def_573, type, sK192: $i > $i > $o).
% 7.65/2.38 thf(func_def_580, type, sK199: $i > $i).
% 7.65/2.38 thf(func_def_581, type, sK200: ($i > $i) > $i > $i > $i).
% 7.65/2.38 thf(func_def_582, type, sK201: $i > ($i > $i) > $i > $i).
% 7.65/2.38 thf(func_def_592, type, sK211: ($i > $o) > $i).
% 7.65/2.38 thf(func_def_593, type, sK212: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_594, type, sK213: $i > $o).
% 7.65/2.38 thf(func_def_598, type, sK217: $i > $i > $o).
% 7.65/2.38 thf(func_def_600, type, sK219: $i > ($i > $i > $o) > $i).
% 7.65/2.38 thf(func_def_601, type, sK220: $i > $i > ($i > $i > $o) > $i).
% 7.65/2.38 thf(func_def_602, type, sK221: $i > ($i > $i > $o) > $i).
% 7.65/2.38 thf(func_def_603, type, sK222: $i > ($i > $i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_608, type, sK227: $i > $o).
% 7.65/2.38 thf(func_def_610, type, sK229: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_611, type, sK230: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_617, type, sK236: $i > $o).
% 7.65/2.38 thf(func_def_619, type, sK238: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_620, type, sK239: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_626, type, sK245: ($i > $o) > ($i > $o) > $i > $i > $i).
% 7.65/2.38 thf(func_def_627, type, sK246: ($i > $o) > ($i > $o) > $i > $i > $i).
% 7.65/2.38 thf(func_def_630, type, sK249: $i > $o).
% 7.65/2.38 thf(func_def_631, type, sK250: $i > $o).
% 7.65/2.38 thf(func_def_642, type, sK261: ($i > $o) > $i).
% 7.65/2.38 thf(func_def_643, type, sK262: ($i > $o) > $i).
% 7.65/2.38 thf(func_def_644, type, sK263: $i > $o).
% 7.65/2.38 thf(func_def_646, type, sK265: $i > $i > ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_647, type, sK266: $i > $i > ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_648, type, sK267: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_652, type, sK271: $i > $o).
% 7.65/2.38 thf(func_def_662, type, sK281: ($i > $o) > $i).
% 7.65/2.38 thf(func_def_663, type, sK282: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_664, type, sK283: $i > $o).
% 7.65/2.38 thf(func_def_668, type, sK287: $i > $o).
% 7.65/2.38 thf(func_def_675, type, sK294: $i > $i).
% 7.65/2.38 thf(func_def_676, type, sK295: $i > $i).
% 7.65/2.38 thf(func_def_687, type, sK306: $i > $o).
% 7.65/2.38 thf(func_def_688, type, sK307: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_691, type, sK310: $i > $i > $o).
% 7.65/2.38 thf(func_def_699, type, sK318: $i > $i > $i).
% 7.65/2.38 thf(func_def_701, type, sK320: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_702, type, sK321: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_706, type, sK325: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_707, type, sK326: ($i > $o) > $i).
% 7.65/2.38 thf(func_def_708, type, sK327: $i > $o).
% 7.65/2.38 thf(func_def_721, type, sK340: $i > $o).
% 7.65/2.38 thf(func_def_723, type, sK342: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_724, type, sK343: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_760, type, sK379: $i > $i).
% 7.65/2.38 thf(func_def_766, type, sK385: $i > $i).
% 7.65/2.38 thf(func_def_767, type, sK386: ($i > $i) > $i > $i > $i).
% 7.65/2.38 thf(func_def_778, type, sK397: $i > $i > $i).
% 7.65/2.38 thf(func_def_779, type, sK398: $i > $i > $i).
% 7.65/2.38 thf(func_def_785, type, sK404: $i > $i > $i).
% 7.65/2.38 thf(func_def_795, type, sK414: $i > $i > $o).
% 7.65/2.38 thf(func_def_797, type, sK416: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_817, type, sK436: $i > $o).
% 7.65/2.38 thf(func_def_818, type, sK437: $i > $o).
% 7.65/2.38 thf(func_def_819, type, sK438: ($i > $o) > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_820, type, sK439: ($i > $o) > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_838, type, sK457: $i > $i).
% 7.65/2.38 thf(func_def_840, type, sK459: $i > ($i > $i) > $i > $i).
% 7.65/2.38 thf(func_def_860, type, sK479: $i > $i > $i).
% 7.65/2.38 thf(func_def_868, type, sK487: $i > $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_869, type, sK488: $i > $o).
% 7.65/2.38 thf(func_def_872, type, sK491: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_873, type, sK492: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_874, type, sK493: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_875, type, sK494: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_880, type, sK499: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_881, type, sK500: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_888, type, sK507: $i > $i > $i).
% 7.65/2.38 thf(func_def_889, type, sK508: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_890, type, sK509: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_891, type, sK510: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_892, type, sK511: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_897, type, sK516: $i > $i).
% 7.65/2.38 thf(func_def_899, type, sK518: $i > ($i > $i) > $i > $i).
% 7.65/2.38 thf(func_def_957, type, sK576: $i > $o).
% 7.65/2.38 thf(func_def_958, type, sK577: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_959, type, sK578: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_969, type, sK588: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_970, type, sK589: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_976, type, sK595: $i > $i > $i).
% 7.65/2.38 thf(func_def_985, type, sK604: $i > $o).
% 7.65/2.38 thf(func_def_995, type, sK614: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_996, type, sK615: ($i > $o) > $i).
% 7.65/2.38 thf(func_def_997, type, sK616: $i > $o).
% 7.65/2.38 thf(func_def_1006, type, sK625: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1007, type, sK626: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1011, type, sK630: $i > $i).
% 7.65/2.38 thf(func_def_1014, type, sK633: $i > $i > ($i > $i) > $i).
% 7.65/2.38 thf(func_def_1018, type, sK637: $i > $i > $o).
% 7.65/2.38 thf(func_def_1052, type, sK671: $i > $o).
% 7.65/2.38 thf(func_def_1053, type, sK672: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1054, type, sK673: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1076, type, sK695: $i > $i).
% 7.65/2.38 thf(func_def_1079, type, sK698: $i > $i > ($i > $i) > $i).
% 7.65/2.38 thf(func_def_1080, type, sK699: $i > $i > ($i > $i) > $i).
% 7.65/2.38 thf(func_def_1086, type, sK705: $i > $i > $o).
% 7.65/2.38 thf(func_def_1101, type, sK720: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1102, type, sK721: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1103, type, sK722: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1104, type, sK723: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1112, type, sK731: $i > $o).
% 7.65/2.38 thf(func_def_1115, type, sK734: $i > ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1116, type, sK735: $i > ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1117, type, sK736: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1120, type, sK739: $i > $o).
% 7.65/2.38 thf(func_def_1130, type, sK749: $i > $i > $o).
% 7.65/2.38 thf(func_def_1140, type, sK759: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1141, type, sK760: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1142, type, sK761: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1144, type, sK763: $i > $o).
% 7.65/2.38 thf(func_def_1149, type, sK768: $i > $o).
% 7.65/2.38 thf(func_def_1150, type, sK769: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1151, type, sK770: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1167, type, sK786: $i > $i > $o).
% 7.65/2.38 thf(func_def_1176, type, sK795: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1177, type, sK796: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1178, type, sK797: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1179, type, sK798: $i > $o).
% 7.65/2.38 thf(func_def_1181, type, sK800: $i > $o).
% 7.65/2.38 thf(func_def_1187, type, sK806: $i > $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_1195, type, sK814: $i > $o).
% 7.65/2.38 thf(func_def_1196, type, sK815: $i > $o).
% 7.65/2.38 thf(func_def_1197, type, sK816: ($i > $o) > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1198, type, sK817: ($i > $o) > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1199, type, sK818: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1234, type, sK853: $i > ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1235, type, sK854: $i > ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1236, type, sK855: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1239, type, sK858: $i > $o).
% 7.65/2.38 thf(func_def_1246, type, sK865: $i > $i > ($i > $i > $o) > $i).
% 7.65/2.38 thf(func_def_1247, type, sK866: $i > ($i > $i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1248, type, sK867: ($i > $i > $o) > $i > $i > $i).
% 7.65/2.38 thf(func_def_1249, type, sK868: ($i > $i > $o) > $i > $i > $i).
% 7.65/2.38 thf(func_def_1250, type, sK869: $i > $i > $o).
% 7.65/2.38 thf(func_def_1274, type, sK893: $i > $i).
% 7.65/2.38 thf(func_def_1297, type, sK916: $i > $i > $i).
% 7.65/2.38 thf(func_def_1303, type, sK922: $i > $i > $i).
% 7.65/2.38 thf(func_def_1305, type, sK924: $i > $o).
% 7.65/2.38 thf(func_def_1306, type, sK925: ($i > $o) > $i > $i).
% 7.65/2.38 thf(func_def_1307, type, sK926: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1312, type, sK931: $i > $i > $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1313, type, sK932: $i > $i > $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1314, type, sK933: $i > ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1318, type, sK937: $i > $o).
% 7.65/2.38 thf(func_def_1322, type, sK941: $i > $i).
% 7.65/2.38 thf(func_def_1323, type, sK942: $i > $i > $i > $o).
% 7.65/2.38 thf(func_def_1324, type, sK943: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1353, type, sK972: $i > $i).
% 7.65/2.38 thf(func_def_1359, type, sK978: $i > $i).
% 7.65/2.38 thf(func_def_1360, type, sK979: $i > $i).
% 7.65/2.38 thf(func_def_1361, type, sK980: ($i > $i) > $i > $i).
% 7.65/2.38 thf(func_def_1362, type, sK981: ($i > $i) > $i > $i > $i).
% 7.65/2.38 thf(func_def_1385, type, sK1004: $i > $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1392, type, sK1011: $i > $i).
% 7.65/2.38 thf(func_def_1394, type, sK1013: ($i > $i) > $i > $i > $i).
% 7.65/2.38 thf(func_def_1395, type, sK1014: $i > $i > $i).
% 7.65/2.38 thf(func_def_1396, type, sK1015: $i > $i > $i).
% 7.65/2.38 thf(func_def_1397, type, sK1016: $i > $i > $i).
% 7.65/2.38 thf(func_def_1398, type, sK1017: $i > $i > $i).
% 7.65/2.38 thf(func_def_1399, type, sK1018: $i > $i > $i).
% 7.65/2.38 thf(func_def_1400, type, sK1019: $i > $i > $i).
% 7.65/2.38 thf(func_def_1401, type, sK1020: $i > $i > $i).
% 7.65/2.38 thf(func_def_1402, type, sK1021: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1403, type, sK1022: $i > $i).
% 7.65/2.38 thf(func_def_1404, type, sK1023: $i > $i).
% 7.65/2.38 thf(func_def_1405, type, sK1024: $i > $i).
% 7.65/2.38 thf(func_def_1406, type, sK1025: $i > $i).
% 7.65/2.38 thf(func_def_1407, type, sK1026: $i > $i > $i).
% 7.65/2.38 thf(func_def_1408, type, sK1027: $i > $i > $i).
% 7.65/2.38 thf(func_def_1409, type, sK1028: $i > $i > $i).
% 7.65/2.38 thf(func_def_1410, type, sK1029: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1411, type, sK1030: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1413, type, sK1032: $i > $i).
% 7.65/2.38 thf(func_def_1420, type, sK1039: $i > $o > $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1438, type, sK1057: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1445, type, sK1064: $i > $i).
% 7.65/2.38 thf(func_def_1446, type, sK1065: $i > $i).
% 7.65/2.38 thf(func_def_1453, type, sK1072: $o > $i > $i > $i).
% 7.65/2.38 thf(func_def_1457, type, sK1076: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1458, type, sK1077: $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1462, type, sK1081: $i > $o).
% 7.65/2.38 thf(func_def_1486, type, sK1105: $i > $i).
% 7.65/2.38 thf(func_def_1488, type, sK1107: $i > ($i > $i) > $i > $i).
% 7.65/2.38 thf(func_def_1489, type, sK1108: ($i > $o) > $i).
% 7.65/2.38 thf(func_def_1490, type, sK1109: $i > $o).
% 7.65/2.38 thf(func_def_1508, type, sK1127: $i > $i > $i > $i > $i).
% 7.65/2.38 thf(func_def_1529, type, ph1148: !>[X0: $tType]:(X0)).
% 7.65/2.38 thf(f9344,plain,(
% 7.65/2.38 $false),
% 7.65/2.38 inference(subsumption_resolution,[status(thm)],[f9333,f4563])).
% 7.65/2.38 thf(f4563,plain,(
% 7.65/2.38 ($true != (in @ sK702 @ (funcSet @ sK700 @ sK701)))),
% 7.65/2.38 inference(cnf_transformation,[status(thm)],[f2603])).
% 7.65/2.38 thf(f2603,plain,(
% 7.65/2.38 (setukpairinjL1 = $true) & (vacuousDall = $true) & (dsetconstrER = $true) & (demorgan2b = $true) & (funcext = $true) & (breln1invprop = $true) & (symdiffIneg2 = $true) & (setminusLsub = $true) & (setadjoinIL = $true) & (nonemptyI = $true) & (cartprodpairmemEL = $true) & (exuE1 = $true) & (setminusSubset2 = $true) & (binintersectER = $true) & (breln1unionCommutes = $true) & (cartprodmempaircEq = $true) & (demorgan2b2 = $true) & (contrasubsetT1 = $true) & (image1Equiv = $true) & (doubleComplementE1 = $true) & (lamp = $true) & (wellorderingAx = $true) & (symdiffE = $true) & (setbeta = $true) & (breln1unionE = $true) & (binintersectSubset3 = $true) & (notequalI1 = $true) & (setukpairinjR = $true) & (complementTnotintersectT = $true) & (singletonsuniq = $true) & (breln1unionI = $true) & (funcext2 = $true) & (brelnall2 = $true) & (kfstpairEq = $true) & (setunion__Cong = $true) & (apProp = $true) & (infuncsetfunc = $true) & (($true != (in @ sK702 @ (funcSet @ sK700 @ sK701))) & ($true = (in @ sK702 @ (injFuncSet @ sK700 @ sK701)))) & (quantDeMorgan2 = $true) & (omegaIndAx = $true) & (setoftrueEq = $true) & (singletonsubset = $true) & (setunionE = $true) & (emptysetsubset = $true) & (subbreln1 = $true) & (powersetI = $true) & (contrasubsetT = $true) & (setOfPairsIsBReln = $true) & (dpsetconstrI = $true) & (contrasubsetT3 = $true) & (emptyinPowerset = $true) & (subPowSU = $true) & (breln1unionIL = $true) & (in__Cong = $true) & (binintersectLsub = $true) & (emptyInPowerset = $true) & (setextAx = $true) & (singletonsswitch = $true) & (exuE3e = $true) & (woz2B = $true) & (complementImpComplementIntersect = $true) & (iffalse = $true) & (beta2 = $true) & (breln1invE = $true) & (ifSingleton = $true) & (setadjoinE = $true) & (funcGraphProp4 = $true) & (intersectInPowersetIntersectUnions = $true) & (symdiffIneg1 = $true) & (iftrueProp1 = $true) & (upairsubunion = $true) & (exuI3 = $true) & (setminusER = $true) & (emptysetAx = $true) & (ifp = $true) & (setukpairIL = $true) & (bs114d = $true) & (omega0Ax = $true) & (nonemptyI1 = $true) & (demorgan2a1 = $true) & (lam2lamEq = $true) & (powersetT_lem = $true) & (complementT_lem = $true) & (kpairp = $true) & (exuE3u = $true) & (notinsingleton = $true) & (eqimpsubset1 = $true) & (breln1all1 = $true) & (woz13rule4 = $true) & (kpairiskpair = $true) & (ex1I = $true) & (setukpairinjR12 = $true) & (setukpairinjR2 = $true) & (exuI1 = $true) & (binunionTIRcontra = $true) & (upairset2IR = $true) & (lam2p = $true) & (binunionTE = $true) & (demorgan1a = $true) & (demorgan2a2 = $true) & (setminusSubset1 = $true) & (setextsub = $true) & (dsetconstrEL = $true) & (setOfPairsIsBReln1 = $true) & (setunionsingleton1 = $true) & (binunionRsub = $true) & (binunionIR = $true) & (dsetconstrI = $true) & (prop2setE = $true) & (demorgan2a = $true) & (powerset__Cong = $true) & (exuEu = $true) & (choice2fnsingleton = $true) & (binintersectEL = $true) & (breln1compI = $true) & (symdiffI1 = $true) & (secondinupair = $true) & (woz1_5 = $true) & (complementTI1 = $true) & (cartprodfstin = $true) & (upairsetE = $true) & (inIntersectImpInUnion2 = $true) & (descrp = $true) & (setminusELneg = $true) & (doubleComplementEq = $true) & (dpsetconstrEL1 = $true) & (eta1 = $true) & (eta2 = $true) & (eqimpsubset2 = $true) & (quantDeMorgan3 = $true) & (funcImageSingleton = $true) & (subsetemptysetimpeq = $true) & (nonemptyE1 = $true) & (prop2setI = $true) & (binintersectSubset4 = $true) & (woz1_2 = $true) & (binunionEcases = $true) & (ksndpairEq = $true) & (subsetTrans = $true) & (cartprodmempair = $true) & (woz13rule0 = $true) & (setadjoinSub2 = $true) & (complementInPowersetComplementIntersect = $true) & (setminusERneg = $true) & (singletoninpowunion = $true) & (inComplementUnionImpInComplement1 = $true) & (sepInPowerset = $true) & (funcGraphProp3 = $true) & (omegaSAx = $true) & (prop2set2propI = $true) & (disjointsetsI1 = $true) & (ubforcartprodlem3 = $true) & (app = $true) & (powersetsubset = $true) & (inIntersectImpInIntersectUnions = $true) & (ex1E2 = $true) & (emptysetimpfalse = $true) & (setukpairinjL = $true) & (kpairsurjEq = $true) & (dpsetconstrSub = $true) & (binintersectRsub = $true) & (woz13rule1 = $true) & (binunionLsub = $true) & (binunionE = $true) & (breln1compprop = $true) & (woz2Ex = $true) & (demorgan2 = $true) & (subsetI1 = $true) & (cartprodfstpairEq = $true) & (setextT = $true) & (quantDeMorgan4 = $true) & (contraSubsetComplement = $true) & (demorgan1b = $true) & (doubleComplementSub2 = $true) & (demorgan1 = $true) & (setukpairinjL2 = $true) & (setunionsingleton2 = $true) & (quantDeMorgan1 = $true) & (dsetconstr__Cong = $true) & (iftrue = $true) & (binunionTEcontra = $true) & (setunionE2 = $true) & (binintersectTERcontra = $true) & (upairsetIR = $true) & (ex1I2 = $true) & (complementTcontraSubset = $true) & (ex1E1 = $true) & (cartprodpairsurjEq = $true) & (binintersectSubset5 = $true) & (image1I = $true) & (cartprodpairin = $true) & (upairset2E = $true) & (ksndsingleton = $true) & (upairinpowunion = $true) & (binintersectT_lem = $true) & (dpsetconstrEL2 = $true) & (setminusT_lem = $true) & (woz1_1 = $true) & (breln1unionEcases = $true) & (iffalseProp2 = $true) & (sepSubset = $true) & (singletoninpowerset = $true) & (notinemptyset = $true) & (subsetI2 = $true) & (upairsetIL = $true) & (descr__Cong = $true) & (binintersectTELcontra = $true) & (subsetE = $true) & (breln1SetBreln1 = $true) & (binintersectSubset2 = $true) & (emptyset__Cong = $true) & (binunionTILcontra = $true) & (setminusIRneg = $true) & (setunionAx = $true) & (setadjoinSub = $true) & (eqbreln1 = $true) & (eqbreln = $true) & (image1Ex1 = $true) & (setadjoinAx = $true) & (ubforcartprodlem1 = $true) & (setminusEL = $true) & (subsetRefl = $true) & (theeq = $true) & (notdexE = $true) & (woz1_3 = $true) & (inComplementUnionImpNotIn1 = $true) & (powersetTE1 = $true) & (image1Ex = $true) & (subsetTI = $true) & (inCongP = $true) & (funcGraphProp1 = $true) & (funcGraphProp2 = $true) & (cartprodpairmemER = $true) & (omega__Cong = $true) & (setunionI = $true) & (complementUnionInPowersetComplement = $true) & (inIntersectImpInUnion = $true) & (brelnall1 = $true) & (woz2A = $true) & (uniqinunit = $true) & (contrasubsetT2 = $true) & (noeltsimpempty = $true) & (subbreln = $true) & (cartprodmempair1 = $true) & (setext = $true) & (singletonprop = $true) & (powersetE1 = $true) & (breln1unionprop = $true) & (breln1unionIR = $true) & (binunionIL = $true) & (cartprodsndpairEq = $true) & (exuE2 = $true) & (woz13rule2 = $true) & (emptyE1 = $true) & (theprop = $true) & (setukpairinjR11 = $true) & (emptysetE = $true) & (setukpairIR = $true) & (emptyinunitempty = $true) & (notdallE = $true) & (ap2apEq1 = $true) & (exu__Cong = $true) & (setunionsingleton = $true) & (emptyI = $true) & (subsetE2 = $true) & (setminusILneg = $true) & (ap2p = $true) & (subset2powerset = $true) & (ap2apEq2 = $true) & (setukpairinjR1 = $true) & (powersetE = $true) & (image1E = $true) & (powersetAx = $true) & (binintersectI = $true) & (eqinunit = $true) & (iftrueProp2 = $true) & (dpsetconstrER = $true) & (iftrueorfalse = $true) & (dpsetconstrERa = $true) & (ubforcartprodlem2 = $true) & (lamProp = $true) & (breln1all2 = $true) & (upairequniteq = $true) & (complementSubsetComplementIntersect = $true) & (woz1_4 = $true) & (powersetI1 = $true) & (funcextLem = $true) & (symdiffI2 = $true) & (inPowerset = $true) & (woz13rule3 = $true) & (replAx = $true) & (kfstsingleton = $true) & (setadjoin__Cong = $true) & (funcinfuncset = $true) & (setadjoinIR = $true) & (breln1compEex = $true) & (injFuncInInjFuncSet = $true) & (setminusI = $true) & (beta1 = $true) & (breln1invI = $true) & (notequalI2 = $true) & (powersetTI1 = $true) & (nonemptyImpWitness = $true) & (breln1compE = $true) & (setadjoinOr = $true) & (iffalseProp1 = $true) & (exuI2 = $true) & (doubleComplementSub1 = $true) & (notsubsetI = $true) & (complementTE1 = $true) & (binintersectSubset1 = $true) & (foundationAx = $true) & (woz2W = $true) & (doubleComplementI1 = $true) & (binunionT_lem = $true) & (cartprodsndin = $true)),
% 7.65/2.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK700,sK701,sK702])],[f1388,f2602])).
% 7.65/2.38 thf(f2602,plain,(
% 7.65/2.38 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ (funcSet @ X0 @ X1))) & ($true = (in @ X2 @ (injFuncSet @ X0 @ X1)))) => (($true != (in @ sK702 @ (funcSet @ sK700 @ sK701))) & ($true = (in @ sK702 @ (injFuncSet @ sK700 @ sK701))))),
% 7.65/2.38 introduced(definition,[],[choice_axiom])).
% 7.65/2.38 thf(f1388,plain,(
% 7.65/2.38 (setukpairinjL1 = $true) & (vacuousDall = $true) & (dsetconstrER = $true) & (demorgan2b = $true) & (funcext = $true) & (breln1invprop = $true) & (symdiffIneg2 = $true) & (setminusLsub = $true) & (setadjoinIL = $true) & (nonemptyI = $true) & (cartprodpairmemEL = $true) & (exuE1 = $true) & (setminusSubset2 = $true) & (binintersectER = $true) & (breln1unionCommutes = $true) & (cartprodmempaircEq = $true) & (demorgan2b2 = $true) & (contrasubsetT1 = $true) & (image1Equiv = $true) & (doubleComplementE1 = $true) & (lamp = $true) & (wellorderingAx = $true) & (symdiffE = $true) & (setbeta = $true) & (breln1unionE = $true) & (binintersectSubset3 = $true) & (notequalI1 = $true) & (setukpairinjR = $true) & (complementTnotintersectT = $true) & (singletonsuniq = $true) & (breln1unionI = $true) & (funcext2 = $true) & (brelnall2 = $true) & (kfstpairEq = $true) & (setunion__Cong = $true) & (apProp = $true) & (infuncsetfunc = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ (funcSet @ X0 @ X1))) & ($true = (in @ X2 @ (injFuncSet @ X0 @ X1)))) & (quantDeMorgan2 = $true) & (omegaIndAx = $true) & (setoftrueEq = $true) & (singletonsubset = $true) & (setunionE = $true) & (emptysetsubset = $true) & (subbreln1 = $true) & (powersetI = $true) & (contrasubsetT = $true) & (setOfPairsIsBReln = $true) & (dpsetconstrI = $true) & (contrasubsetT3 = $true) & (emptyinPowerset = $true) & (subPowSU = $true) & (breln1unionIL = $true) & (in__Cong = $true) & (binintersectLsub = $true) & (emptyInPowerset = $true) & (setextAx = $true) & (singletonsswitch = $true) & (exuE3e = $true) & (woz2B = $true) & (complementImpComplementIntersect = $true) & (iffalse = $true) & (beta2 = $true) & (breln1invE = $true) & (ifSingleton = $true) & (setadjoinE = $true) & (funcGraphProp4 = $true) & (intersectInPowersetIntersectUnions = $true) & (symdiffIneg1 = $true) & (iftrueProp1 = $true) & (upairsubunion = $true) & (exuI3 = $true) & (setminusER = $true) & (emptysetAx = $true) & (ifp = $true) & (setukpairIL = $true) & (bs114d = $true) & (omega0Ax = $true) & (nonemptyI1 = $true) & (demorgan2a1 = $true) & (lam2lamEq = $true) & (powersetT_lem = $true) & (complementT_lem = $true) & (kpairp = $true) & (exuE3u = $true) & (notinsingleton = $true) & (eqimpsubset1 = $true) & (breln1all1 = $true) & (woz13rule4 = $true) & (kpairiskpair = $true) & (ex1I = $true) & (setukpairinjR12 = $true) & (setukpairinjR2 = $true) & (exuI1 = $true) & (binunionTIRcontra = $true) & (upairset2IR = $true) & (lam2p = $true) & (binunionTE = $true) & (demorgan1a = $true) & (demorgan2a2 = $true) & (setminusSubset1 = $true) & (setextsub = $true) & (dsetconstrEL = $true) & (setOfPairsIsBReln1 = $true) & (setunionsingleton1 = $true) & (binunionRsub = $true) & (binunionIR = $true) & (dsetconstrI = $true) & (prop2setE = $true) & (demorgan2a = $true) & (powerset__Cong = $true) & (exuEu = $true) & (choice2fnsingleton = $true) & (binintersectEL = $true) & (breln1compI = $true) & (symdiffI1 = $true) & (secondinupair = $true) & (woz1_5 = $true) & (complementTI1 = $true) & (cartprodfstin = $true) & (upairsetE = $true) & (inIntersectImpInUnion2 = $true) & (descrp = $true) & (setminusELneg = $true) & (doubleComplementEq = $true) & (dpsetconstrEL1 = $true) & (eta1 = $true) & (eta2 = $true) & (eqimpsubset2 = $true) & (quantDeMorgan3 = $true) & (funcImageSingleton = $true) & (subsetemptysetimpeq = $true) & (nonemptyE1 = $true) & (prop2setI = $true) & (binintersectSubset4 = $true) & (woz1_2 = $true) & (binunionEcases = $true) & (ksndpairEq = $true) & (subsetTrans = $true) & (cartprodmempair = $true) & (woz13rule0 = $true) & (setadjoinSub2 = $true) & (complementInPowersetComplementIntersect = $true) & (setminusERneg = $true) & (singletoninpowunion = $true) & (inComplementUnionImpInComplement1 = $true) & (sepInPowerset = $true) & (funcGraphProp3 = $true) & (omegaSAx = $true) & (prop2set2propI = $true) & (disjointsetsI1 = $true) & (ubforcartprodlem3 = $true) & (app = $true) & (powersetsubset = $true) & (inIntersectImpInIntersectUnions = $true) & (ex1E2 = $true) & (emptysetimpfalse = $true) & (setukpairinjL = $true) & (kpairsurjEq = $true) & (dpsetconstrSub = $true) & (binintersectRsub = $true) & (woz13rule1 = $true) & (binunionLsub = $true) & (binunionE = $true) & (breln1compprop = $true) & (woz2Ex = $true) & (demorgan2 = $true) & (subsetI1 = $true) & (cartprodfstpairEq = $true) & (setextT = $true) & (quantDeMorgan4 = $true) & (contraSubsetComplement = $true) & (demorgan1b = $true) & (doubleComplementSub2 = $true) & (demorgan1 = $true) & (setukpairinjL2 = $true) & (setunionsingleton2 = $true) & (quantDeMorgan1 = $true) & (dsetconstr__Cong = $true) & (iftrue = $true) & (binunionTEcontra = $true) & (setunionE2 = $true) & (binintersectTERcontra = $true) & (upairsetIR = $true) & (ex1I2 = $true) & (complementTcontraSubset = $true) & (ex1E1 = $true) & (cartprodpairsurjEq = $true) & (binintersectSubset5 = $true) & (image1I = $true) & (cartprodpairin = $true) & (upairset2E = $true) & (ksndsingleton = $true) & (upairinpowunion = $true) & (binintersectT_lem = $true) & (dpsetconstrEL2 = $true) & (setminusT_lem = $true) & (woz1_1 = $true) & (breln1unionEcases = $true) & (iffalseProp2 = $true) & (sepSubset = $true) & (singletoninpowerset = $true) & (notinemptyset = $true) & (subsetI2 = $true) & (upairsetIL = $true) & (descr__Cong = $true) & (binintersectTELcontra = $true) & (subsetE = $true) & (breln1SetBreln1 = $true) & (binintersectSubset2 = $true) & (emptyset__Cong = $true) & (binunionTILcontra = $true) & (setminusIRneg = $true) & (setunionAx = $true) & (setadjoinSub = $true) & (eqbreln1 = $true) & (eqbreln = $true) & (image1Ex1 = $true) & (setadjoinAx = $true) & (ubforcartprodlem1 = $true) & (setminusEL = $true) & (subsetRefl = $true) & (theeq = $true) & (notdexE = $true) & (woz1_3 = $true) & (inComplementUnionImpNotIn1 = $true) & (powersetTE1 = $true) & (image1Ex = $true) & (subsetTI = $true) & (inCongP = $true) & (funcGraphProp1 = $true) & (funcGraphProp2 = $true) & (cartprodpairmemER = $true) & (omega__Cong = $true) & (setunionI = $true) & (complementUnionInPowersetComplement = $true) & (inIntersectImpInUnion = $true) & (brelnall1 = $true) & (woz2A = $true) & (uniqinunit = $true) & (contrasubsetT2 = $true) & (noeltsimpempty = $true) & (subbreln = $true) & (cartprodmempair1 = $true) & (setext = $true) & (singletonprop = $true) & (powersetE1 = $true) & (breln1unionprop = $true) & (breln1unionIR = $true) & (binunionIL = $true) & (cartprodsndpairEq = $true) & (exuE2 = $true) & (woz13rule2 = $true) & (emptyE1 = $true) & (theprop = $true) & (setukpairinjR11 = $true) & (emptysetE = $true) & (setukpairIR = $true) & (emptyinunitempty = $true) & (notdallE = $true) & (ap2apEq1 = $true) & (exu__Cong = $true) & (setunionsingleton = $true) & (emptyI = $true) & (subsetE2 = $true) & (setminusILneg = $true) & (ap2p = $true) & (subset2powerset = $true) & (ap2apEq2 = $true) & (setukpairinjR1 = $true) & (powersetE = $true) & (image1E = $true) & (powersetAx = $true) & (binintersectI = $true) & (eqinunit = $true) & (iftrueProp2 = $true) & (dpsetconstrER = $true) & (iftrueorfalse = $true) & (dpsetconstrERa = $true) & (ubforcartprodlem2 = $true) & (lamProp = $true) & (breln1all2 = $true) & (upairequniteq = $true) & (complementSubsetComplementIntersect = $true) & (woz1_4 = $true) & (powersetI1 = $true) & (funcextLem = $true) & (symdiffI2 = $true) & (inPowerset = $true) & (woz13rule3 = $true) & (replAx = $true) & (kfstsingleton = $true) & (setadjoin__Cong = $true) & (funcinfuncset = $true) & (setadjoinIR = $true) & (breln1compEex = $true) & (injFuncInInjFuncSet = $true) & (setminusI = $true) & (beta1 = $true) & (breln1invI = $true) & (notequalI2 = $true) & (powersetTI1 = $true) & (nonemptyImpWitness = $true) & (breln1compE = $true) & (setadjoinOr = $true) & (iffalseProp1 = $true) & (exuI2 = $true) & (doubleComplementSub1 = $true) & (notsubsetI = $true) & (complementTE1 = $true) & (binintersectSubset1 = $true) & (foundationAx = $true) & (woz2W = $true) & (doubleComplementI1 = $true) & (binunionT_lem = $true) & (cartprodsndin = $true)),
% 7.65/2.38 inference(flattening,[status(thm)],[f1387])).
% 7.65/2.38 thf(f1387,plain,(
% 7.65/2.38 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ (funcSet @ X0 @ X1))) & ($true = (in @ X2 @ (injFuncSet @ X0 @ X1)))) & (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)),
% 7.65/2.38 inference(ennf_transformation,[status(thm)],[f365])).
% 7.65/2.38 thf(f365,plain,(
% 7.65/2.38 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (injFuncSet @ X0 @ X1))) => ($true = (in @ X2 @ (funcSet @ X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.65/2.38 inference(fool_elimination,[status(thm)],[f364])).
% 7.65/2.38 thf(f364,plain,(
% 7.65/2.38 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (injFuncSet @ X0 @ X1)) => (in @ X2 @ (funcSet @ X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.65/2.38 inference(rectify,[status(thm)],[f320])).
% 7.65/2.38 thf(f320,negated_conjecture,(
% 7.65/2.38 ~(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 => ! [X3 : $i,X4 : $i,X12 : $i] : ((in @ X12 @ (injFuncSet @ X3 @ X4)) => (in @ X12 @ (funcSet @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.65/2.38 inference(negated_conjecture,[status(cth)],[f319])).
% 7.65/2.38 thf(f319,conjecture,(
% 7.65/2.38 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 => ! [X3 : $i,X4 : $i,X12 : $i] : ((in @ X12 @ (injFuncSet @ X3 @ X4)) => (in @ X12 @ (funcSet @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.65/2.38 file('/export/starexec/sandbox/tmp/tmp.an1PBkiMQH/DTF2THF_18125.p',injFuncSetFuncIn)).
% 7.65/2.38 thf(f9333,plain,(
% 7.65/2.38 ($true = (in @ sK702 @ (funcSet @ sK700 @ sK701)))),
% 7.65/2.38 inference(trivial_inequality_removal,[status(thm)],[f9321])).
% 7.65/2.38 thf(f9321,plain,(
% 7.65/2.38 ($true != $true) | ($true = (in @ sK702 @ (funcSet @ sK700 @ sK701)))),
% 7.65/2.38 inference(superposition,[status(thm)],[f6742,f6541])).
% 7.65/2.38 thf(f6541,plain,(
% 7.65/2.38 ($true = (in @ sK702 @ (dsetconstr @ (funcSet @ sK700 @ sK701) @ (injective @ sK700 @ sK701))))),
% 7.65/2.38 inference(beta_eta_normalization,[status(thm)],[f5964])).
% 7.65/2.38 thf(f5964,plain,(
% 7.65/2.38 ($true = (in @ sK702 @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))) @ sK700 @ sK701)))),
% 7.65/2.38 inference(definition_unfolding,[status(thm)],[f4562,f3862])).
% 7.65/2.38 thf(f3862,plain,(
% 7.65/2.38 (injFuncSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))))),
% 7.65/2.38 inference(cnf_transformation,[status(thm)],[f618])).
% 7.65/2.38 thf(f618,plain,(
% 7.65/2.38 (injFuncSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))))),
% 7.65/2.38 inference(fool_elimination,[status(thm)],[f617])).
% 7.65/2.38 thf(f617,plain,(
% 7.65/2.38 (injFuncSet = (^[X0 : $i, X1 : $i] : (dsetconstr @ (funcSet @ X0 @ X1) @ (^[X2 : $i] : (injective @ X0 @ X1 @ X2)))))),
% 7.65/2.38 inference(rectify,[status(thm)],[f317])).
% 7.65/2.38 thf(f317,axiom,(
% 7.65/2.38 (injFuncSet = (^[X3 : $i, X4 : $i] : (dsetconstr @ (funcSet @ X3 @ X4) @ (^[X12 : $i] : (injective @ X3 @ X4 @ X12)))))),
% 7.65/2.38 file('/export/starexec/sandbox/tmp/tmp.an1PBkiMQH/DTF2THF_18125.p',injFuncSet)).
% 7.65/2.38 thf(f4562,plain,(
% 7.65/2.38 ($true = (in @ sK702 @ (injFuncSet @ sK700 @ sK701)))),
% 7.65/2.38 inference(cnf_transformation,[status(thm)],[f2603])).
% 7.65/2.38 thf(f6742,plain,(
% 7.65/2.38 ( ! [X2 : $i > $o,X0 : $i,X1 : $i] : (($true != (in @ X1 @ (dsetconstr @ X0 @ X2))) | ($true = (in @ X1 @ X0))) )),
% 7.65/2.38 inference(beta_eta_normalization,[status(thm)],[f6741])).
% 7.65/2.38 thf(f6741,plain,(
% 7.65/2.38 ( ! [X2 : $i > $o,X0 : $i,X1 : $i] : (($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | ($true = (in @ X1 @ X0))) )),
% 7.65/2.38 inference(trivial_inequality_removal,[status(thm)],[f5824])).
% 7.65/2.38 thf(f5824,plain,(
% 7.65/2.38 ( ! [X2 : $i > $o,X0 : $i,X1 : $i] : (($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | ($true = (in @ X1 @ X0)) | ($true != $true)) )),
% 7.65/2.38 inference(definition_unfolding,[status(thm)],[f4093,f4497])).
% 7.65/2.38 thf(f4497,plain,(
% 7.65/2.38 (dsetconstrEL = $true)),
% 7.65/2.38 inference(cnf_transformation,[status(thm)],[f2603])).
% 7.65/2.38 thf(f4093,plain,(
% 7.65/2.38 ( ! [X2 : $i > $o,X0 : $i,X1 : $i] : (($true = (in @ X1 @ X0)) | ($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | (dsetconstrEL != $true)) )),
% 7.65/2.38 inference(cnf_transformation,[status(thm)],[f2378])).
% 7.65/2.38 thf(f2378,plain,(
% 7.65/2.38 (! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true = (in @ X1 @ X0)) | ($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0)))))) | (dsetconstrEL != $true)) & ((dsetconstrEL = $true) | (($true != (in @ sK575 @ sK574)) & ($true = (in @ sK575 @ (dsetconstr @ sK574 @ (^[Y0 : $i]: (sK576 @ Y0)))))))),
% 7.65/2.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK574,sK575,sK576])],[f2376,f2377])).
% 7.65/2.38 thf(f2377,plain,(
% 7.65/2.38 ? [X3 : $i,X4 : $i,X5 : $i > $o] : (((in @ X4 @ X3) != $true) & ($true = (in @ X4 @ (dsetconstr @ X3 @ (^[Y0 : $i]: (X5 @ Y0)))))) => (($true != (in @ sK575 @ sK574)) & ($true = (in @ sK575 @ (dsetconstr @ sK574 @ (^[Y0 : $i]: (sK576 @ Y0))))))),
% 7.65/2.38 introduced(definition,[],[choice_axiom])).
% 7.65/2.38 thf(f2376,plain,(
% 7.65/2.38 (! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true = (in @ X1 @ X0)) | ($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0)))))) | (dsetconstrEL != $true)) & ((dsetconstrEL = $true) | ? [X3 : $i,X4 : $i,X5 : $i > $o] : (((in @ X4 @ X3) != $true) & ($true = (in @ X4 @ (dsetconstr @ X3 @ (^[Y0 : $i]: (X5 @ Y0)))))))),
% 7.65/2.38 inference(rectify,[status(thm)],[f2375])).
% 7.65/2.38 thf(f2375,plain,(
% 7.65/2.38 (! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true = (in @ X1 @ X0)) | ($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0)))))) | (dsetconstrEL != $true)) & ((dsetconstrEL = $true) | ? [X0 : $i,X1 : $i,X2 : $i > $o] : (($true != (in @ X1 @ X0)) & ($true = (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0)))))))),
% 7.65/2.38 inference(nnf_transformation,[status(thm)],[f1154])).
% 7.65/2.38 thf(f1154,plain,(
% 7.65/2.38 ! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true = (in @ X1 @ X0)) | ($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0)))))) <=> (dsetconstrEL = $true)),
% 7.65/2.38 inference(ennf_transformation,[status(thm)],[f682])).
% 7.65/2.38 thf(f682,plain,(
% 7.65/2.38 (dsetconstrEL = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true = (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) => ($true = (in @ X1 @ X0)))),
% 7.65/2.38 inference(fool_elimination,[status(thm)],[f681])).
% 7.65/2.38 thf(f681,plain,(
% 7.65/2.38 (! [X0 : $i,X1 : $i,X2 : $i > $o] : ((in @ X1 @ (dsetconstr @ X0 @ (^[X3 : $i] : (X2 @ X3)))) => (in @ X1 @ X0)) = dsetconstrEL)),
% 7.65/2.38 inference(rectify,[status(thm)],[f15])).
% 7.65/2.38 thf(f15,axiom,(
% 7.65/2.38 (! [X3 : $i,X1 : $i,X0 : $i > $o] : ((in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))) => (in @ X1 @ X3)) = dsetconstrEL)),
% 7.65/2.38 file('/export/starexec/sandbox/tmp/tmp.an1PBkiMQH/DTF2THF_18125.p',dsetconstrEL)).
% 7.65/2.38 % SZS output end Proof for DTF2THF_18125
% 7.65/2.38 % (18402)------------------------------
% 7.65/2.38 % (18402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.65/2.38 % (18402)Termination reason: Refutation
% 7.65/2.38
% 7.65/2.38 % (18402)Memory used [KB]: 13048
% 7.65/2.38 % (18402)Time elapsed: 0.430 s
% 7.65/2.38 % (18402)Instructions burned: 887 (million)
% 7.65/2.38 % (18402)------------------------------
% 7.65/2.38 % (18402)------------------------------
% 7.65/2.38 % (18316)Success in time 0.841 s
%------------------------------------------------------------------------------