%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU776^1 : 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 : n001.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 09:37:05 AM UTC 2026
% Result : Theorem 10.00s 2.98s
% Output : Refutation 10.00s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 8
% Syntax : Number of formulae : 41 ( 15 unt; 0 typ; 0 def)
% Number of atoms : 4956 (1315 equ; 0 cnn)
% Maximal formula atoms : 297 ( 120 avg)
% Number of connectives : 2816 ( 67 ~; 43 |; 928 &; 577 @)
% ( 3 <=>;1198 =>; 0 <=; 0 <~>)
% Maximal formula depth : 306 ( 56 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 30 ( 30 >; 0 *; 0 +; 0 <<)
% Number of symbols : 602 ( 599 usr; 302 con; 0-4 aty)
% Number of variables : 173 ( 62 ^; 76 !; 35 ?; 173 :)
% 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_354,type,
sP1: $i > $i > ( $i > $o ) > $o ).
thf(func_def_356,type,
sP3: ( $i > $o ) > $o ).
thf(func_def_357,type,
sP4: $i > $o ).
thf(func_def_358,type,
sP5: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_359,type,
sP6: ( $i > $i > $o ) > $i > $o ).
thf(func_def_361,type,
sP8: ( $i > $o ) > $i > $o ).
thf(func_def_363,type,
sP10: $i > $i > $i > $o ).
thf(func_def_364,type,
sP11: $i > $i > $i > $o ).
thf(func_def_365,type,
sP12: $i > $i > $i > $o ).
thf(func_def_366,type,
sP13: $i > $i > $o ).
thf(func_def_367,type,
sP14: $i > $i > $o ).
thf(func_def_368,type,
sP15: $i > $i > $o ).
thf(func_def_369,type,
sP16: $i > $i > $o ).
thf(func_def_370,type,
sP17: $i > $o ).
thf(func_def_371,type,
sP18: $i > $i > $o ).
thf(func_def_373,type,
sP20: $i > $i > $i > $o ).
thf(func_def_374,type,
sP21: $i > $o ).
thf(func_def_375,type,
sP22: $i > $o ).
thf(func_def_377,type,
sP24: ( $i > $o ) > $i > $o ).
thf(func_def_379,type,
sP26: $i > $i > $i > $i > $o ).
thf(func_def_380,type,
sP27: $i > $i > $i > $i > $o ).
thf(func_def_381,type,
sP28: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_382,type,
sP29: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_383,type,
sP30: $i > $i > $i > $o ).
thf(func_def_384,type,
sP31: $i > $i > ( $i > $i ) > $o ).
thf(func_def_386,type,
sP33: $i > $i > $i > $i > $o ).
thf(func_def_387,type,
sP34: $i > $i > $i > $o ).
thf(func_def_388,type,
sP35: $i > ( $i > $o ) > $o ).
thf(func_def_389,type,
sP36: $i > ( $i > $o ) > $i > $o ).
thf(func_def_390,type,
sP37: ( $i > $o ) > $o ).
thf(func_def_391,type,
sP38: $i > $i > ( $i > $i ) > $o ).
thf(func_def_393,type,
sP40: $i > $i > $o ).
thf(func_def_395,type,
sP42: $o > $i > $i > $i > $o ).
thf(func_def_396,type,
sP43: $i > $i > $i > $o > $o ).
thf(func_def_397,type,
sP44: $i > ( $i > $o ) > $o ).
thf(func_def_398,type,
sP45: $i > ( $i > $o ) > $o ).
thf(func_def_399,type,
sP46: $i > $i > $o > $o ).
thf(func_def_403,type,
sP50: $i > ( $i > $o ) > $o ).
thf(func_def_404,type,
sP51: $i > ( $i > $o ) > $i > $i > $o ).
thf(func_def_405,type,
sP52: $i > ( $i > $o ) > $o ).
thf(func_def_406,type,
sP53: $i > ( $i > $o ) > $i > $o ).
thf(func_def_408,type,
sP55: $i > ( $i > $o ) > $o ).
thf(func_def_410,type,
sP57: $i > ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_411,type,
sP58: $o > $i > $i > $o ).
thf(func_def_412,type,
sP59: $i > $i > $i > $o ).
thf(func_def_413,type,
sP60: $i > $o > $i > $o ).
thf(func_def_415,type,
sP62: $i > $i > $o ).
thf(func_def_417,type,
sP64: $i > $i > $i > $o ).
thf(func_def_418,type,
sP65: $i > $i > $i > $o ).
thf(func_def_419,type,
sP66: ( $i > $o ) > $o ).
thf(func_def_420,type,
sP67: ( $i > $o ) > $o ).
thf(func_def_421,type,
sP68: $i > $i > $i > $o ).
thf(func_def_422,type,
sP69: $i > $i > $i > $o ).
thf(func_def_425,type,
sP72: $i > $i > $i > $o ).
thf(func_def_426,type,
sP73: $i > $i > $i > $o ).
thf(func_def_427,type,
sP74: $i > $i > $i > $o ).
thf(func_def_429,type,
sP76: ( $i > $o ) > $o ).
thf(func_def_431,type,
sP78: $i > ( $i > $o ) > $o ).
thf(func_def_434,type,
sP81: $o > $i > $i > $o ).
thf(func_def_435,type,
sP82: $i > $i > $i > $o ).
thf(func_def_436,type,
sP83: $i > ( $i > $o ) > $o ).
thf(func_def_437,type,
sP84: $i > ( $i > $o ) > $o ).
thf(func_def_438,type,
sP85: ( $i > $o ) > $i > $o ).
thf(func_def_441,type,
sP88: ( $i > $o ) > $i > $o ).
thf(func_def_443,type,
sP90: ( $i > $o ) > $i > $o ).
thf(func_def_444,type,
sP91: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_445,type,
sP92: $i > $i > $i > $i > $o ).
thf(func_def_453,type,
sK100: $i > $i ).
thf(func_def_456,type,
sK103: $i > $i > ( $i > $i ) > $i ).
thf(func_def_461,type,
sK108: $i > $i > ( $i > $o ) > $i ).
thf(func_def_463,type,
sK110: $i > $o ).
thf(func_def_469,type,
sK116: $i > $i ).
thf(func_def_472,type,
sK119: $i > $i > ( $i > $i ) > $i ).
thf(func_def_474,type,
sK121: $i > $o ).
thf(func_def_493,type,
sK140: $i > $i > $o ).
thf(func_def_494,type,
sK141: $i > $i > $o ).
thf(func_def_499,type,
sK146: $i > $o ).
thf(func_def_520,type,
sK167: $i > ( $i > $o ) > $i ).
thf(func_def_521,type,
sK168: ( $i > $o ) > $i ).
thf(func_def_522,type,
sK169: $i > $o ).
thf(func_def_531,type,
sK178: $i > $i > $i ).
thf(func_def_532,type,
sK179: $i > $i ).
thf(func_def_536,type,
sK183: $i > $i > $o ).
thf(func_def_538,type,
sK185: $i > ( $i > $i > $o ) > $i ).
thf(func_def_539,type,
sK186: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_540,type,
sK187: ( $i > $i > $o ) > $i > $i ).
thf(func_def_541,type,
sK188: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_546,type,
sK193: $i > $o ).
thf(func_def_548,type,
sK195: $i > ( $i > $o ) > $i ).
thf(func_def_549,type,
sK196: ( $i > $o ) > $i > $i ).
thf(func_def_552,type,
sK199: $i > $i > $i > $i ).
thf(func_def_553,type,
sK200: $i > $i > $i > $i ).
thf(func_def_557,type,
sK204: $i > $i ).
thf(func_def_558,type,
sK205: $i > $i ).
thf(func_def_559,type,
sK206: $i > $i ).
thf(func_def_560,type,
sK207: $i > $i ).
thf(func_def_561,type,
sK208: $i > $i > $i ).
thf(func_def_562,type,
sK209: $i > $i > $i ).
thf(func_def_563,type,
sK210: $i > $i > $i ).
thf(func_def_564,type,
sK211: $i > $i > $i ).
thf(func_def_565,type,
sK212: $i > $i > $i > $i ).
thf(func_def_566,type,
sK213: $i > $i > $i ).
thf(func_def_567,type,
sK214: $i > $i > $i ).
thf(func_def_568,type,
sK215: $i > $i > $i ).
thf(func_def_569,type,
sK216: $i > $i > $i ).
thf(func_def_570,type,
sK217: $i > $i > $i ).
thf(func_def_571,type,
sK218: $i > $i > $i ).
thf(func_def_572,type,
sK219: $i > $i > $i > $i ).
thf(func_def_573,type,
sK220: $i > $i > $i > $i ).
thf(func_def_574,type,
sK221: $i > $i ).
thf(func_def_587,type,
sK234: $i > $i > $i ).
thf(func_def_588,type,
sK235: $i > $i > $i ).
thf(func_def_592,type,
sK239: $i > $o ).
thf(func_def_595,type,
sK242: $i > $i > $i > $i ).
thf(func_def_596,type,
sK243: $i > $i > $i > $i > $i ).
thf(func_def_624,type,
sK271: $i > $i ).
thf(func_def_625,type,
sK272: $i > $i ).
thf(func_def_631,type,
sK278: $i > $o ).
thf(func_def_632,type,
sK279: ( $i > $o ) > $i > $i ).
thf(func_def_633,type,
sK280: $i > $o ).
thf(func_def_635,type,
sK282: $i > ( $i > $o ) > $i ).
thf(func_def_636,type,
sK283: ( $i > $o ) > $i > $i ).
thf(func_def_637,type,
sK284: $i > $i > $i > $i > $i ).
thf(func_def_638,type,
sK285: $i > $i > $i > $i > $i ).
thf(func_def_639,type,
sK286: $i > $i > $i > $i > $i ).
thf(func_def_640,type,
sK287: $i > $i > $i > $i > $i ).
thf(func_def_645,type,
sK292: $i > $o ).
thf(func_def_660,type,
sK307: $i > $o ).
thf(func_def_666,type,
sK313: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_667,type,
sK314: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_668,type,
sK315: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_669,type,
sK316: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_671,type,
sK318: $i > $i > $o ).
thf(func_def_683,type,
sK330: $i > $o ).
thf(func_def_706,type,
sK353: $i > $i > $i > $i ).
thf(func_def_707,type,
sK354: $i > $i > $i > $i ).
thf(func_def_736,type,
sK383: $i > $i ).
thf(func_def_738,type,
sK385: $i > ( $i > $i ) > $i > $i ).
thf(func_def_739,type,
sK386: $i > $i > ( $i > $i ) > $i ).
thf(func_def_755,type,
sK402: $i > $i > $i > $i ).
thf(func_def_756,type,
sK403: $i > $i > $i > $i > $i ).
thf(func_def_757,type,
sK404: $i > $i > $i > $i > $i ).
thf(func_def_758,type,
sK405: $i > $i > $i > $i > $i ).
thf(func_def_765,type,
sK412: $i > ( $i > $o ) > $i > $i ).
thf(func_def_766,type,
sK413: $i > ( $i > $o ) > $i > $i ).
thf(func_def_767,type,
sK414: $i > ( $i > $o ) > $i ).
thf(func_def_770,type,
sK417: $i > $o ).
thf(func_def_775,type,
sK422: ( $i > $o ) > $i ).
thf(func_def_776,type,
sK423: ( $i > $o ) > $i ).
thf(func_def_777,type,
sK424: $i > $o ).
thf(func_def_782,type,
sK429: $i > $i ).
thf(func_def_785,type,
sK432: $i > $i > ( $i > $i ) > $i ).
thf(func_def_786,type,
sK433: $i > $i > ( $i > $i ) > $i ).
thf(func_def_789,type,
sK436: $i > $i > $i ).
thf(func_def_798,type,
sK445: $i > $i > $i ).
thf(func_def_828,type,
sK475: $i > ( $i > $o ) > $i ).
thf(func_def_829,type,
sK476: $i > ( $i > $o ) > $i ).
thf(func_def_830,type,
sK477: $i > ( $i > $o ) > $i ).
thf(func_def_831,type,
sK478: $i > $o ).
thf(func_def_892,type,
sK539: $i > $i > $i ).
thf(func_def_897,type,
sK544: $i > $i > $i ).
thf(func_def_898,type,
sK545: $i > ( $i > $o ) > $i > $i > $i ).
thf(func_def_899,type,
sK546: $i > ( $i > $o ) > $i > $i > $i ).
thf(func_def_900,type,
sK547: $i > ( $i > $o ) > $i ).
thf(func_def_904,type,
sK551: $i > $o ).
thf(func_def_925,type,
sK572: $i > ( $i > $o ) > $i > $i ).
thf(func_def_926,type,
sK573: $i > ( $i > $o ) > $i > $i ).
thf(func_def_927,type,
sK574: $i > ( $i > $o ) > $i ).
thf(func_def_930,type,
sK577: $i > $o ).
thf(func_def_935,type,
sK582: $i > $i ).
thf(func_def_938,type,
sK585: $i > $i > ( $i > $i ) > $i ).
thf(func_def_941,type,
sK588: $i > $o ).
thf(func_def_943,type,
sK590: $i > ( $i > $o ) > $i ).
thf(func_def_944,type,
sK591: $i > ( $i > $o ) > $i ).
thf(func_def_949,type,
sK596: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_950,type,
sK597: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_953,type,
sK600: $i > $o ).
thf(func_def_954,type,
sK601: $i > $o ).
thf(func_def_957,type,
sK604: $i > $i > $i > $o ).
thf(func_def_968,type,
sK615: $i > $i > $o ).
thf(func_def_973,type,
sK620: $i > $o > $i > $i ).
thf(func_def_982,type,
sK629: $i > $i > $i ).
thf(func_def_1027,type,
sK674: $i > $i > $o ).
thf(func_def_1029,type,
sK676: $i > $o ).
thf(func_def_1035,type,
sK682: $i > $i > $i ).
thf(func_def_1040,type,
sK687: $i > $i > $i > $i ).
thf(func_def_1041,type,
sK688: $i > $i > $i > $i > $i ).
thf(func_def_1082,type,
sK729: $i > $i > $i > $i ).
thf(func_def_1089,type,
sK736: ( $i > $o ) > $i ).
thf(func_def_1090,type,
sK737: $i > ( $i > $o ) > $i ).
thf(func_def_1091,type,
sK738: $i > $o ).
thf(func_def_1097,type,
sK744: $i > ( $i > $o ) > $i ).
thf(func_def_1098,type,
sK745: ( $i > $o ) > $i ).
thf(func_def_1099,type,
sK746: $i > $o ).
thf(func_def_1101,type,
sK748: $i > $i > $i > $i ).
thf(func_def_1102,type,
sK749: $i > $i > $i > $i ).
thf(func_def_1134,type,
sK781: $i > $o ).
thf(func_def_1151,type,
sK798: $i > $i > $i ).
thf(func_def_1156,type,
sK803: $i > $o ).
thf(func_def_1157,type,
sK804: $i > $i > $i > $i ).
thf(func_def_1161,type,
sK808: $i > $i > $i > $i ).
thf(func_def_1162,type,
sK809: $i > $i > $i > $i ).
thf(func_def_1163,type,
sK810: $i > $i > $i > $i ).
thf(func_def_1164,type,
sK811: $i > $i > $i > $i ).
thf(func_def_1180,type,
sK827: $i > $o ).
thf(func_def_1184,type,
sK831: $i > $o ).
thf(func_def_1185,type,
sK832: $i > $o ).
thf(func_def_1186,type,
sK833: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1187,type,
sK834: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1191,type,
sK838: ( $i > $o ) > $i ).
thf(func_def_1192,type,
sK839: $i > ( $i > $o ) > $i ).
thf(func_def_1193,type,
sK840: $i > $o ).
thf(func_def_1196,type,
sK843: $i > $i > $o ).
thf(func_def_1200,type,
sK847: $i > $i > $i ).
thf(func_def_1202,type,
sK849: $i > $o ).
thf(func_def_1203,type,
sK850: ( $i > $o ) > $i > $i ).
thf(func_def_1204,type,
sK851: $i > ( $i > $o ) > $i ).
thf(func_def_1208,type,
sK855: $i > $i ).
thf(func_def_1211,type,
sK858: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1218,type,
sK865: $i > $i > $i > $i ).
thf(func_def_1219,type,
sK866: $i > $i > $i > $o ).
thf(func_def_1225,type,
sK872: $i > ( $i > $o ) > $i ).
thf(func_def_1226,type,
sK873: $i > ( $i > $o ) > $i ).
thf(func_def_1227,type,
sK874: $i > ( $i > $o ) > $i ).
thf(func_def_1229,type,
sK876: $i > $o ).
thf(func_def_1233,type,
sK880: $i > $o ).
thf(func_def_1236,type,
sK883: $i > $o ).
thf(func_def_1237,type,
sK884: ( $i > $o ) > $i ).
thf(func_def_1263,type,
sK910: $i > $o ).
thf(func_def_1264,type,
sK911: ( $i > $o ) > $i > $i ).
thf(func_def_1265,type,
sK912: ( $i > $o ) > $i > $i ).
thf(func_def_1266,type,
sK913: $i > $o ).
thf(func_def_1267,type,
sK914: $i > $o ).
thf(func_def_1268,type,
sK915: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1269,type,
sK916: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1285,type,
sK932: $i > $i ).
thf(func_def_1293,type,
sK940: $i > $o ).
thf(func_def_1295,type,
sK942: $i > ( $i > $o ) > $i ).
thf(func_def_1296,type,
sK943: ( $i > $o ) > $i > $i ).
thf(func_def_1307,type,
sK954: $i > $o ).
thf(func_def_1308,type,
sK955: $i > $i ).
thf(func_def_1314,type,
sK961: $i > $i > $o ).
thf(func_def_1322,type,
sK969: $i > $i ).
thf(func_def_1327,type,
sK974: $i > $o ).
thf(func_def_1329,type,
sK976: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1330,type,
sK977: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1331,type,
sK978: ( $i > $o ) > $i > $i ).
thf(func_def_1335,type,
sK982: $i > $o ).
thf(func_def_1347,type,
sK994: $i > $i > $i > $i > $i ).
thf(func_def_1348,type,
sK995: $i > $i > $i > $i > $i ).
thf(func_def_1359,type,
sK1006: $i > $i > $o ).
thf(func_def_1372,type,
sK1019: $i > $i ).
thf(func_def_1375,type,
sK1022: $i > $i > $o ).
thf(func_def_1395,type,
ph1042:
!>[X0: $tType] : X0 ).
thf(f10357,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f10356,f4133]) ).
thf(f4133,plain,
( $true
= ( in @ sK640 @ sK639 ) ),
inference(cnf_transformation,[status(thm)],[f2414]) ).
thf(f2414,plain,
( ( cartprodfstpairEq = $true )
& ( singletonprop = $true )
& ( prop2setE = $true )
& ( eqbreln = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyE1 = $true )
& ( eta1 = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( binintersectSubset1 = $true )
& ( sepSubset = $true )
& ( binintersectER = $true )
& ( iffalseProp1 = $true )
& ( singletonsuniq = $true )
& ( cartprodsndpairEq = $true )
& ( setminusIRneg = $true )
& ( notsubsetI = $true )
& ( setukpairIR = $true )
& ( binintersectT_lem = $true )
& ( apProp = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binintersectRsub = $true )
& ( eqinunit = $true )
& ( kfstsingleton = $true )
& ( iftrueorfalse = $true )
& ( sepInPowerset = $true )
& ( upairsubunion = $true )
& ( notequalI2 = $true )
& ( setminusEL = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( setext = $true )
& ( powerset__Cong = $true )
& ( contrasubsetT1 = $true )
& ( binintersectEL = $true )
& ( doubleComplementE1 = $true )
& ( exuE3e = $true )
& ( doubleComplementI1 = $true )
& ( setadjoinE = $true )
& ( singletoninpowerset = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( demorgan2a = $true )
& ( lamProp = $true )
& ( exuI1 = $true )
& ( ifp = $true )
& ( emptyinunitempty = $true )
& ( woz1_5 = $true )
& ( symdiffI1 = $true )
& ( notdexE = $true )
& ( nonemptyImpWitness = $true )
& ( emptysetsubset = $true )
& ( exu__Cong = $true )
& ( complementTnotintersectT = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( upairinpowunion = $true )
& ( setadjoinIR = $true )
& ( emptyE1 = $true )
& ( doubleComplementSub1 = $true )
& ( setbeta = $true )
& ( cartprodpairsurjEq = $true )
& ( symdiffE = $true )
& ( binunionIR = $true )
& ( woz1_3 = $true )
& ( contrasubsetT2 = $true )
& ( iffalse = $true )
& ( cartprodpairmemEL = $true )
& ( subsetI1 = $true )
& ( doubleComplementEq = $true )
& ( setminusSubset2 = $true )
& ( setminusI = $true )
& ( ap2apEq2 = $true )
& ( notinsingleton = $true )
& ( demorgan2a2 = $true )
& ( breln1all2 = $true )
& ( binunionIL = $true )
& ( complementUnionInPowersetComplement = $true )
& ( setminusERneg = $true )
& ( woz1_1 = $true )
& ( setadjoinSub = $true )
& ( wellorderingAx = $true )
& ( demorgan2a1 = $true )
& ( singletonsubset = $true )
& ( setukpairinjR1 = $true )
& ( contrasubsetT3 = $true )
& ( doubleComplementSub2 = $true )
& ( setunion__Cong = $true )
& ( breln1invprop = $true )
& ( binunionRsub = $true )
& ( setadjoinOr = $true )
& ( ex1I = $true )
& ( setminusLsub = $true )
& ( powersetI = $true )
& ( brelnall2 = $true )
& ( demorgan1a = $true )
& ( foundationAx = $true )
& ( powersetsubset = $true )
& ( upairsetIL = $true )
& ( dpsetconstrEL2 = $true )
& ( exuE3u = $true )
& ( dpsetconstrEL1 = $true )
& ( noeltsimpempty = $true )
& ( setadjoin__Cong = $true )
& ( brelnall1 = $true )
& ( app = $true )
& ( binunionTEcontra = $true )
& ( exuEu = $true )
& ( dpsetconstrSub = $true )
& ( subsetemptysetimpeq = $true )
& ( woz1_2 = $true )
& ( dsetconstrER = $true )
& ( iftrueProp1 = $true )
& ( ( breln1 @ sK639 @ sK638 )
= $true )
& ( $true
= ( in @ sK640 @ sK639 ) )
& ( $true
= ( in @ sK641 @ sK639 ) )
& ( $true
!= ( in @ ( kpair @ sK641 @ sK640 ) @ ( breln1invset @ sK639 @ sK638 ) ) )
& ( $true
= ( in @ ( kpair @ sK640 @ sK641 ) @ sK638 ) )
& ( cartprodmempair = $true )
& ( notequalI1 = $true )
& ( binintersectTERcontra = $true )
& ( cartprodpairmemER = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan1 = $true )
& ( nonemptyI1 = $true )
& ( demorgan2b2 = $true )
& ( setunionE2 = $true )
& ( emptysetAx = $true )
& ( demorgan1 = $true )
& ( infuncsetfunc = $true )
& ( upairset2E = $true )
& ( funcextLem = $true )
& ( funcext2 = $true )
& ( complementT_lem = $true )
& ( setextsub = $true )
& ( demorgan2 = $true )
& ( cartprodmempair1 = $true )
& ( iftrueProp2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binintersectLsub = $true )
& ( inIntersectImpInUnion2 = $true )
& ( singletoninpowunion = $true )
& ( contraSubsetComplement = $true )
& ( binintersectSubset5 = $true )
& ( kpairp = $true )
& ( funcGraphProp2 = $true )
& ( subPowSU = $true )
& ( binintersectSubset3 = $true )
& ( binintersectTELcontra = $true )
& ( funcGraphProp4 = $true )
& ( powersetE1 = $true )
& ( binintersectI = $true )
& ( cartprodfstin = $true )
& ( eqbreln1 = $true )
& ( singletonsswitch = $true )
& ( ap2p = $true )
& ( omegaIndAx = $true )
& ( subsetI2 = $true )
& ( complementTI1 = $true )
& ( setukpairinjL = $true )
& ( setOfPairsIsBReln = $true )
& ( binunionEcases = $true )
& ( exuE1 = $true )
& ( funcinfuncset = $true )
& ( complementTE1 = $true )
& ( binunionTIRcontra = $true )
& ( kpairiskpair = $true )
& ( ubforcartprodlem2 = $true )
& ( beta1 = $true )
& ( dsetconstrI = $true )
& ( setminusILneg = $true )
& ( subsetTrans = $true )
& ( vacuousDall = $true )
& ( powersetE = $true )
& ( powersetTE1 = $true )
& ( woz13rule0 = $true )
& ( replAx = $true )
& ( powersetT_lem = $true )
& ( eqimpsubset2 = $true )
& ( setminusSubset1 = $true )
& ( notinemptyset = $true )
& ( cartprodsndin = $true )
& ( eqimpsubset1 = $true )
& ( setunionE = $true )
& ( inPowerset = $true )
& ( breln1all1 = $true )
& ( setminusELneg = $true )
& ( subsetRefl = $true )
& ( emptyset__Cong = $true )
& ( contrasubsetT = $true )
& ( dsetconstrEL = $true )
& ( funcext = $true )
& ( ifSingleton = $true )
& ( exuE2 = $true )
& ( quantDeMorgan3 = $true )
& ( omegaSAx = $true )
& ( binunionE = $true )
& ( ex1E1 = $true )
& ( powersetTI1 = $true )
& ( bs114d = $true )
& ( setextT = $true )
& ( woz13rule3 = $true )
& ( setadjoinSub2 = $true )
& ( funcGraphProp3 = $true )
& ( subbreln = $true )
& ( breln1SetBreln1 = $true )
& ( descrp = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( binunionTE = $true )
& ( ksndsingleton = $true )
& ( demorgan1b = $true )
& ( complementImpComplementIntersect = $true )
& ( emptyI = $true )
& ( binintersectSubset2 = $true )
& ( theprop = $true )
& ( kfstpairEq = $true )
& ( binunionTILcontra = $true )
& ( descr__Cong = $true )
& ( prop2set2propI = $true )
& ( symdiffIneg2 = $true )
& ( subsetE = $true )
& ( setminusT_lem = $true )
& ( theeq = $true )
& ( iftrue = $true )
& ( exuI3 = $true )
& ( kpairsurjEq = $true )
& ( upairset2IR = $true )
& ( iffalseProp2 = $true )
& ( dsetconstr__Cong = $true )
& ( ubforcartprodlem3 = $true )
& ( lam2lamEq = $true )
& ( woz13rule4 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( notdallE = $true )
& ( ubforcartprodlem1 = $true )
& ( funcGraphProp1 = $true )
& ( upairequniteq = $true )
& ( disjointsetsI1 = $true )
& ( cartprodmempaircEq = $true )
& ( emptyinPowerset = $true )
& ( setunionsingleton = $true )
& ( binunionT_lem = $true )
& ( omega__Cong = $true )
& ( symdiffI2 = $true )
& ( cartprodpairin = $true )
& ( upairsetIR = $true )
& ( setminusER = $true )
& ( beta2 = $true )
& ( emptyInPowerset = $true )
& ( setukpairinjR12 = $true )
& ( lamp = $true )
& ( powersetI1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setunionI = $true )
& ( in__Cong = $true )
& ( ap2apEq1 = $true )
& ( symdiffIneg1 = $true )
& ( emptysetE = $true )
& ( setadjoinAx = $true )
& ( nonemptyI = $true )
& ( choice2fnsingleton = $true )
& ( uniqinunit = $true )
& ( setukpairinjL2 = $true )
& ( ex1E2 = $true )
& ( demorgan2b = $true )
& ( binintersectSubset4 = $true )
& ( prop2setI = $true )
& ( subsetE2 = $true )
& ( subset2powerset = $true )
& ( lam2p = $true )
& ( setadjoinIL = $true )
& ( dpsetconstrERa = $true )
& ( setunionsingleton1 = $true )
& ( setunionAx = $true )
& ( funcImageSingleton = $true )
& ( subbreln1 = $true )
& ( ex1I2 = $true )
& ( ksndpairEq = $true )
& ( dpsetconstrER = $true )
& ( upairsetE = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( woz13rule2 = $true )
& ( secondinupair = $true )
& ( dpsetconstrI = $true )
& ( woz13rule1 = $true )
& ( complementTcontraSubset = $true )
& ( setukpairIL = $true )
& ( woz1_4 = $true )
& ( binunionLsub = $true )
& ( omega0Ax = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan2 = $true )
& ( inCongP = $true )
& ( eta2 = $true )
& ( exuI2 = $true )
& ( setextAx = $true )
& ( subsetTI = $true )
& ( setunionsingleton2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK638,sK639,sK640,sK641])],[f1079,f2413,f2412,f2411]) ).
thf(f2411,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( breln1 @ X1 @ X0 ) )
& ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( $true
!= ( in @ ( kpair @ X3 @ X2 ) @ ( breln1invset @ X1 @ X0 ) ) )
& ( $true
= ( in @ ( kpair @ X2 @ X3 ) @ X0 ) ) ) ) )
=> ( ( ( breln1 @ sK639 @ sK638 )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ sK639 )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK639 ) )
& ( $true
!= ( in @ ( kpair @ X3 @ X2 ) @ ( breln1invset @ sK639 @ sK638 ) ) )
& ( ( in @ ( kpair @ X2 @ X3 ) @ sK638 )
= $true ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2412,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ sK639 )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK639 ) )
& ( $true
!= ( in @ ( kpair @ X3 @ X2 ) @ ( breln1invset @ sK639 @ sK638 ) ) )
& ( ( in @ ( kpair @ X2 @ X3 ) @ sK638 )
= $true ) ) )
=> ( ( $true
= ( in @ sK640 @ sK639 ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK639 ) )
& ( $true
!= ( in @ ( kpair @ X3 @ sK640 ) @ ( breln1invset @ sK639 @ sK638 ) ) )
& ( $true
= ( in @ ( kpair @ sK640 @ X3 ) @ sK638 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2413,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK639 ) )
& ( $true
!= ( in @ ( kpair @ X3 @ sK640 ) @ ( breln1invset @ sK639 @ sK638 ) ) )
& ( $true
= ( in @ ( kpair @ sK640 @ X3 ) @ sK638 ) ) )
=> ( ( $true
= ( in @ sK641 @ sK639 ) )
& ( $true
!= ( in @ ( kpair @ sK641 @ sK640 ) @ ( breln1invset @ sK639 @ sK638 ) ) )
& ( $true
= ( in @ ( kpair @ sK640 @ sK641 ) @ sK638 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1079,plain,
( ( cartprodfstpairEq = $true )
& ( singletonprop = $true )
& ( prop2setE = $true )
& ( eqbreln = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyE1 = $true )
& ( eta1 = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( binintersectSubset1 = $true )
& ( sepSubset = $true )
& ( binintersectER = $true )
& ( iffalseProp1 = $true )
& ( singletonsuniq = $true )
& ( cartprodsndpairEq = $true )
& ( setminusIRneg = $true )
& ( notsubsetI = $true )
& ( setukpairIR = $true )
& ( binintersectT_lem = $true )
& ( apProp = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binintersectRsub = $true )
& ( eqinunit = $true )
& ( kfstsingleton = $true )
& ( iftrueorfalse = $true )
& ( sepInPowerset = $true )
& ( upairsubunion = $true )
& ( notequalI2 = $true )
& ( setminusEL = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( setext = $true )
& ( powerset__Cong = $true )
& ( contrasubsetT1 = $true )
& ( binintersectEL = $true )
& ( doubleComplementE1 = $true )
& ( exuE3e = $true )
& ( doubleComplementI1 = $true )
& ( setadjoinE = $true )
& ( singletoninpowerset = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( demorgan2a = $true )
& ( lamProp = $true )
& ( exuI1 = $true )
& ( ifp = $true )
& ( emptyinunitempty = $true )
& ( woz1_5 = $true )
& ( symdiffI1 = $true )
& ( notdexE = $true )
& ( nonemptyImpWitness = $true )
& ( emptysetsubset = $true )
& ( exu__Cong = $true )
& ( complementTnotintersectT = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( upairinpowunion = $true )
& ( setadjoinIR = $true )
& ( emptyE1 = $true )
& ( doubleComplementSub1 = $true )
& ( setbeta = $true )
& ( cartprodpairsurjEq = $true )
& ( symdiffE = $true )
& ( binunionIR = $true )
& ( woz1_3 = $true )
& ( contrasubsetT2 = $true )
& ( iffalse = $true )
& ( cartprodpairmemEL = $true )
& ( subsetI1 = $true )
& ( doubleComplementEq = $true )
& ( setminusSubset2 = $true )
& ( setminusI = $true )
& ( ap2apEq2 = $true )
& ( notinsingleton = $true )
& ( demorgan2a2 = $true )
& ( breln1all2 = $true )
& ( binunionIL = $true )
& ( complementUnionInPowersetComplement = $true )
& ( setminusERneg = $true )
& ( woz1_1 = $true )
& ( setadjoinSub = $true )
& ( wellorderingAx = $true )
& ( demorgan2a1 = $true )
& ( singletonsubset = $true )
& ( setukpairinjR1 = $true )
& ( contrasubsetT3 = $true )
& ( doubleComplementSub2 = $true )
& ( setunion__Cong = $true )
& ( breln1invprop = $true )
& ( binunionRsub = $true )
& ( setadjoinOr = $true )
& ( ex1I = $true )
& ( setminusLsub = $true )
& ( powersetI = $true )
& ( brelnall2 = $true )
& ( demorgan1a = $true )
& ( foundationAx = $true )
& ( powersetsubset = $true )
& ( upairsetIL = $true )
& ( dpsetconstrEL2 = $true )
& ( exuE3u = $true )
& ( dpsetconstrEL1 = $true )
& ( noeltsimpempty = $true )
& ( setadjoin__Cong = $true )
& ( brelnall1 = $true )
& ( app = $true )
& ( binunionTEcontra = $true )
& ( exuEu = $true )
& ( dpsetconstrSub = $true )
& ( subsetemptysetimpeq = $true )
& ( woz1_2 = $true )
& ( dsetconstrER = $true )
& ( iftrueProp1 = $true )
& ? [X0: $i,X1: $i] :
( ( $true
= ( breln1 @ X1 @ X0 ) )
& ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( $true
!= ( in @ ( kpair @ X3 @ X2 ) @ ( breln1invset @ X1 @ X0 ) ) )
& ( $true
= ( in @ ( kpair @ X2 @ X3 ) @ X0 ) ) ) ) )
& ( cartprodmempair = $true )
& ( notequalI1 = $true )
& ( binintersectTERcontra = $true )
& ( cartprodpairmemER = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan1 = $true )
& ( nonemptyI1 = $true )
& ( demorgan2b2 = $true )
& ( setunionE2 = $true )
& ( emptysetAx = $true )
& ( demorgan1 = $true )
& ( infuncsetfunc = $true )
& ( upairset2E = $true )
& ( funcextLem = $true )
& ( funcext2 = $true )
& ( complementT_lem = $true )
& ( setextsub = $true )
& ( demorgan2 = $true )
& ( cartprodmempair1 = $true )
& ( iftrueProp2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binintersectLsub = $true )
& ( inIntersectImpInUnion2 = $true )
& ( singletoninpowunion = $true )
& ( contraSubsetComplement = $true )
& ( binintersectSubset5 = $true )
& ( kpairp = $true )
& ( funcGraphProp2 = $true )
& ( subPowSU = $true )
& ( binintersectSubset3 = $true )
& ( binintersectTELcontra = $true )
& ( funcGraphProp4 = $true )
& ( powersetE1 = $true )
& ( binintersectI = $true )
& ( cartprodfstin = $true )
& ( eqbreln1 = $true )
& ( singletonsswitch = $true )
& ( ap2p = $true )
& ( omegaIndAx = $true )
& ( subsetI2 = $true )
& ( complementTI1 = $true )
& ( setukpairinjL = $true )
& ( setOfPairsIsBReln = $true )
& ( binunionEcases = $true )
& ( exuE1 = $true )
& ( funcinfuncset = $true )
& ( complementTE1 = $true )
& ( binunionTIRcontra = $true )
& ( kpairiskpair = $true )
& ( ubforcartprodlem2 = $true )
& ( beta1 = $true )
& ( dsetconstrI = $true )
& ( setminusILneg = $true )
& ( subsetTrans = $true )
& ( vacuousDall = $true )
& ( powersetE = $true )
& ( powersetTE1 = $true )
& ( woz13rule0 = $true )
& ( replAx = $true )
& ( powersetT_lem = $true )
& ( eqimpsubset2 = $true )
& ( setminusSubset1 = $true )
& ( notinemptyset = $true )
& ( cartprodsndin = $true )
& ( eqimpsubset1 = $true )
& ( setunionE = $true )
& ( inPowerset = $true )
& ( breln1all1 = $true )
& ( setminusELneg = $true )
& ( subsetRefl = $true )
& ( emptyset__Cong = $true )
& ( contrasubsetT = $true )
& ( dsetconstrEL = $true )
& ( funcext = $true )
& ( ifSingleton = $true )
& ( exuE2 = $true )
& ( quantDeMorgan3 = $true )
& ( omegaSAx = $true )
& ( binunionE = $true )
& ( ex1E1 = $true )
& ( powersetTI1 = $true )
& ( bs114d = $true )
& ( setextT = $true )
& ( woz13rule3 = $true )
& ( setadjoinSub2 = $true )
& ( funcGraphProp3 = $true )
& ( subbreln = $true )
& ( breln1SetBreln1 = $true )
& ( descrp = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( binunionTE = $true )
& ( ksndsingleton = $true )
& ( demorgan1b = $true )
& ( complementImpComplementIntersect = $true )
& ( emptyI = $true )
& ( binintersectSubset2 = $true )
& ( theprop = $true )
& ( kfstpairEq = $true )
& ( binunionTILcontra = $true )
& ( descr__Cong = $true )
& ( prop2set2propI = $true )
& ( symdiffIneg2 = $true )
& ( subsetE = $true )
& ( setminusT_lem = $true )
& ( theeq = $true )
& ( iftrue = $true )
& ( exuI3 = $true )
& ( kpairsurjEq = $true )
& ( upairset2IR = $true )
& ( iffalseProp2 = $true )
& ( dsetconstr__Cong = $true )
& ( ubforcartprodlem3 = $true )
& ( lam2lamEq = $true )
& ( woz13rule4 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( notdallE = $true )
& ( ubforcartprodlem1 = $true )
& ( funcGraphProp1 = $true )
& ( upairequniteq = $true )
& ( disjointsetsI1 = $true )
& ( cartprodmempaircEq = $true )
& ( emptyinPowerset = $true )
& ( setunionsingleton = $true )
& ( binunionT_lem = $true )
& ( omega__Cong = $true )
& ( symdiffI2 = $true )
& ( cartprodpairin = $true )
& ( upairsetIR = $true )
& ( setminusER = $true )
& ( beta2 = $true )
& ( emptyInPowerset = $true )
& ( setukpairinjR12 = $true )
& ( lamp = $true )
& ( powersetI1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setunionI = $true )
& ( in__Cong = $true )
& ( ap2apEq1 = $true )
& ( symdiffIneg1 = $true )
& ( emptysetE = $true )
& ( setadjoinAx = $true )
& ( nonemptyI = $true )
& ( choice2fnsingleton = $true )
& ( uniqinunit = $true )
& ( setukpairinjL2 = $true )
& ( ex1E2 = $true )
& ( demorgan2b = $true )
& ( binintersectSubset4 = $true )
& ( prop2setI = $true )
& ( subsetE2 = $true )
& ( subset2powerset = $true )
& ( lam2p = $true )
& ( setadjoinIL = $true )
& ( dpsetconstrERa = $true )
& ( setunionsingleton1 = $true )
& ( setunionAx = $true )
& ( funcImageSingleton = $true )
& ( subbreln1 = $true )
& ( ex1I2 = $true )
& ( ksndpairEq = $true )
& ( dpsetconstrER = $true )
& ( upairsetE = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( woz13rule2 = $true )
& ( secondinupair = $true )
& ( dpsetconstrI = $true )
& ( woz13rule1 = $true )
& ( complementTcontraSubset = $true )
& ( setukpairIL = $true )
& ( woz1_4 = $true )
& ( binunionLsub = $true )
& ( omega0Ax = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan2 = $true )
& ( inCongP = $true )
& ( eta2 = $true )
& ( exuI2 = $true )
& ( setextAx = $true )
& ( subsetTI = $true )
& ( setunionsingleton2 = $true ) ),
inference(flattening,[status(thm)],[f1078]) ).
thf(f1078,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X2 ) @ ( breln1invset @ X1 @ X0 ) ) )
& ( $true
= ( in @ ( kpair @ X2 @ X3 ) @ X0 ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( in @ X2 @ X1 )
= $true ) )
& ( $true
= ( breln1 @ X1 @ X0 ) ) )
& ( 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)],[f464]) ).
thf(f464,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 )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( breln1 @ X1 @ X0 ) )
=> ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( $true
= ( in @ ( kpair @ X2 @ X3 ) @ X0 ) )
=> ( $true
= ( in @ ( kpair @ X3 @ X2 ) @ ( breln1invset @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f463]) ).
thf(f463,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
=> ! [X0: $i,X1: $i] :
( ( breln1 @ X1 @ X0 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X0 )
=> ( in @ ( kpair @ X3 @ X2 ) @ ( breln1invset @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f299]) ).
thf(f299,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
=> ! [X13: $i,X3: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
=> ( in @ ( kpair @ X2 @ X1 ) @ ( breln1invset @ X3 @ X13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f298]) ).
thf(f298,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
=> ! [X13: $i,X3: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
=> ( in @ ( kpair @ X2 @ X1 ) @ ( breln1invset @ X3 @ X13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ExEgOH0aih/DTF2THF_12985.p',breln1invI) ).
thf(f10356,plain,
( $true
!= ( in @ sK640 @ sK639 ) ),
inference(subsumption_resolution,[status(thm)],[f10355,f4130]) ).
thf(f4130,plain,
( $true
= ( in @ ( kpair @ sK640 @ sK641 ) @ sK638 ) ),
inference(cnf_transformation,[status(thm)],[f2414]) ).
thf(f10355,plain,
( ( $true
!= ( in @ ( kpair @ sK640 @ sK641 ) @ sK638 ) )
| ( $true
!= ( in @ sK640 @ sK639 ) ) ),
inference(subsumption_resolution,[status(thm)],[f10350,f4132]) ).
thf(f4132,plain,
( $true
= ( in @ sK641 @ sK639 ) ),
inference(cnf_transformation,[status(thm)],[f2414]) ).
thf(f10350,plain,
( ( $true
!= ( in @ sK641 @ sK639 ) )
| ( $true
!= ( in @ sK640 @ sK639 ) )
| ( $true
!= ( in @ ( kpair @ sK640 @ sK641 ) @ sK638 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f10349]) ).
thf(f10349,plain,
( ( $true
!= ( in @ sK640 @ sK639 ) )
| ( $true
!= ( in @ sK641 @ sK639 ) )
| ( $true
!= ( ^ [Y0: $i,Y1: $i] : ( in @ ( kpair @ Y1 @ Y0 ) @ sK638 )
@ sK641
@ sK640 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f10340]) ).
thf(f10340,plain,
( ( $true
!= ( ^ [Y0: $i,Y1: $i] : ( in @ ( kpair @ Y1 @ Y0 ) @ sK638 )
@ sK641
@ sK640 ) )
| ( $true
!= ( in @ sK640 @ sK639 ) )
| ( $true
!= ( in @ sK641 @ sK639 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f6237,f6008]) ).
thf(f6008,plain,
! [X2: $i,X3: $i,X0: $i > $i > $o,X1: $i,X4: $i] :
( ( $true
= ( in @ ( kpair @ X2 @ X4 ) @ ( dpsetconstr @ X3 @ X1 @ X0 ) ) )
| ( $true
!= ( X0 @ X2 @ X4 ) )
| ( ( in @ X2 @ X3 )
!= $true )
| ( $true
!= ( in @ X4 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f6007]) ).
thf(f6007,plain,
! [X2: $i,X3: $i,X0: $i > $i > $o,X1: $i,X4: $i] :
( ( $true
!= ( X0 @ X2 @ X4 ) )
| ( ( in @ X2 @ X3 )
!= $true )
| ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( in @ X4 @ X1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5854]) ).
thf(f5854,plain,
! [X2: $i,X3: $i,X0: $i > $i > $o,X1: $i,X4: $i] :
( ( $true
!= ( X0 @ X2 @ X4 ) )
| ( ( in @ X2 @ X3 )
!= $true )
| ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
| ( $true != $true )
| ( $true
!= ( in @ X4 @ X1 ) ) ),
inference(definition_unfolding,[status(thm)],[f4805,f3963]) ).
thf(f3963,plain,
dpsetconstrI = $true,
inference(cnf_transformation,[status(thm)],[f2414]) ).
thf(f4805,plain,
! [X2: $i,X3: $i,X0: $i > $i > $o,X1: $i,X4: $i] :
( ( $true
!= ( in @ X4 @ X1 ) )
| ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( X0 @ X2 @ X4 ) )
| ( ( in @ X2 @ X3 )
!= $true )
| ( dpsetconstrI != $true ) ),
inference(cnf_transformation,[status(thm)],[f3062]) ).
thf(f3062,plain,
( ( ! [X0: $i > $i > $o,X1: $i,X2: $i,X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ X4 @ X1 ) )
| ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( X0 @ X2 @ X4 ) ) )
| ( ( in @ X2 @ X3 )
!= $true ) )
| ( dpsetconstrI != $true ) )
& ( ( dpsetconstrI = $true )
| ( ( $true
= ( in @ sK1010 @ sK1007 ) )
& ( $true
!= ( in @ ( kpair @ sK1008 @ sK1010 )
@ ( dpsetconstr @ sK1009 @ sK1007
@ ^ [Y0: $i,Y1: $i] : ( sK1006 @ Y0 @ Y1 ) ) ) )
& ( $true
= ( sK1006 @ sK1008 @ sK1010 ) )
& ( ( in @ sK1008 @ sK1009 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1006,sK1007,sK1008,sK1009,sK1010])],[f3059,f3061,f3060]) ).
thf(f3060,plain,
( ? [X5: $i > $i > $o,X6: $i,X7: $i,X8: $i] :
( ? [X9: $i] :
( ( $true
= ( in @ X9 @ X6 ) )
& ( $true
!= ( in @ ( kpair @ X7 @ X9 )
@ ( dpsetconstr @ X8 @ X6
@ ^ [Y0: $i,Y1: $i] : ( X5 @ Y0 @ Y1 ) ) ) )
& ( $true
= ( X5 @ X7 @ X9 ) ) )
& ( ( in @ X7 @ X8 )
= $true ) )
=> ( ? [X9: $i] :
( ( $true
= ( in @ X9 @ sK1007 ) )
& ( $true
!= ( in @ ( kpair @ sK1008 @ X9 )
@ ( dpsetconstr @ sK1009 @ sK1007
@ ^ [Y0: $i,Y1: $i] : ( sK1006 @ Y0 @ Y1 ) ) ) )
& ( $true
= ( sK1006 @ sK1008 @ X9 ) ) )
& ( ( in @ sK1008 @ sK1009 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f3061,plain,
( ? [X9: $i] :
( ( $true
= ( in @ X9 @ sK1007 ) )
& ( $true
!= ( in @ ( kpair @ sK1008 @ X9 )
@ ( dpsetconstr @ sK1009 @ sK1007
@ ^ [Y0: $i,Y1: $i] : ( sK1006 @ Y0 @ Y1 ) ) ) )
& ( $true
= ( sK1006 @ sK1008 @ X9 ) ) )
=> ( ( $true
= ( in @ sK1010 @ sK1007 ) )
& ( $true
!= ( in @ ( kpair @ sK1008 @ sK1010 )
@ ( dpsetconstr @ sK1009 @ sK1007
@ ^ [Y0: $i,Y1: $i] : ( sK1006 @ Y0 @ Y1 ) ) ) )
& ( $true
= ( sK1006 @ sK1008 @ sK1010 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f3059,plain,
( ( ! [X0: $i > $i > $o,X1: $i,X2: $i,X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ X4 @ X1 ) )
| ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( X0 @ X2 @ X4 ) ) )
| ( ( in @ X2 @ X3 )
!= $true ) )
| ( dpsetconstrI != $true ) )
& ( ( dpsetconstrI = $true )
| ? [X5: $i > $i > $o,X6: $i,X7: $i,X8: $i] :
( ? [X9: $i] :
( ( $true
= ( in @ X9 @ X6 ) )
& ( $true
!= ( in @ ( kpair @ X7 @ X9 )
@ ( dpsetconstr @ X8 @ X6
@ ^ [Y0: $i,Y1: $i] : ( X5 @ Y0 @ Y1 ) ) ) )
& ( $true
= ( X5 @ X7 @ X9 ) ) )
& ( ( in @ X7 @ X8 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f3058]) ).
thf(f3058,plain,
( ( ! [X0: $i > $i > $o,X1: $i,X2: $i,X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ X4 @ X1 ) )
| ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( X0 @ X2 @ X4 ) ) )
| ( ( in @ X2 @ X3 )
!= $true ) )
| ( dpsetconstrI != $true ) )
& ( ( dpsetconstrI = $true )
| ? [X0: $i > $i > $o,X1: $i,X2: $i,X3: $i] :
( ? [X4: $i] :
( ( $true
= ( in @ X4 @ X1 ) )
& ( $true
!= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
& ( $true
= ( X0 @ X2 @ X4 ) ) )
& ( ( in @ X2 @ X3 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1114]) ).
thf(f1114,plain,
( ! [X0: $i > $i > $o,X1: $i,X2: $i,X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ X4 @ X1 ) )
| ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( X0 @ X2 @ X4 ) ) )
| ( ( in @ X2 @ X3 )
!= $true ) )
<=> ( dpsetconstrI = $true ) ),
inference(flattening,[status(thm)],[f1113]) ).
thf(f1113,plain,
( ! [X0: $i > $i > $o,X1: $i,X2: $i,X3: $i] :
( ! [X4: $i] :
( ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( X0 @ X2 @ X4 ) )
| ( $true
!= ( in @ X4 @ X1 ) ) )
| ( ( in @ X2 @ X3 )
!= $true ) )
<=> ( dpsetconstrI = $true ) ),
inference(ennf_transformation,[status(thm)],[f631]) ).
thf(f631,plain,
( ! [X0: $i > $i > $o,X1: $i,X2: $i,X3: $i] :
( ( ( in @ X2 @ X3 )
= $true )
=> ! [X4: $i] :
( ( $true
= ( in @ X4 @ X1 ) )
=> ( ( $true
= ( X0 @ X2 @ X4 ) )
=> ( $true
= ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) ) ) ) ) )
<=> ( dpsetconstrI = $true ) ),
inference(fool_elimination,[status(thm)],[f630]) ).
thf(f630,plain,
! [X0: $i > $i > $o,X1: $i,X2: $i,X3: $i] :
( ( ( in @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X0 @ X2 @ X4 )
=> ( in @ ( kpair @ X2 @ X4 )
@ ( dpsetconstr @ X3 @ X1
@ ^ [X5: $i,X6: $i] : ( X0 @ X5 @ X6 ) ) ) ) ) )
= dpsetconstrI ),
inference(rectify,[status(thm)],[f185]) ).
thf(f185,axiom,
! [X0: $i > $i > $o,X4: $i,X1: $i,X3: $i] :
( ( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( X0 @ X1 @ X2 )
=> ( in @ ( kpair @ X1 @ X2 )
@ ( dpsetconstr @ X3 @ X4
@ ^ [X8: $i,X10: $i] : ( X0 @ X8 @ X10 ) ) ) ) ) )
= dpsetconstrI ),
file('/export/starexec/sandbox/tmp/tmp.ExEgOH0aih/DTF2THF_12985.p',dpsetconstrI) ).
thf(f6237,plain,
( $true
!= ( in @ ( kpair @ sK641 @ sK640 )
@ ( dpsetconstr @ sK639 @ sK639
@ ^ [Y0: $i,Y1: $i] : ( in @ ( kpair @ Y1 @ Y0 ) @ sK638 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f5426]) ).
thf(f5426,plain,
( ( in @ ( kpair @ sK641 @ sK640 )
@ ( ^ [Y0: $i,Y1: $i] :
( dpsetconstr @ Y0 @ Y0
@ ^ [Y2: $i,Y3: $i] : ( in @ ( kpair @ Y3 @ Y2 ) @ Y1 ) )
@ sK639
@ sK638 ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f4131,f3198]) ).
thf(f3198,plain,
( breln1invset
= ( ^ [Y0: $i,Y1: $i] :
( dpsetconstr @ Y0 @ Y0
@ ^ [Y2: $i,Y3: $i] : ( in @ ( kpair @ Y3 @ Y2 ) @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f823]) ).
thf(f823,plain,
( breln1invset
= ( ^ [Y0: $i,Y1: $i] :
( dpsetconstr @ Y0 @ Y0
@ ^ [Y2: $i,Y3: $i] : ( in @ ( kpair @ Y3 @ Y2 ) @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f822]) ).
thf(f822,plain,
( ( ^ [X0: $i,X1: $i] :
( dpsetconstr @ X0 @ X0
@ ^ [X2: $i,X3: $i] : ( in @ ( kpair @ X3 @ X2 ) @ X1 ) ) )
= breln1invset ),
inference(rectify,[status(thm)],[f296]) ).
thf(f296,axiom,
( ( ^ [X3: $i,X13: $i] :
( dpsetconstr @ X3 @ X3
@ ^ [X1: $i,X2: $i] : ( in @ ( kpair @ X2 @ X1 ) @ X13 ) ) )
= breln1invset ),
file('/export/starexec/sandbox/tmp/tmp.ExEgOH0aih/DTF2THF_12985.p',breln1invset) ).
thf(f4131,plain,
( $true
!= ( in @ ( kpair @ sK641 @ sK640 ) @ ( breln1invset @ sK639 @ sK638 ) ) ),
inference(cnf_transformation,[status(thm)],[f2414]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU776^1 : TPTP v9.2.1. Released v3.7.0.
% 0.10/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.32 % Computer : n001.cluster.edu
% 0.16/0.32 % Model : x86_64 x86_64
% 0.16/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.32 % Memory : 8042.1875MB
% 0.16/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.32 % CPULimit : 300
% 0.16/0.32 % WCLimit : 300
% 0.16/0.32 % DateTime : Tue Feb 24 12:34:20 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
% 0.29/0.44 ---- Original DTF file ---
% 0.29/0.44 thf(spec,logic,$$dhol).
% 0.29/0.44 %------------------------------------------------------------------------------
% 0.29/0.44 % File : SEU776^1 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.44 % Domain : Set Theory
% 0.29/0.44 % Problem : Binary Relations on a Set
% 0.29/0.44 % Version : Especial.
% 0.29/0.44 % English : (! A:i.! R:i.breln1 A R -> (! x:i.in x A -> (! y:i.in y A ->
% 0.29/0.44 % in (kpair x y) R -> in (kpair y x) (breln1invset A R))))
% 0.29/0.44
% 0.29/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.44 % Source : [Bro08]
% 0.29/0.44 % Names : ZFC278g [Bro08]
% 0.29/0.44
% 0.29/0.44 % Status : Theorem
% 0.29/0.44 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.62 v8.1.0, 0.55 v7.5.0, 0.43 v7.4.0, 0.56 v7.2.0, 0.50 v7.1.0, 0.62 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.86 v5.5.0, 0.83 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.29/0.44 % Syntax : Number of formulae : 637 ( 297 unt; 339 typ; 297 def)
% 0.29/0.44 % Number of atoms : 1793 ( 427 equ; 0 cnn)
% 0.29/0.44 % Maximal formula atoms : 297 ( 6 avg)
% 0.29/0.44 % Number of connectives : 3871 ( 73 ~; 12 |; 46 &;2807 @)
% 0.29/0.44 % ( 14 <=>; 919 =>; 0 <=; 0 <~>)
% 0.29/0.44 % Maximal formula depth : 305 ( 2 avg)
% 0.29/0.44 % Number of types : 2 ( 0 usr)
% 0.29/0.44 % Number of type conns : 154 ( 154 >; 0 *; 0 +; 0 <<)
% 0.29/0.44 % Number of symbols : 342 ( 339 usr; 296 con; 0-4 aty)
% 0.29/0.44 % Number of variables : 1031 ( 80 ^; 915 !; 36 ?;1031 :)
% 0.29/0.44 % SPC : TH0_THM_EQU_NAR
% 0.29/0.44
% 0.29/0.44 % Comments : http://mathgate.info/detsetitem.php?id=336
% 0.29/0.44 %------------------------------------------------------------------------------
% 0.29/0.44 thf(in_type,type,
% 0.29/0.44 in: $i > $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(exu_type,type,
% 0.29/0.44 exu: ( $i > $o ) > $o ).
% 0.29/0.44
% 0.29/0.44 thf(exu,definition,
% 0.29/0.44 ( exu
% 0.29/0.44 = ( ^ [Xphi: $i > $o] :
% 0.29/0.44 ? [Xx: $i] :
% 0.29/0.44 ( ( Xphi @ Xx )
% 0.29/0.44 & ! [Xy: $i] :
% 0.29/0.44 ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setextAx_type,type,
% 0.29/0.44 setextAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setextAx,definition,
% 0.29/0.44 ( setextAx
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 <=> ( in @ Xx @ B ) )
% 0.29/0.44 => ( A = B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptyset_type,type,
% 0.29/0.44 emptyset: $i ).
% 0.29/0.44
% 0.29/0.44 thf(emptysetAx_type,type,
% 0.29/0.44 emptysetAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptysetAx,definition,
% 0.29/0.44 ( emptysetAx
% 0.29/0.44 = ( ! [Xx: $i] :
% 0.29/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoin_type,type,
% 0.29/0.44 setadjoin: $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinAx_type,type,
% 0.29/0.44 setadjoinAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinAx,definition,
% 0.29/0.44 ( setadjoinAx
% 0.29/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.44 <=> ( ( Xy = Xx )
% 0.29/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powerset_type,type,
% 0.29/0.44 powerset: $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(powersetAx_type,type,
% 0.29/0.44 powersetAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetAx,definition,
% 0.29/0.44 ( powersetAx
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.44 <=> ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setunion_type,type,
% 0.29/0.44 setunion: $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(setunionAx_type,type,
% 0.29/0.44 setunionAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setunionAx,definition,
% 0.29/0.44 ( setunionAx
% 0.29/0.44 = ( ! [A: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.44 <=> ? [B: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(omega_type,type,
% 0.29/0.44 omega: $i ).
% 0.29/0.44
% 0.29/0.44 thf(omega0Ax_type,type,
% 0.29/0.44 omega0Ax: $o ).
% 0.29/0.44
% 0.29/0.44 thf(omega0Ax,definition,
% 0.29/0.44 ( omega0Ax
% 0.29/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(omegaSAx_type,type,
% 0.29/0.44 omegaSAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(omegaSAx,definition,
% 0.29/0.44 ( omegaSAx
% 0.29/0.44 = ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ omega )
% 0.29/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(omegaIndAx_type,type,
% 0.29/0.44 omegaIndAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(omegaIndAx,definition,
% 0.29/0.44 ( omegaIndAx
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ( ( in @ emptyset @ A )
% 0.29/0.44 & ! [Xx: $i] :
% 0.29/0.44 ( ( ( in @ Xx @ omega )
% 0.29/0.44 & ( in @ Xx @ A ) )
% 0.29/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ omega )
% 0.29/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(replAx_type,type,
% 0.29/0.44 replAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(replAx,definition,
% 0.29/0.44 ( replAx
% 0.29/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( exu
% 0.29/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.29/0.44 => ? [B: $i] :
% 0.29/0.44 ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 <=> ? [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(foundationAx_type,type,
% 0.29/0.44 foundationAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(foundationAx,definition,
% 0.29/0.44 ( foundationAx
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.44 => ? [B: $i] :
% 0.29/0.44 ( ( in @ B @ A )
% 0.29/0.44 & ~ ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(wellorderingAx_type,type,
% 0.29/0.44 wellorderingAx: $o ).
% 0.29/0.44
% 0.29/0.44 thf(wellorderingAx,definition,
% 0.29/0.44 ( wellorderingAx
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ? [B: $i] :
% 0.29/0.44 ( ! [C: $i] :
% 0.29/0.44 ( ( in @ C @ B )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ C )
% 0.29/0.44 => ( in @ Xx @ A ) ) )
% 0.29/0.44 & ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( ( in @ Xx @ A )
% 0.29/0.44 & ( in @ Xy @ A ) )
% 0.29/0.44 => ( ! [C: $i] :
% 0.29/0.44 ( ( in @ C @ B )
% 0.29/0.44 => ( ( in @ Xx @ C )
% 0.29/0.44 <=> ( in @ Xy @ C ) ) )
% 0.29/0.44 => ( Xx = Xy ) ) )
% 0.29/0.44 & ! [C: $i,D: $i] :
% 0.29/0.44 ( ( ( in @ C @ B )
% 0.29/0.44 & ( in @ D @ B ) )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ C )
% 0.29/0.44 => ( in @ Xx @ D ) )
% 0.29/0.44 | ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ D )
% 0.29/0.44 => ( in @ Xx @ C ) ) ) )
% 0.29/0.44 & ! [C: $i] :
% 0.29/0.44 ( ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ C )
% 0.29/0.44 => ( in @ Xx @ A ) )
% 0.29/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.29/0.44 => ? [D: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ D @ B )
% 0.29/0.44 & ( in @ Xx @ C )
% 0.29/0.44 & ~ ? [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ D )
% 0.29/0.44 & ( in @ Xy @ C ) )
% 0.29/0.44 & ! [E: $i] :
% 0.29/0.44 ( ( in @ E @ B )
% 0.29/0.44 => ( ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ E )
% 0.29/0.44 => ( in @ Xy @ D ) )
% 0.29/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(descr_type,type,
% 0.29/0.44 descr: ( $i > $o ) > $i ).
% 0.29/0.44
% 0.29/0.44 thf(descrp_type,type,
% 0.29/0.44 descrp: $o ).
% 0.29/0.44
% 0.29/0.44 thf(descrp,definition,
% 0.29/0.44 ( descrp
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ( Xphi
% 0.29/0.44 @ ( descr
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstr_type,type,
% 0.29/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstrI_type,type,
% 0.29/0.44 dsetconstrI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstrI,definition,
% 0.29/0.44 ( dsetconstrI
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 => ( in @ Xx
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstrEL_type,type,
% 0.29/0.44 dsetconstrEL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstrEL,definition,
% 0.29/0.44 ( dsetconstrEL
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstrER_type,type,
% 0.29/0.44 dsetconstrER: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstrER,definition,
% 0.29/0.44 ( dsetconstrER
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exuE1_type,type,
% 0.29/0.44 exuE1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exuE1,definition,
% 0.29/0.44 ( exuE1
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ? [Xx: $i] :
% 0.29/0.44 ( ( Xphi @ Xx )
% 0.29/0.44 & ! [Xy: $i] :
% 0.29/0.44 ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(prop2set_type,type,
% 0.29/0.44 prop2set: $o > $i ).
% 0.29/0.44
% 0.29/0.44 thf(prop2setE_type,type,
% 0.29/0.44 prop2setE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(prop2setE,definition,
% 0.29/0.44 ( prop2setE
% 0.29/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.29/0.44 => Xphi ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptysetE_type,type,
% 0.29/0.44 emptysetE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptysetE,definition,
% 0.29/0.44 ( emptysetE
% 0.29/0.44 = ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ emptyset )
% 0.29/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptysetimpfalse_type,type,
% 0.29/0.44 emptysetimpfalse: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptysetimpfalse,definition,
% 0.29/0.44 ( emptysetimpfalse
% 0.29/0.44 = ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ emptyset )
% 0.29/0.44 => $false ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(notinemptyset_type,type,
% 0.29/0.44 notinemptyset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(notinemptyset,definition,
% 0.29/0.44 ( notinemptyset
% 0.29/0.44 = ( ! [Xx: $i] :
% 0.29/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exuE3e_type,type,
% 0.29/0.44 exuE3e: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exuE3e,definition,
% 0.29/0.44 ( exuE3e
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setext_type,type,
% 0.29/0.44 setext: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setext,definition,
% 0.29/0.44 ( setext
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ Xx @ B ) )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ A ) )
% 0.29/0.44 => ( A = B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptyI_type,type,
% 0.29/0.44 emptyI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptyI,definition,
% 0.29/0.44 ( emptyI
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ~ ( in @ Xx @ A )
% 0.29/0.44 => ( A = emptyset ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(noeltsimpempty_type,type,
% 0.29/0.44 noeltsimpempty: $o ).
% 0.29/0.44
% 0.29/0.44 thf(noeltsimpempty,definition,
% 0.29/0.44 ( noeltsimpempty
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ~ ( in @ Xx @ A )
% 0.29/0.44 => ( A = emptyset ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setbeta_type,type,
% 0.29/0.44 setbeta: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setbeta,definition,
% 0.29/0.44 ( setbeta
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(nonempty_type,type,
% 0.29/0.44 nonempty: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(nonempty,definition,
% 0.29/0.44 ( nonempty
% 0.29/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(nonemptyE1_type,type,
% 0.29/0.44 nonemptyE1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(nonemptyE1,definition,
% 0.29/0.44 ( nonemptyE1
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ( nonempty @ A )
% 0.29/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(nonemptyI_type,type,
% 0.29/0.44 nonemptyI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(nonemptyI,definition,
% 0.29/0.44 ( nonemptyI
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 => ( nonempty
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(nonemptyI1_type,type,
% 0.29/0.44 nonemptyI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(nonemptyI1,definition,
% 0.29/0.44 ( nonemptyI1
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinIL_type,type,
% 0.29/0.44 setadjoinIL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinIL,definition,
% 0.29/0.44 ( setadjoinIL
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptyinunitempty_type,type,
% 0.29/0.44 emptyinunitempty: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptyinunitempty,definition,
% 0.29/0.44 ( emptyinunitempty
% 0.29/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinIR_type,type,
% 0.29/0.44 setadjoinIR: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinIR,definition,
% 0.29/0.44 ( setadjoinIR
% 0.29/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinE_type,type,
% 0.29/0.44 setadjoinE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinE,definition,
% 0.29/0.44 ( setadjoinE
% 0.29/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.44 => ! [Xphi: $o] :
% 0.29/0.44 ( ( ( Xy = Xx )
% 0.29/0.44 => Xphi )
% 0.29/0.44 => ( ( ( in @ Xy @ A )
% 0.29/0.44 => Xphi )
% 0.29/0.44 => Xphi ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinOr_type,type,
% 0.29/0.44 setadjoinOr: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinOr,definition,
% 0.29/0.44 ( setadjoinOr
% 0.29/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.44 => ( ( Xy = Xx )
% 0.29/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setoftrueEq_type,type,
% 0.29/0.44 setoftrueEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setoftrueEq,definition,
% 0.29/0.44 ( setoftrueEq
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xx: $i] : $true )
% 0.29/0.44 = A ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powersetI_type,type,
% 0.29/0.44 powersetI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetI,definition,
% 0.29/0.44 ( powersetI
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ A ) )
% 0.29/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptyinPowerset_type,type,
% 0.29/0.44 emptyinPowerset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptyinPowerset,definition,
% 0.29/0.44 ( emptyinPowerset
% 0.29/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptyInPowerset_type,type,
% 0.29/0.44 emptyInPowerset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptyInPowerset,definition,
% 0.29/0.44 ( emptyInPowerset
% 0.29/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powersetE_type,type,
% 0.29/0.44 powersetE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetE,definition,
% 0.29/0.44 ( powersetE
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.44 => ( ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setunionI_type,type,
% 0.29/0.44 setunionI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setunionI,definition,
% 0.29/0.44 ( setunionI
% 0.29/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 => ( ( in @ B @ A )
% 0.29/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setunionE_type,type,
% 0.29/0.44 setunionE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setunionE,definition,
% 0.29/0.44 ( setunionE
% 0.29/0.44 = ( ! [A: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.44 => ! [Xphi: $o] :
% 0.29/0.44 ( ! [B: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 => ( ( in @ B @ A )
% 0.29/0.44 => Xphi ) )
% 0.29/0.44 => Xphi ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subPowSU_type,type,
% 0.29/0.44 subPowSU: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subPowSU,definition,
% 0.29/0.44 ( subPowSU
% 0.29/0.44 = ( ! [A: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exuE2_type,type,
% 0.29/0.44 exuE2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exuE2,definition,
% 0.29/0.44 ( exuE2
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ? [Xx: $i] :
% 0.29/0.44 ! [Xy: $i] :
% 0.29/0.44 ( ( Xphi @ Xy )
% 0.29/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(nonemptyImpWitness_type,type,
% 0.29/0.44 nonemptyImpWitness: $o ).
% 0.29/0.44
% 0.29/0.44 thf(nonemptyImpWitness,definition,
% 0.29/0.44 ( nonemptyImpWitness
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ( nonempty @ A )
% 0.29/0.44 => ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & $true ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(uniqinunit_type,type,
% 0.29/0.44 uniqinunit: $o ).
% 0.29/0.44
% 0.29/0.44 thf(uniqinunit,definition,
% 0.29/0.44 ( uniqinunit
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(notinsingleton_type,type,
% 0.29/0.44 notinsingleton: $o ).
% 0.29/0.44
% 0.29/0.44 thf(notinsingleton,definition,
% 0.29/0.44 ( notinsingleton
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx != Xy )
% 0.29/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(eqinunit_type,type,
% 0.29/0.44 eqinunit: $o ).
% 0.29/0.44
% 0.29/0.44 thf(eqinunit,definition,
% 0.29/0.44 ( eqinunit
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx = Xy )
% 0.29/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(singletonsswitch_type,type,
% 0.29/0.44 singletonsswitch: $o ).
% 0.29/0.44
% 0.29/0.44 thf(singletonsswitch,definition,
% 0.29/0.44 ( singletonsswitch
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(upairsetE_type,type,
% 0.29/0.44 upairsetE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairsetE,definition,
% 0.29/0.44 ( upairsetE
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.44 => ( ( Xz = Xx )
% 0.29/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(upairsetIL_type,type,
% 0.29/0.44 upairsetIL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairsetIL,definition,
% 0.29/0.44 ( upairsetIL
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(upairsetIR_type,type,
% 0.29/0.44 upairsetIR: $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairsetIR,definition,
% 0.29/0.44 ( upairsetIR
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptyE1_type,type,
% 0.29/0.44 emptyE1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptyE1,definition,
% 0.29/0.44 ( emptyE1
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ( Xphi @ Xx ) )
% 0.29/0.44 => ( ( ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 = emptyset )
% 0.29/0.44 => $false ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(vacuousDall_type,type,
% 0.29/0.44 vacuousDall: $o ).
% 0.29/0.44
% 0.29/0.44 thf(vacuousDall,definition,
% 0.29/0.44 ( vacuousDall
% 0.29/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ emptyset )
% 0.29/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(quantDeMorgan1_type,type,
% 0.29/0.44 quantDeMorgan1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(quantDeMorgan1,definition,
% 0.29/0.44 ( quantDeMorgan1
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ~ ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( Xphi @ Xx ) )
% 0.29/0.44 => ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(quantDeMorgan2_type,type,
% 0.29/0.44 quantDeMorgan2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(quantDeMorgan2,definition,
% 0.29/0.44 ( quantDeMorgan2
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ~ ( Xphi @ Xx ) )
% 0.29/0.44 => ~ ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(quantDeMorgan3_type,type,
% 0.29/0.44 quantDeMorgan3: $o ).
% 0.29/0.44
% 0.29/0.44 thf(quantDeMorgan3,definition,
% 0.29/0.44 ( quantDeMorgan3
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ~ ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ( Xphi @ Xx ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(quantDeMorgan4_type,type,
% 0.29/0.44 quantDeMorgan4: $o ).
% 0.29/0.44
% 0.29/0.44 thf(quantDeMorgan4,definition,
% 0.29/0.44 ( quantDeMorgan4
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ~ ( Xphi @ Xx ) )
% 0.29/0.44 => ~ ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(prop2setI_type,type,
% 0.29/0.44 prop2setI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(prop2setI,definition,
% 0.29/0.44 ( prop2setI
% 0.29/0.44 = ( ! [Xphi: $o] :
% 0.29/0.44 ( Xphi
% 0.29/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(set2prop_type,type,
% 0.29/0.44 set2prop: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(prop2set2propI_type,type,
% 0.29/0.44 prop2set2propI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(prop2set2propI,definition,
% 0.29/0.44 ( prop2set2propI
% 0.29/0.44 = ( ! [Xphi: $o] :
% 0.29/0.44 ( Xphi
% 0.29/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(notdexE_type,type,
% 0.29/0.44 notdexE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(notdexE,definition,
% 0.29/0.44 ( notdexE
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ~ ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ( Xphi @ Xx ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(notdallE_type,type,
% 0.29/0.44 notdallE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(notdallE,definition,
% 0.29/0.44 ( notdallE
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ~ ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( Xphi @ Xx ) )
% 0.29/0.44 => ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exuI1_type,type,
% 0.29/0.44 exuI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exuI1,definition,
% 0.29/0.44 ( exuI1
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ? [Xx: $i] :
% 0.29/0.44 ( ( Xphi @ Xx )
% 0.29/0.44 & ! [Xy: $i] :
% 0.29/0.44 ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) )
% 0.29/0.44 => ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exuI3_type,type,
% 0.29/0.44 exuI3: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exuI3,definition,
% 0.29/0.44 ( exuI3
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.29/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xphi @ Xx )
% 0.29/0.44 => ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) )
% 0.29/0.44 => ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exuI2_type,type,
% 0.29/0.44 exuI2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exuI2,definition,
% 0.29/0.44 ( exuI2
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ? [Xx: $i] :
% 0.29/0.44 ! [Xy: $i] :
% 0.29/0.44 ( ( Xphi @ Xy )
% 0.29/0.44 <=> ( Xy = Xx ) )
% 0.29/0.44 => ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(inCongP_type,type,
% 0.29/0.44 inCongP: $o ).
% 0.29/0.44
% 0.29/0.44 thf(inCongP,definition,
% 0.29/0.44 ( inCongP
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( A = B )
% 0.29/0.44 => ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx = Xy )
% 0.29/0.44 => ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(in__Cong_type,type,
% 0.29/0.44 in__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(in__Cong,definition,
% 0.29/0.44 ( in__Cong
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( A = B )
% 0.29/0.44 => ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx = Xy )
% 0.29/0.44 => ( ( in @ Xx @ A )
% 0.29/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exuE3u_type,type,
% 0.29/0.44 exuE3u: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exuE3u,definition,
% 0.29/0.44 ( exuE3u
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xphi @ Xx )
% 0.29/0.44 => ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exu__Cong_type,type,
% 0.29/0.44 exu__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exu__Cong,definition,
% 0.29/0.44 ( exu__Cong
% 0.29/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx = Xy )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.44 => ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 <=> ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptyset__Cong_type,type,
% 0.29/0.44 emptyset__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptyset__Cong,definition,
% 0.29/0.44 ( emptyset__Cong
% 0.29/0.44 = ( emptyset = emptyset ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoin__Cong_type,type,
% 0.29/0.44 setadjoin__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoin__Cong,definition,
% 0.29/0.44 ( setadjoin__Cong
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx = Xy )
% 0.29/0.44 => ! [Xz: $i,Xu: $i] :
% 0.29/0.44 ( ( Xz = Xu )
% 0.29/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.29/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powerset__Cong_type,type,
% 0.29/0.44 powerset__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powerset__Cong,definition,
% 0.29/0.44 ( powerset__Cong
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( A = B )
% 0.29/0.44 => ( ( powerset @ A )
% 0.29/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setunion__Cong_type,type,
% 0.29/0.44 setunion__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setunion__Cong,definition,
% 0.29/0.44 ( setunion__Cong
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( A = B )
% 0.29/0.44 => ( ( setunion @ A )
% 0.29/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(omega__Cong_type,type,
% 0.29/0.44 omega__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(omega__Cong,definition,
% 0.29/0.44 ( omega__Cong
% 0.29/0.44 = ( omega = omega ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(exuEu_type,type,
% 0.29/0.44 exuEu: $o ).
% 0.29/0.44
% 0.29/0.44 thf(exuEu,definition,
% 0.29/0.44 ( exuEu
% 0.29/0.44 = ( ! [Xphi: $i > $o] :
% 0.29/0.44 ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xphi @ Xx )
% 0.29/0.44 => ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(descr__Cong_type,type,
% 0.29/0.44 descr__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(descr__Cong,definition,
% 0.29/0.44 ( descr__Cong
% 0.29/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx = Xy )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.44 => ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ( ( exu
% 0.29/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.29/0.44 => ( ( descr
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 = ( descr
% 0.29/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstr__Cong_type,type,
% 0.29/0.44 dsetconstr__Cong: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dsetconstr__Cong,definition,
% 0.29/0.44 ( dsetconstr__Cong
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( A = B )
% 0.29/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( Xx = Xy )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.29/0.44 => ( ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 = ( dsetconstr @ B
% 0.29/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subset_type,type,
% 0.29/0.44 subset: $i > $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(disjoint_type,type,
% 0.29/0.44 disjoint: $i > $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(setsmeet_type,type,
% 0.29/0.44 setsmeet: $i > $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetI1_type,type,
% 0.29/0.44 subsetI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetI1,definition,
% 0.29/0.44 ( subsetI1
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ Xx @ B ) )
% 0.29/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(eqimpsubset2_type,type,
% 0.29/0.44 eqimpsubset2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(eqimpsubset2,definition,
% 0.29/0.44 ( eqimpsubset2
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( A = B )
% 0.29/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(eqimpsubset1_type,type,
% 0.29/0.44 eqimpsubset1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(eqimpsubset1,definition,
% 0.29/0.44 ( eqimpsubset1
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( A = B )
% 0.29/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subsetI2_type,type,
% 0.29/0.44 subsetI2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetI2,definition,
% 0.29/0.44 ( subsetI2
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ Xx @ B ) )
% 0.29/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(emptysetsubset_type,type,
% 0.29/0.44 emptysetsubset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(emptysetsubset,definition,
% 0.29/0.44 ( emptysetsubset
% 0.29/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subsetE_type,type,
% 0.29/0.44 subsetE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetE,definition,
% 0.29/0.44 ( subsetE
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subsetE2_type,type,
% 0.29/0.44 subsetE2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetE2,definition,
% 0.29/0.44 ( subsetE2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( ~ ( in @ Xx @ B )
% 0.29/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(notsubsetI_type,type,
% 0.29/0.44 notsubsetI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(notsubsetI,definition,
% 0.29/0.44 ( notsubsetI
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ B )
% 0.29/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(notequalI1_type,type,
% 0.29/0.44 notequalI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(notequalI1,definition,
% 0.29/0.44 ( notequalI1
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ~ ( subset @ A @ B )
% 0.29/0.44 => ( A != B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(notequalI2_type,type,
% 0.29/0.44 notequalI2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(notequalI2,definition,
% 0.29/0.44 ( notequalI2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ B )
% 0.29/0.44 => ( A != B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subsetRefl_type,type,
% 0.29/0.44 subsetRefl: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetRefl,definition,
% 0.29/0.44 ( subsetRefl
% 0.29/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subsetTrans_type,type,
% 0.29/0.44 subsetTrans: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetTrans,definition,
% 0.29/0.44 ( subsetTrans
% 0.29/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( ( subset @ B @ C )
% 0.29/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinSub_type,type,
% 0.29/0.44 setadjoinSub: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinSub,definition,
% 0.29/0.44 ( setadjoinSub
% 0.29/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinSub2_type,type,
% 0.29/0.44 setadjoinSub2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setadjoinSub2,definition,
% 0.29/0.44 ( setadjoinSub2
% 0.29/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subset2powerset_type,type,
% 0.29/0.44 subset2powerset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subset2powerset,definition,
% 0.29/0.44 ( subset2powerset
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setextsub_type,type,
% 0.29/0.44 setextsub: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setextsub,definition,
% 0.29/0.44 ( setextsub
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( ( subset @ B @ A )
% 0.29/0.44 => ( A = B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subsetemptysetimpeq_type,type,
% 0.29/0.44 subsetemptysetimpeq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetemptysetimpeq,definition,
% 0.29/0.44 ( subsetemptysetimpeq
% 0.29/0.44 = ( ! [A: $i] :
% 0.29/0.44 ( ( subset @ A @ emptyset )
% 0.29/0.44 => ( A = emptyset ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powersetI1_type,type,
% 0.29/0.44 powersetI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetI1,definition,
% 0.29/0.44 ( powersetI1
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( subset @ B @ A )
% 0.29/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powersetE1_type,type,
% 0.29/0.44 powersetE1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetE1,definition,
% 0.29/0.44 ( powersetE1
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(inPowerset_type,type,
% 0.29/0.44 inPowerset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(inPowerset,definition,
% 0.29/0.44 ( inPowerset
% 0.29/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powersetsubset_type,type,
% 0.29/0.44 powersetsubset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetsubset,definition,
% 0.29/0.44 ( powersetsubset
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(sepInPowerset_type,type,
% 0.29/0.44 sepInPowerset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(sepInPowerset,definition,
% 0.29/0.44 ( sepInPowerset
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( in
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(sepSubset_type,type,
% 0.29/0.44 sepSubset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(sepSubset,definition,
% 0.29/0.44 ( sepSubset
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( subset
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 @ A ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunion_type,type,
% 0.29/0.44 binunion: $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(binunionIL_type,type,
% 0.29/0.44 binunionIL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionIL,definition,
% 0.29/0.44 ( binunionIL
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(upairset2IR_type,type,
% 0.29/0.44 upairset2IR: $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairset2IR,definition,
% 0.29/0.44 ( upairset2IR
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionIR_type,type,
% 0.29/0.44 binunionIR: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionIR,definition,
% 0.29/0.44 ( binunionIR
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionEcases_type,type,
% 0.29/0.44 binunionEcases: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionEcases,definition,
% 0.29/0.44 ( binunionEcases
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.29/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.29/0.44 => ( ( ( in @ Xx @ A )
% 0.29/0.44 => Xphi )
% 0.29/0.44 => ( ( ( in @ Xx @ B )
% 0.29/0.44 => Xphi )
% 0.29/0.44 => Xphi ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionE_type,type,
% 0.29/0.44 binunionE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionE,definition,
% 0.29/0.44 ( binunionE
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.29/0.44 => ( ( in @ Xx @ A )
% 0.29/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionLsub_type,type,
% 0.29/0.44 binunionLsub: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionLsub,definition,
% 0.29/0.44 ( binunionLsub
% 0.29/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionRsub_type,type,
% 0.29/0.44 binunionRsub: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionRsub,definition,
% 0.29/0.44 ( binunionRsub
% 0.29/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersect_type,type,
% 0.29/0.44 binintersect: $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectI_type,type,
% 0.29/0.44 binintersectI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectI,definition,
% 0.29/0.44 ( binintersectI
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset5_type,type,
% 0.29/0.44 binintersectSubset5: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset5,definition,
% 0.29/0.44 ( binintersectSubset5
% 0.29/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.44 ( ( subset @ C @ A )
% 0.29/0.44 => ( ( subset @ C @ B )
% 0.29/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectEL_type,type,
% 0.29/0.44 binintersectEL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectEL,definition,
% 0.29/0.44 ( binintersectEL
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.29/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectLsub_type,type,
% 0.29/0.44 binintersectLsub: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectLsub,definition,
% 0.29/0.44 ( binintersectLsub
% 0.29/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset2_type,type,
% 0.29/0.44 binintersectSubset2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset2,definition,
% 0.29/0.44 ( binintersectSubset2
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( ( binintersect @ A @ B )
% 0.29/0.44 = A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset3_type,type,
% 0.29/0.44 binintersectSubset3: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset3,definition,
% 0.29/0.44 ( binintersectSubset3
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( ( binintersect @ A @ B )
% 0.29/0.44 = B )
% 0.29/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectER_type,type,
% 0.29/0.44 binintersectER: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectER,definition,
% 0.29/0.44 ( binintersectER
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.29/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(disjointsetsI1_type,type,
% 0.29/0.44 disjointsetsI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(disjointsetsI1,definition,
% 0.29/0.44 ( disjointsetsI1
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ~ ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ( in @ Xx @ B ) )
% 0.29/0.44 => ( ( binintersect @ A @ B )
% 0.29/0.44 = emptyset ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectRsub_type,type,
% 0.29/0.44 binintersectRsub: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectRsub,definition,
% 0.29/0.44 ( binintersectRsub
% 0.29/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset4_type,type,
% 0.29/0.44 binintersectSubset4: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset4,definition,
% 0.29/0.44 ( binintersectSubset4
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( subset @ B @ A )
% 0.29/0.44 => ( ( binintersect @ A @ B )
% 0.29/0.44 = B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset1_type,type,
% 0.29/0.44 binintersectSubset1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectSubset1,definition,
% 0.29/0.44 ( binintersectSubset1
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( ( binintersect @ A @ B )
% 0.29/0.44 = A )
% 0.29/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(bs114d_type,type,
% 0.29/0.44 bs114d: $o ).
% 0.29/0.44
% 0.29/0.44 thf(bs114d,definition,
% 0.29/0.44 ( bs114d
% 0.29/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.29/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(regular_type,type,
% 0.29/0.44 regular: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminus_type,type,
% 0.29/0.44 setminus: $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(setminusI_type,type,
% 0.29/0.44 setminusI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusI,definition,
% 0.29/0.44 ( setminusI
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusEL_type,type,
% 0.29/0.44 setminusEL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusEL,definition,
% 0.29/0.44 ( setminusEL
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusER_type,type,
% 0.29/0.44 setminusER: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusER,definition,
% 0.29/0.44 ( setminusER
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusSubset2_type,type,
% 0.29/0.44 setminusSubset2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusSubset2,definition,
% 0.29/0.44 ( setminusSubset2
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( subset @ A @ B )
% 0.29/0.44 => ( ( setminus @ A @ B )
% 0.29/0.44 = emptyset ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusERneg_type,type,
% 0.29/0.44 setminusERneg: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusERneg,definition,
% 0.29/0.44 ( setminusERneg
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.44 => ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusELneg_type,type,
% 0.29/0.44 setminusELneg: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusELneg,definition,
% 0.29/0.44 ( setminusELneg
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.44 => ( ~ ( in @ Xx @ B )
% 0.29/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusILneg_type,type,
% 0.29/0.44 setminusILneg: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusILneg,definition,
% 0.29/0.44 ( setminusILneg
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ~ ( in @ Xx @ A )
% 0.29/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusIRneg_type,type,
% 0.29/0.44 setminusIRneg: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusIRneg,definition,
% 0.29/0.44 ( setminusIRneg
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ B )
% 0.29/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusLsub_type,type,
% 0.29/0.44 setminusLsub: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusLsub,definition,
% 0.29/0.44 ( setminusLsub
% 0.29/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusSubset1_type,type,
% 0.29/0.44 setminusSubset1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusSubset1,definition,
% 0.29/0.44 ( setminusSubset1
% 0.29/0.44 = ( ! [A: $i,B: $i] :
% 0.29/0.44 ( ( ( setminus @ A @ B )
% 0.29/0.44 = emptyset )
% 0.29/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(symdiff_type,type,
% 0.29/0.44 symdiff: $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffE_type,type,
% 0.29/0.44 symdiffE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffE,definition,
% 0.29/0.44 ( symdiffE
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.29/0.44 => ! [Xphi: $o] :
% 0.29/0.44 ( ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ B )
% 0.29/0.44 => Xphi ) )
% 0.29/0.44 => ( ( ~ ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ B )
% 0.29/0.44 => Xphi ) )
% 0.29/0.44 => Xphi ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffI1_type,type,
% 0.29/0.44 symdiffI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffI1,definition,
% 0.29/0.44 ( symdiffI1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffI2_type,type,
% 0.29/0.44 symdiffI2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffI2,definition,
% 0.29/0.44 ( symdiffI2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ~ ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ B )
% 0.29/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffIneg1_type,type,
% 0.29/0.44 symdiffIneg1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffIneg1,definition,
% 0.29/0.44 ( symdiffIneg1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ B )
% 0.29/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffIneg2_type,type,
% 0.29/0.44 symdiffIneg2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(symdiffIneg2,definition,
% 0.29/0.44 ( symdiffIneg2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ~ ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ B )
% 0.29/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(iskpair_type,type,
% 0.29/0.44 iskpair: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(secondinupair_type,type,
% 0.29/0.44 secondinupair: $o ).
% 0.29/0.44
% 0.29/0.44 thf(secondinupair,definition,
% 0.29/0.44 ( secondinupair
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairIL_type,type,
% 0.29/0.44 setukpairIL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairIL,definition,
% 0.29/0.44 ( setukpairIL
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairIR_type,type,
% 0.29/0.44 setukpairIR: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairIR,definition,
% 0.29/0.44 ( setukpairIR
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(kpairiskpair_type,type,
% 0.29/0.44 kpairiskpair: $o ).
% 0.29/0.44
% 0.29/0.44 thf(kpairiskpair,definition,
% 0.29/0.44 ( kpairiskpair
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(kpair_type,type,
% 0.29/0.44 kpair: $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(kpairp_type,type,
% 0.29/0.44 kpairp: $o ).
% 0.29/0.44
% 0.29/0.44 thf(kpairp,definition,
% 0.29/0.44 ( kpairp
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprod_type,type,
% 0.29/0.44 cartprod: $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(singletonsubset_type,type,
% 0.29/0.44 singletonsubset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(singletonsubset,definition,
% 0.29/0.44 ( singletonsubset
% 0.29/0.44 = ( ! [A: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(singletoninpowerset_type,type,
% 0.29/0.44 singletoninpowerset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(singletoninpowerset,definition,
% 0.29/0.44 ( singletoninpowerset
% 0.29/0.44 = ( ! [A: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(singletoninpowunion_type,type,
% 0.29/0.44 singletoninpowunion: $o ).
% 0.29/0.44
% 0.29/0.44 thf(singletoninpowunion,definition,
% 0.29/0.44 ( singletoninpowunion
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(upairset2E_type,type,
% 0.29/0.44 upairset2E: $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairset2E,definition,
% 0.29/0.44 ( upairset2E
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.44 => ( ( Xz = Xx )
% 0.29/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(upairsubunion_type,type,
% 0.29/0.44 upairsubunion: $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairsubunion,definition,
% 0.29/0.44 ( upairsubunion
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(upairinpowunion_type,type,
% 0.29/0.44 upairinpowunion: $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairinpowunion,definition,
% 0.29/0.44 ( upairinpowunion
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ubforcartprodlem1_type,type,
% 0.29/0.44 ubforcartprodlem1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ubforcartprodlem1,definition,
% 0.29/0.44 ( ubforcartprodlem1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ubforcartprodlem2_type,type,
% 0.29/0.44 ubforcartprodlem2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ubforcartprodlem2,definition,
% 0.29/0.44 ( ubforcartprodlem2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ubforcartprodlem3_type,type,
% 0.29/0.44 ubforcartprodlem3: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ubforcartprodlem3,definition,
% 0.29/0.44 ( ubforcartprodlem3
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodpairin_type,type,
% 0.29/0.44 cartprodpairin: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodpairin,definition,
% 0.29/0.44 ( cartprodpairin
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodmempair1_type,type,
% 0.29/0.44 cartprodmempair1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodmempair1,definition,
% 0.29/0.44 ( cartprodmempair1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.44 => ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ? [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 & ( Xu
% 0.29/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodmempair_type,type,
% 0.29/0.44 cartprodmempair: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodmempair,definition,
% 0.29/0.44 ( cartprodmempair
% 0.29/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.44 => ( iskpair @ Xu ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setunionE2_type,type,
% 0.29/0.44 setunionE2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setunionE2,definition,
% 0.29/0.44 ( setunionE2
% 0.29/0.44 = ( ! [A: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.44 => ? [X: $i] :
% 0.29/0.44 ( ( in @ X @ A )
% 0.29/0.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setunionsingleton1_type,type,
% 0.29/0.44 setunionsingleton1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setunionsingleton1,definition,
% 0.29/0.44 ( setunionsingleton1
% 0.29/0.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setunionsingleton2_type,type,
% 0.29/0.44 setunionsingleton2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setunionsingleton2,definition,
% 0.29/0.44 ( setunionsingleton2
% 0.29/0.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setunionsingleton_type,type,
% 0.29/0.44 setunionsingleton: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setunionsingleton,definition,
% 0.29/0.44 ( setunionsingleton
% 0.29/0.44 = ( ! [Xx: $i] :
% 0.29/0.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.29/0.44 = Xx ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(singleton_type,type,
% 0.29/0.44 singleton: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(singletonprop_type,type,
% 0.29/0.44 singletonprop: $o ).
% 0.29/0.44
% 0.29/0.44 thf(singletonprop,definition,
% 0.29/0.44 ( singletonprop
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 => ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) )
% 0.29/0.44 => ( ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ( Xphi @ Xx ) )
% 0.29/0.44 => ( singleton
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ex1_type,type,
% 0.29/0.44 ex1: $i > ( $i > $o ) > $o ).
% 0.29/0.44
% 0.29/0.44 thf(ex1E1_type,type,
% 0.29/0.44 ex1E1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ex1E1,definition,
% 0.29/0.44 ( ex1E1
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ( ex1 @ A
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ex1I_type,type,
% 0.29/0.44 ex1I: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ex1I,definition,
% 0.29/0.44 ( ex1I
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 => ( ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xy = Xx ) ) )
% 0.29/0.44 => ( ex1 @ A
% 0.29/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ex1I2_type,type,
% 0.29/0.44 ex1I2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ex1I2,definition,
% 0.29/0.44 ( ex1I2
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 => ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) )
% 0.29/0.44 => ( ? [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 & ( Xphi @ Xx ) )
% 0.29/0.44 => ( ex1 @ A
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(singletonsuniq_type,type,
% 0.29/0.44 singletonsuniq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(singletonsuniq,definition,
% 0.29/0.44 ( singletonsuniq
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( ( setadjoin @ Xx @ emptyset )
% 0.29/0.44 = ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(atmost1p_type,type,
% 0.29/0.44 atmost1p: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(atleast2p_type,type,
% 0.29/0.44 atleast2p: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(atmost2p_type,type,
% 0.29/0.44 atmost2p: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairsetp_type,type,
% 0.29/0.44 upairsetp: $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjL1_type,type,
% 0.29/0.44 setukpairinjL1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjL1,definition,
% 0.29/0.44 ( setukpairinjL1
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.44 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.29/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(kfstsingleton_type,type,
% 0.29/0.44 kfstsingleton: $o ).
% 0.29/0.44
% 0.29/0.44 thf(kfstsingleton,definition,
% 0.29/0.44 ( kfstsingleton
% 0.29/0.44 = ( ! [Xu: $i] :
% 0.29/0.44 ( ( iskpair @ Xu )
% 0.29/0.44 => ( singleton
% 0.29/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.29/0.44 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(theprop_type,type,
% 0.29/0.44 theprop: $o ).
% 0.29/0.44
% 0.29/0.44 thf(theprop,definition,
% 0.29/0.44 ( theprop
% 0.29/0.44 = ( ! [X: $i] :
% 0.29/0.44 ( ( singleton @ X )
% 0.29/0.44 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(kfst_type,type,
% 0.29/0.44 kfst: $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(kfstpairEq_type,type,
% 0.29/0.44 kfstpairEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(kfstpairEq,definition,
% 0.29/0.44 ( kfstpairEq
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.29/0.44 = Xx ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodfstin_type,type,
% 0.29/0.44 cartprodfstin: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodfstin,definition,
% 0.29/0.44 ( cartprodfstin
% 0.29/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.44 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjL2_type,type,
% 0.29/0.44 setukpairinjL2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjL2,definition,
% 0.29/0.44 ( setukpairinjL2
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.29/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.29/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjL_type,type,
% 0.29/0.44 setukpairinjL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjL,definition,
% 0.29/0.44 ( setukpairinjL
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.29/0.44 = ( kpair @ Xz @ Xu ) )
% 0.29/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR11_type,type,
% 0.29/0.44 setukpairinjR11: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR11,definition,
% 0.29/0.44 ( setukpairinjR11
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx = Xy )
% 0.29/0.44 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR12_type,type,
% 0.29/0.44 setukpairinjR12: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR12,definition,
% 0.29/0.44 ( setukpairinjR12
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xx = Xy )
% 0.29/0.44 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.29/0.44 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR1_type,type,
% 0.29/0.44 setukpairinjR1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR1,definition,
% 0.29/0.44 ( setukpairinjR1
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.29/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.29/0.44 => ( ( Xz = Xu )
% 0.29/0.44 => ( Xy = Xu ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(upairequniteq_type,type,
% 0.29/0.44 upairequniteq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(upairequniteq,definition,
% 0.29/0.44 ( upairequniteq
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.44 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.44 = ( setadjoin @ Xz @ emptyset ) )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR2_type,type,
% 0.29/0.44 setukpairinjR2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR2,definition,
% 0.29/0.44 ( setukpairinjR2
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.29/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.29/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR_type,type,
% 0.29/0.44 setukpairinjR: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setukpairinjR,definition,
% 0.29/0.44 ( setukpairinjR
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.29/0.44 = ( kpair @ Xz @ Xu ) )
% 0.29/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ksndsingleton_type,type,
% 0.29/0.44 ksndsingleton: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ksndsingleton,definition,
% 0.29/0.44 ( ksndsingleton
% 0.29/0.44 = ( ! [Xu: $i] :
% 0.29/0.44 ( ( iskpair @ Xu )
% 0.29/0.44 => ( singleton
% 0.29/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.29/0.44 @ ^ [Xx: $i] :
% 0.29/0.44 ( Xu
% 0.29/0.44 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ksnd_type,type,
% 0.29/0.44 ksnd: $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(ksndpairEq_type,type,
% 0.29/0.44 ksndpairEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ksndpairEq,definition,
% 0.29/0.44 ( ksndpairEq
% 0.29/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.29/0.44 = Xy ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(kpairsurjEq_type,type,
% 0.29/0.44 kpairsurjEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(kpairsurjEq,definition,
% 0.29/0.44 ( kpairsurjEq
% 0.29/0.44 = ( ! [Xu: $i] :
% 0.29/0.44 ( ( iskpair @ Xu )
% 0.29/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.29/0.44 = Xu ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodsndin_type,type,
% 0.29/0.44 cartprodsndin: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodsndin,definition,
% 0.29/0.44 ( cartprodsndin
% 0.29/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.44 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodpairmemEL_type,type,
% 0.29/0.44 cartprodpairmemEL: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodpairmemEL,definition,
% 0.29/0.44 ( cartprodpairmemEL
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.29/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodpairmemER_type,type,
% 0.29/0.44 cartprodpairmemER: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodpairmemER,definition,
% 0.29/0.44 ( cartprodpairmemER
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.29/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodmempaircEq_type,type,
% 0.29/0.44 cartprodmempaircEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodmempaircEq,definition,
% 0.29/0.44 ( cartprodmempaircEq
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( kpair @ Xx @ Xy )
% 0.29/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodfstpairEq_type,type,
% 0.29/0.44 cartprodfstpairEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodfstpairEq,definition,
% 0.29/0.44 ( cartprodfstpairEq
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.29/0.44 = Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodsndpairEq_type,type,
% 0.29/0.44 cartprodsndpairEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodsndpairEq,definition,
% 0.29/0.44 ( cartprodsndpairEq
% 0.29/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.29/0.44 = Xy ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodpairsurjEq_type,type,
% 0.29/0.44 cartprodpairsurjEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(cartprodpairsurjEq,definition,
% 0.29/0.44 ( cartprodpairsurjEq
% 0.29/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.29/0.44 = Xu ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(breln_type,type,
% 0.29/0.44 breln: $i > $i > $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(breln,definition,
% 0.29/0.44 ( breln
% 0.29/0.44 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstr_type,type,
% 0.29/0.44 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrI_type,type,
% 0.29/0.44 dpsetconstrI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrI,definition,
% 0.29/0.44 ( dpsetconstrI
% 0.29/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( Xphi @ Xx @ Xy )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.44 @ ( dpsetconstr @ A @ B
% 0.29/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrSub_type,type,
% 0.29/0.44 dpsetconstrSub: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrSub,definition,
% 0.29/0.44 ( dpsetconstrSub
% 0.29/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.29/0.44 ( subset
% 0.29/0.44 @ ( dpsetconstr @ A @ B
% 0.29/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.29/0.44 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setOfPairsIsBReln_type,type,
% 0.29/0.44 setOfPairsIsBReln: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setOfPairsIsBReln,definition,
% 0.29/0.44 ( setOfPairsIsBReln
% 0.29/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.29/0.44 ( breln @ A @ B
% 0.29/0.44 @ ( dpsetconstr @ A @ B
% 0.29/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrERa_type,type,
% 0.29/0.44 dpsetconstrERa: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrERa,definition,
% 0.29/0.44 ( dpsetconstrERa
% 0.29/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.44 @ ( dpsetconstr @ A @ B
% 0.29/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.29/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrEL1_type,type,
% 0.29/0.44 dpsetconstrEL1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrEL1,definition,
% 0.29/0.44 ( dpsetconstrEL1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.44 @ ( dpsetconstr @ A @ B
% 0.29/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.29/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrEL2_type,type,
% 0.29/0.44 dpsetconstrEL2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrEL2,definition,
% 0.29/0.44 ( dpsetconstrEL2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.44 @ ( dpsetconstr @ A @ B
% 0.29/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.29/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrER_type,type,
% 0.29/0.44 dpsetconstrER: $o ).
% 0.29/0.44
% 0.29/0.44 thf(dpsetconstrER,definition,
% 0.29/0.44 ( dpsetconstrER
% 0.29/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.44 @ ( dpsetconstr @ A @ B
% 0.29/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.29/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(func_type,type,
% 0.29/0.44 func: $i > $i > $i > $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcSet_type,type,
% 0.29/0.44 funcSet: $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(funcImageSingleton_type,type,
% 0.29/0.44 funcImageSingleton: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcImageSingleton,definition,
% 0.29/0.44 ( funcImageSingleton
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( singleton
% 0.29/0.44 @ ( dsetconstr @ B
% 0.29/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(apProp_type,type,
% 0.29/0.44 apProp: $o ).
% 0.29/0.44
% 0.29/0.44 thf(apProp,definition,
% 0.29/0.44 ( apProp
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in
% 0.29/0.44 @ ( setunion
% 0.29/0.44 @ ( dsetconstr @ B
% 0.29/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.29/0.44 @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ap_type,type,
% 0.29/0.44 ap: $i > $i > $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(app_type,type,
% 0.29/0.44 app: $o ).
% 0.29/0.44
% 0.29/0.44 thf(app,definition,
% 0.29/0.44 ( app
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(infuncsetfunc_type,type,
% 0.29/0.44 infuncsetfunc: $o ).
% 0.29/0.44
% 0.29/0.44 thf(infuncsetfunc,definition,
% 0.29/0.44 ( infuncsetfunc
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.44 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ap2p_type,type,
% 0.29/0.44 ap2p: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ap2p,definition,
% 0.29/0.44 ( ap2p
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(funcinfuncset_type,type,
% 0.29/0.44 funcinfuncset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcinfuncset,definition,
% 0.29/0.44 ( funcinfuncset
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(lamProp_type,type,
% 0.29/0.44 lamProp: $o ).
% 0.29/0.44
% 0.29/0.44 thf(lamProp,definition,
% 0.29/0.44 ( lamProp
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.44 => ( func @ A @ B
% 0.29/0.44 @ ( dpsetconstr @ A @ B
% 0.29/0.44 @ ^ [Xx: $i,Xy: $i] :
% 0.29/0.44 ( ( Xf @ Xx )
% 0.29/0.44 = Xy ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(lam_type,type,
% 0.29/0.44 lam: $i > $i > ( $i > $i ) > $i ).
% 0.29/0.44
% 0.29/0.44 thf(lamp_type,type,
% 0.29/0.44 lamp: $o ).
% 0.29/0.44
% 0.29/0.44 thf(lamp,definition,
% 0.29/0.44 ( lamp
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.44 => ( func @ A @ B
% 0.29/0.44 @ ( lam @ A @ B
% 0.29/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(lam2p_type,type,
% 0.29/0.44 lam2p: $o ).
% 0.29/0.44
% 0.29/0.44 thf(lam2p,definition,
% 0.29/0.44 ( lam2p
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.44 => ( in
% 0.29/0.44 @ ( lam @ A @ B
% 0.29/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.29/0.44 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(brelnall1_type,type,
% 0.29/0.44 brelnall1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(brelnall1,definition,
% 0.29/0.44 ( brelnall1
% 0.29/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.29/0.44 ( ( breln @ A @ B @ R )
% 0.29/0.44 => ! [Xphi: $i > $o] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ R )
% 0.29/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(brelnall2_type,type,
% 0.29/0.44 brelnall2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(brelnall2,definition,
% 0.29/0.44 ( brelnall2
% 0.29/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.29/0.44 ( ( breln @ A @ B @ R )
% 0.29/0.44 => ! [Xphi: $i > $o] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ R )
% 0.29/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ex1E2_type,type,
% 0.29/0.44 ex1E2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ex1E2,definition,
% 0.29/0.44 ( ex1E2
% 0.29/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.44 ( ( ex1 @ A
% 0.29/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( ( Xphi @ Xx )
% 0.29/0.44 => ( ( Xphi @ Xy )
% 0.29/0.44 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(funcGraphProp1_type,type,
% 0.29/0.44 funcGraphProp1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcGraphProp1,definition,
% 0.29/0.44 ( funcGraphProp1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(funcGraphProp3_type,type,
% 0.29/0.44 funcGraphProp3: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcGraphProp3,definition,
% 0.29/0.44 ( funcGraphProp3
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(funcGraphProp2_type,type,
% 0.29/0.44 funcGraphProp2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcGraphProp2,definition,
% 0.29/0.44 ( funcGraphProp2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.29/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(funcextLem_type,type,
% 0.29/0.44 funcextLem: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcextLem,definition,
% 0.29/0.44 ( funcextLem
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ! [Xg: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xg )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(funcGraphProp4_type,type,
% 0.29/0.44 funcGraphProp4: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcGraphProp4,definition,
% 0.29/0.44 ( funcGraphProp4
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.29/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subbreln_type,type,
% 0.29/0.44 subbreln: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subbreln,definition,
% 0.29/0.44 ( subbreln
% 0.29/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.29/0.44 ( ( breln @ A @ B @ R )
% 0.29/0.44 => ! [S: $i] :
% 0.29/0.44 ( ( breln @ A @ B @ S )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.29/0.44 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(eqbreln_type,type,
% 0.29/0.44 eqbreln: $o ).
% 0.29/0.44
% 0.29/0.44 thf(eqbreln,definition,
% 0.29/0.44 ( eqbreln
% 0.29/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.29/0.44 ( ( breln @ A @ B @ R )
% 0.29/0.44 => ! [S: $i] :
% 0.29/0.44 ( ( breln @ A @ B @ S )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ B )
% 0.29/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.29/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.29/0.44 => ( R = S ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(funcext_type,type,
% 0.29/0.44 funcext: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcext,definition,
% 0.29/0.44 ( funcext
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ! [Xg: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xg )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.29/0.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(funcext2_type,type,
% 0.29/0.44 funcext2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(funcext2,definition,
% 0.29/0.44 ( funcext2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.44 => ! [Xg: $i] :
% 0.29/0.44 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.29/0.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ap2apEq1_type,type,
% 0.29/0.44 ap2apEq1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ap2apEq1,definition,
% 0.29/0.44 ( ap2apEq1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ap2apEq2_type,type,
% 0.29/0.44 ap2apEq2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ap2apEq2,definition,
% 0.29/0.44 ( ap2apEq2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(beta1_type,type,
% 0.29/0.44 beta1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(beta1,definition,
% 0.29/0.44 ( beta1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( ap @ A @ B
% 0.29/0.44 @ ( lam @ A @ B
% 0.29/0.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.29/0.44 @ Xx )
% 0.29/0.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(eta1_type,type,
% 0.29/0.44 eta1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(eta1,definition,
% 0.29/0.44 ( eta1
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( func @ A @ B @ Xf )
% 0.29/0.44 => ( ( lam @ A @ B
% 0.29/0.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.29/0.44 = Xf ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(lam2lamEq_type,type,
% 0.29/0.44 lam2lamEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(lam2lamEq,definition,
% 0.29/0.44 ( lam2lamEq
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.44 => ( ( lam @ A @ B
% 0.29/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.29/0.44 = ( lam @ A @ B
% 0.29/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(beta2_type,type,
% 0.29/0.44 beta2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(beta2,definition,
% 0.29/0.44 ( beta2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.44 ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( ap @ A @ B
% 0.29/0.44 @ ( lam @ A @ B
% 0.29/0.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.29/0.44 @ Xx )
% 0.29/0.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(eta2_type,type,
% 0.29/0.44 eta2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(eta2,definition,
% 0.29/0.44 ( eta2
% 0.29/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.44 => ( ( lam @ A @ B
% 0.29/0.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.29/0.44 = Xf ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(iffalseProp1_type,type,
% 0.29/0.44 iffalseProp1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(iffalseProp1,definition,
% 0.29/0.44 ( iffalseProp1
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( ~ Xphi
% 0.29/0.44 => ( in @ Xy
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xz: $i] :
% 0.29/0.44 ( ( Xphi
% 0.29/0.44 & ( Xz = Xx ) )
% 0.29/0.44 | ( ~ Xphi
% 0.29/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(iffalseProp2_type,type,
% 0.29/0.44 iffalseProp2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(iffalseProp2,definition,
% 0.29/0.44 ( iffalseProp2
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( ~ Xphi
% 0.29/0.44 => ( ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xz: $i] :
% 0.29/0.44 ( ( Xphi
% 0.29/0.44 & ( Xz = Xx ) )
% 0.29/0.44 | ( ~ Xphi
% 0.29/0.44 & ( Xz = Xy ) ) ) )
% 0.29/0.44 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(iftrueProp1_type,type,
% 0.29/0.44 iftrueProp1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(iftrueProp1,definition,
% 0.29/0.44 ( iftrueProp1
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( Xphi
% 0.29/0.44 => ( in @ Xx
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xz: $i] :
% 0.29/0.44 ( ( Xphi
% 0.29/0.44 & ( Xz = Xx ) )
% 0.29/0.44 | ( ~ Xphi
% 0.29/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(iftrueProp2_type,type,
% 0.29/0.44 iftrueProp2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(iftrueProp2,definition,
% 0.29/0.44 ( iftrueProp2
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( Xphi
% 0.29/0.44 => ( ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xz: $i] :
% 0.29/0.44 ( ( Xphi
% 0.29/0.44 & ( Xz = Xx ) )
% 0.29/0.44 | ( ~ Xphi
% 0.29/0.44 & ( Xz = Xy ) ) ) )
% 0.29/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(ifSingleton_type,type,
% 0.29/0.44 ifSingleton: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ifSingleton,definition,
% 0.29/0.44 ( ifSingleton
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( singleton
% 0.29/0.44 @ ( dsetconstr @ A
% 0.29/0.44 @ ^ [Xz: $i] :
% 0.29/0.44 ( ( Xphi
% 0.29/0.44 & ( Xz = Xx ) )
% 0.29/0.44 | ( ~ Xphi
% 0.29/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(if_type,type,
% 0.29/0.44 if: $i > $o > $i > $i > $i ).
% 0.29/0.44
% 0.29/0.44 thf(ifp_type,type,
% 0.29/0.44 ifp: $o ).
% 0.29/0.44
% 0.29/0.44 thf(ifp,definition,
% 0.29/0.44 ( ifp
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(theeq_type,type,
% 0.29/0.44 theeq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(theeq,definition,
% 0.29/0.44 ( theeq
% 0.29/0.44 = ( ! [X: $i] :
% 0.29/0.44 ( ( singleton @ X )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ X )
% 0.29/0.44 => ( ( setunion @ X )
% 0.29/0.44 = Xx ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(iftrue_type,type,
% 0.29/0.44 iftrue: $o ).
% 0.29/0.44
% 0.29/0.44 thf(iftrue,definition,
% 0.29/0.44 ( iftrue
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( Xphi
% 0.29/0.44 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.29/0.44 = Xx ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(iffalse_type,type,
% 0.29/0.44 iffalse: $o ).
% 0.29/0.44
% 0.29/0.44 thf(iffalse,definition,
% 0.29/0.44 ( iffalse
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( ~ Xphi
% 0.29/0.44 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.29/0.44 = Xy ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(iftrueorfalse_type,type,
% 0.29/0.44 iftrueorfalse: $o ).
% 0.29/0.44
% 0.29/0.44 thf(iftrueorfalse,definition,
% 0.29/0.44 ( iftrueorfalse
% 0.29/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ! [Xy: $i] :
% 0.29/0.44 ( ( in @ Xy @ A )
% 0.29/0.44 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectT_lem_type,type,
% 0.29/0.44 binintersectT_lem: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectT_lem,definition,
% 0.29/0.44 ( binintersectT_lem
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionT_lem_type,type,
% 0.29/0.44 binunionT_lem: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionT_lem,definition,
% 0.29/0.44 ( binunionT_lem
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powersetT_lem_type,type,
% 0.29/0.44 powersetT_lem: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetT_lem,definition,
% 0.29/0.44 ( powersetT_lem
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setminusT_lem_type,type,
% 0.29/0.44 setminusT_lem: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setminusT_lem,definition,
% 0.29/0.44 ( setminusT_lem
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementT_lem_type,type,
% 0.29/0.44 complementT_lem: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementT_lem,definition,
% 0.29/0.44 ( complementT_lem
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(setextT_type,type,
% 0.29/0.44 setextT: $o ).
% 0.29/0.44
% 0.29/0.44 thf(setextT,definition,
% 0.29/0.44 ( setextT
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ X )
% 0.29/0.44 => ( in @ Xx @ Y ) ) )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ Y )
% 0.29/0.44 => ( in @ Xx @ X ) ) )
% 0.29/0.44 => ( X = Y ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(subsetTI_type,type,
% 0.29/0.44 subsetTI: $o ).
% 0.29/0.44
% 0.29/0.44 thf(subsetTI,definition,
% 0.29/0.44 ( subsetTI
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ X )
% 0.29/0.44 => ( in @ Xx @ Y ) ) )
% 0.29/0.44 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powersetTI1_type,type,
% 0.29/0.44 powersetTI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetTI1,definition,
% 0.29/0.44 ( powersetTI1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ X )
% 0.29/0.44 => ( in @ Xx @ Y ) ) )
% 0.29/0.44 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(powersetTE1_type,type,
% 0.29/0.44 powersetTE1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(powersetTE1,definition,
% 0.29/0.44 ( powersetTE1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ X @ ( powerset @ Y ) )
% 0.29/0.44 => ( ( in @ Xx @ X )
% 0.29/0.44 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementTI1_type,type,
% 0.29/0.44 complementTI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementTI1,definition,
% 0.29/0.44 ( complementTI1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ X )
% 0.29/0.44 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementTE1_type,type,
% 0.29/0.44 complementTE1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementTE1,definition,
% 0.29/0.44 ( complementTE1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.29/0.44 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectTELcontra_type,type,
% 0.29/0.44 binintersectTELcontra: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectTELcontra,definition,
% 0.29/0.44 ( binintersectTELcontra
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ X )
% 0.29/0.44 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectTERcontra_type,type,
% 0.29/0.44 binintersectTERcontra: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binintersectTERcontra,definition,
% 0.29/0.44 ( binintersectTERcontra
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ Y )
% 0.29/0.44 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(contrasubsetT_type,type,
% 0.29/0.44 contrasubsetT: $o ).
% 0.29/0.44
% 0.29/0.44 thf(contrasubsetT,definition,
% 0.29/0.44 ( contrasubsetT
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.29/0.44 => ( ( in @ Xx @ Y )
% 0.29/0.44 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(contrasubsetT1_type,type,
% 0.29/0.44 contrasubsetT1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(contrasubsetT1,definition,
% 0.29/0.44 ( contrasubsetT1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( subset @ X @ Y )
% 0.29/0.44 => ( ~ ( in @ Xx @ Y )
% 0.29/0.44 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(contrasubsetT2_type,type,
% 0.29/0.44 contrasubsetT2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(contrasubsetT2,definition,
% 0.29/0.44 ( contrasubsetT2
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( ( subset @ X @ Y )
% 0.29/0.44 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(contrasubsetT3_type,type,
% 0.29/0.44 contrasubsetT3: $o ).
% 0.29/0.44
% 0.29/0.44 thf(contrasubsetT3,definition,
% 0.29/0.44 ( contrasubsetT3
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.29/0.44 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementI1_type,type,
% 0.29/0.44 doubleComplementI1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementI1,definition,
% 0.29/0.44 ( doubleComplementI1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ X )
% 0.29/0.44 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementE1_type,type,
% 0.29/0.44 doubleComplementE1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementE1,definition,
% 0.29/0.44 ( doubleComplementE1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.29/0.44 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementSub1_type,type,
% 0.29/0.44 doubleComplementSub1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementSub1,definition,
% 0.29/0.44 ( doubleComplementSub1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementSub2_type,type,
% 0.29/0.44 doubleComplementSub2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementSub2,definition,
% 0.29/0.44 ( doubleComplementSub2
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementEq_type,type,
% 0.29/0.44 doubleComplementEq: $o ).
% 0.29/0.44
% 0.29/0.44 thf(doubleComplementEq,definition,
% 0.29/0.44 ( doubleComplementEq
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ( X
% 0.29/0.44 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementTnotintersectT_type,type,
% 0.29/0.44 complementTnotintersectT: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementTnotintersectT,definition,
% 0.29/0.44 ( complementTnotintersectT
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.29/0.44 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementImpComplementIntersect_type,type,
% 0.29/0.44 complementImpComplementIntersect: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementImpComplementIntersect,definition,
% 0.29/0.44 ( complementImpComplementIntersect
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.29/0.44 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementSubsetComplementIntersect_type,type,
% 0.29/0.44 complementSubsetComplementIntersect: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementSubsetComplementIntersect,definition,
% 0.29/0.44 ( complementSubsetComplementIntersect
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementInPowersetComplementIntersect_type,type,
% 0.29/0.44 complementInPowersetComplementIntersect: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementInPowersetComplementIntersect,definition,
% 0.29/0.44 ( complementInPowersetComplementIntersect
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(contraSubsetComplement_type,type,
% 0.29/0.44 contraSubsetComplement: $o ).
% 0.29/0.44
% 0.29/0.44 thf(contraSubsetComplement,definition,
% 0.29/0.44 ( contraSubsetComplement
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ Y )
% 0.29/0.44 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementTcontraSubset_type,type,
% 0.29/0.44 complementTcontraSubset: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementTcontraSubset,definition,
% 0.29/0.44 ( complementTcontraSubset
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.29/0.44 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionTILcontra_type,type,
% 0.29/0.44 binunionTILcontra: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionTILcontra,definition,
% 0.29/0.44 ( binunionTILcontra
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.29/0.44 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionTIRcontra_type,type,
% 0.29/0.44 binunionTIRcontra: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionTIRcontra,definition,
% 0.29/0.44 ( binunionTIRcontra
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.29/0.44 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(inIntersectImpInUnion_type,type,
% 0.29/0.44 inIntersectImpInUnion: $o ).
% 0.29/0.44
% 0.29/0.44 thf(inIntersectImpInUnion,definition,
% 0.29/0.44 ( inIntersectImpInUnion
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Z: $i] :
% 0.29/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.29/0.44 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(inIntersectImpInUnion2_type,type,
% 0.29/0.44 inIntersectImpInUnion2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(inIntersectImpInUnion2,definition,
% 0.29/0.44 ( inIntersectImpInUnion2
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Z: $i] :
% 0.29/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.29/0.44 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(inIntersectImpInIntersectUnions_type,type,
% 0.29/0.44 inIntersectImpInIntersectUnions: $o ).
% 0.29/0.44
% 0.29/0.44 thf(inIntersectImpInIntersectUnions,definition,
% 0.29/0.44 ( inIntersectImpInIntersectUnions
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Z: $i] :
% 0.29/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.29/0.44 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(intersectInPowersetIntersectUnions_type,type,
% 0.29/0.44 intersectInPowersetIntersectUnions: $o ).
% 0.29/0.44
% 0.29/0.44 thf(intersectInPowersetIntersectUnions,definition,
% 0.29/0.44 ( intersectInPowersetIntersectUnions
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Z: $i] :
% 0.29/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.44 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(inComplementUnionImpNotIn1_type,type,
% 0.29/0.44 inComplementUnionImpNotIn1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(inComplementUnionImpNotIn1,definition,
% 0.29/0.44 ( inComplementUnionImpNotIn1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.44 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(inComplementUnionImpInComplement1_type,type,
% 0.29/0.44 inComplementUnionImpInComplement1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(inComplementUnionImpInComplement1,definition,
% 0.29/0.44 ( inComplementUnionImpInComplement1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.44 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionTE_type,type,
% 0.29/0.44 binunionTE: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionTE,definition,
% 0.29/0.44 ( binunionTE
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xphi: $o,Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.29/0.44 => ( ( ( in @ Xx @ X )
% 0.29/0.44 => Xphi )
% 0.29/0.44 => ( ( ( in @ Xx @ Y )
% 0.29/0.44 => Xphi )
% 0.29/0.44 => Xphi ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(binunionTEcontra_type,type,
% 0.29/0.44 binunionTEcontra: $o ).
% 0.29/0.44
% 0.29/0.44 thf(binunionTEcontra,definition,
% 0.29/0.44 ( binunionTEcontra
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ~ ( in @ Xx @ X )
% 0.29/0.44 => ( ~ ( in @ Xx @ Y )
% 0.29/0.44 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(demorgan2a1_type,type,
% 0.29/0.44 demorgan2a1: $o ).
% 0.29/0.44
% 0.29/0.44 thf(demorgan2a1,definition,
% 0.29/0.44 ( demorgan2a1
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.44 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(complementUnionInPowersetComplement_type,type,
% 0.29/0.44 complementUnionInPowersetComplement: $o ).
% 0.29/0.44
% 0.29/0.44 thf(complementUnionInPowersetComplement,definition,
% 0.29/0.44 ( complementUnionInPowersetComplement
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(demorgan2a2_type,type,
% 0.29/0.44 demorgan2a2: $o ).
% 0.29/0.44
% 0.29/0.44 thf(demorgan2a2,definition,
% 0.29/0.44 ( demorgan2a2
% 0.29/0.44 = ( ! [A: $i,X: $i] :
% 0.29/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.44 => ! [Y: $i] :
% 0.29/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.44 => ! [Xx: $i] :
% 0.29/0.44 ( ( in @ Xx @ A )
% 0.29/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.44 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.44
% 0.29/0.44 thf(demorgan1a_type,type,
% 0.29/0.44 demorgan1a: $o ).
% 0.29/0.44
% 0.29/0.44 thf(demorgan1a,definition,
% 0.29/0.44 ( demorgan1a
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.29/0.45 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan1b_type,type,
% 0.29/0.45 demorgan1b: $o ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan1b,definition,
% 0.29/0.45 ( demorgan1b
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.29/0.45 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan1_type,type,
% 0.29/0.45 demorgan1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan1,definition,
% 0.29/0.45 ( demorgan1
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.29/0.45 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan2a_type,type,
% 0.29/0.45 demorgan2a: $o ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan2a,definition,
% 0.29/0.45 ( demorgan2a
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.45 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan2b2_type,type,
% 0.29/0.45 demorgan2b2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan2b2,definition,
% 0.29/0.45 ( demorgan2b2
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.29/0.45 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.29/0.45 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan2b_type,type,
% 0.29/0.45 demorgan2b: $o ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan2b,definition,
% 0.29/0.45 ( demorgan2b
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.29/0.45 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan2_type,type,
% 0.29/0.45 demorgan2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(demorgan2,definition,
% 0.29/0.45 ( demorgan2
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.29/0.45 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule0_type,type,
% 0.29/0.45 woz13rule0: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule0,definition,
% 0.29/0.45 ( woz13rule0
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.29/0.45 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule1_type,type,
% 0.29/0.45 woz13rule1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule1,definition,
% 0.29/0.45 ( woz13rule1
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Z: $i] :
% 0.29/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.45 => ( ( subset @ X @ Z )
% 0.29/0.45 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule2_type,type,
% 0.29/0.45 woz13rule2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule2,definition,
% 0.29/0.45 ( woz13rule2
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Z: $i] :
% 0.29/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.45 => ( ( subset @ Y @ Z )
% 0.29/0.45 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule3_type,type,
% 0.29/0.45 woz13rule3: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule3,definition,
% 0.29/0.45 ( woz13rule3
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Z: $i] :
% 0.29/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.45 => ( ( subset @ X @ Y )
% 0.29/0.45 => ( ( subset @ X @ Z )
% 0.29/0.45 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule4_type,type,
% 0.29/0.45 woz13rule4: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz13rule4,definition,
% 0.29/0.45 ( woz13rule4
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Z: $i] :
% 0.29/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.45 => ! [W: $i] :
% 0.29/0.45 ( ( in @ W @ ( powerset @ A ) )
% 0.29/0.45 => ( ( subset @ X @ Z )
% 0.29/0.45 => ( ( subset @ Y @ W )
% 0.29/0.45 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_1_type,type,
% 0.29/0.45 woz1_1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_1,definition,
% 0.29/0.45 ( woz1_1
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_2_type,type,
% 0.29/0.45 woz1_2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_2,definition,
% 0.29/0.45 ( woz1_2
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Z: $i] :
% 0.29/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.45 => ! [W: $i] :
% 0.29/0.45 ( ( in @ W @ ( powerset @ A ) )
% 0.29/0.45 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.29/0.45 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_3_type,type,
% 0.29/0.45 woz1_3: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_3,definition,
% 0.29/0.45 ( woz1_3
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ! [Z: $i] :
% 0.29/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.45 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_4_type,type,
% 0.29/0.45 woz1_4: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_4,definition,
% 0.29/0.45 ( woz1_4
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.29/0.45 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_5_type,type,
% 0.29/0.45 woz1_5: $o ).
% 0.29/0.45
% 0.29/0.45 thf(woz1_5,definition,
% 0.29/0.45 ( woz1_5
% 0.29/0.45 = ( ! [A: $i,X: $i] :
% 0.29/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.45 => ! [Y: $i] :
% 0.29/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.45 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(breln1_type,type,
% 0.29/0.45 breln1: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(breln1,definition,
% 0.29/0.45 ( breln1
% 0.29/0.45 = ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(breln1all2_type,type,
% 0.29/0.45 breln1all2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(breln1all2,definition,
% 0.29/0.45 ( breln1all2
% 0.29/0.45 = ( ! [A: $i,R: $i] :
% 0.29/0.45 ( ( breln1 @ A @ R )
% 0.29/0.45 => ! [Xphi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.45 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ R )
% 0.29/0.45 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(breln1Set_type,type,
% 0.29/0.45 breln1Set: $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(breln1SetBreln1_type,type,
% 0.29/0.45 breln1SetBreln1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(breln1SetBreln1,definition,
% 0.29/0.45 ( breln1SetBreln1
% 0.29/0.45 = ( ! [A: $i,R: $i] :
% 0.29/0.45 ( ( in @ R @ ( breln1Set @ A ) )
% 0.29/0.45 => ( breln1 @ A @ R ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(transitive_type,type,
% 0.29/0.45 transitive: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(antisymmetric_type,type,
% 0.29/0.45 antisymmetric: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(reflexive_type,type,
% 0.29/0.45 reflexive: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(refltransitive_type,type,
% 0.29/0.45 refltransitive: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(refllinearorder_type,type,
% 0.29/0.45 refllinearorder: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(reflwellordering_type,type,
% 0.29/0.45 reflwellordering: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(choice2fnsingleton_type,type,
% 0.29/0.45 choice2fnsingleton: $o ).
% 0.29/0.45
% 0.29/0.45 thf(choice2fnsingleton,definition,
% 0.29/0.45 ( choice2fnsingleton
% 0.29/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ? [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 & ( Xphi @ Xx @ Xy ) ) )
% 0.29/0.45 => ! [R: $i] :
% 0.29/0.45 ( ( in @ R @ ( breln1Set @ B ) )
% 0.29/0.45 => ( ( reflwellordering @ B @ R )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( singleton
% 0.29/0.45 @ ( dsetconstr @ B
% 0.29/0.45 @ ^ [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xx @ Xy )
% 0.29/0.45 & ! [Xz: $i] :
% 0.29/0.45 ( ( in @ Xz @ B )
% 0.29/0.45 => ( ( Xphi @ Xx @ Xz )
% 0.29/0.45 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setOfPairsIsBReln1_type,type,
% 0.29/0.45 setOfPairsIsBReln1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setOfPairsIsBReln1,definition,
% 0.29/0.45 ( setOfPairsIsBReln1
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.29/0.45 ( breln1 @ A
% 0.29/0.45 @ ( dpsetconstr @ A @ A
% 0.29/0.45 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(breln1all1_type,type,
% 0.29/0.45 breln1all1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(breln1all1,definition,
% 0.29/0.45 ( breln1all1
% 0.29/0.45 = ( ! [A: $i,R: $i] :
% 0.29/0.45 ( ( breln1 @ A @ R )
% 0.29/0.45 => ! [Xphi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.45 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ R )
% 0.29/0.45 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subbreln1_type,type,
% 0.29/0.45 subbreln1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subbreln1,definition,
% 0.29/0.45 ( subbreln1
% 0.29/0.45 = ( ! [A: $i,R: $i] :
% 0.29/0.45 ( ( breln1 @ A @ R )
% 0.29/0.45 => ! [S: $i] :
% 0.29/0.45 ( ( breln1 @ A @ S )
% 0.29/0.45 => ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.29/0.45 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(eqbreln1_type,type,
% 0.29/0.45 eqbreln1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(eqbreln1,definition,
% 0.29/0.45 ( eqbreln1
% 0.29/0.45 = ( ! [A: $i,R: $i] :
% 0.29/0.45 ( ( breln1 @ A @ R )
% 0.29/0.45 => ! [S: $i] :
% 0.29/0.45 ( ( breln1 @ A @ S )
% 0.29/0.45 => ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.29/0.45 => ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.29/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.29/0.45 => ( R = S ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(breln1invset_type,type,
% 0.29/0.45 breln1invset: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(breln1invset,definition,
% 0.29/0.45 ( breln1invset
% 0.29/0.45 = ( ^ [A: $i,R: $i] :
% 0.29/0.45 ( dpsetconstr @ A @ A
% 0.29/0.45 @ ^ [Xx: $i,Xy: $i] : ( in @ ( kpair @ Xy @ Xx ) @ R ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(breln1invprop_type,type,
% 0.29/0.45 breln1invprop: $o ).
% 0.29/0.45
% 0.29/0.45 thf(breln1invprop,definition,
% 0.29/0.45 ( breln1invprop
% 0.29/0.45 = ( ! [A: $i,R: $i] :
% 0.29/0.45 ( ( breln1 @ A @ R )
% 0.29/0.45 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(breln1invI,conjecture,
% 0.29/0.45 ( setextAx
% 0.29/0.45 => ( emptysetAx
% 0.29/0.45 => ( setadjoinAx
% 0.29/0.45 => ( powersetAx
% 0.29/0.45 => ( setunionAx
% 0.29/0.45 => ( omega0Ax
% 0.29/0.45 => ( omegaSAx
% 0.29/0.45 => ( omegaIndAx
% 0.29/0.45 => ( replAx
% 0.29/0.45 => ( foundationAx
% 0.29/0.45 => ( wellorderingAx
% 0.29/0.45 => ( descrp
% 0.29/0.45 => ( dsetconstrI
% 0.29/0.45 => ( dsetconstrEL
% 0.29/0.45 => ( dsetconstrER
% 0.29/0.45 => ( exuE1
% 0.29/0.45 => ( prop2setE
% 0.29/0.45 => ( emptysetE
% 0.29/0.45 => ( emptysetimpfalse
% 0.29/0.45 => ( notinemptyset
% 0.29/0.45 => ( exuE3e
% 0.29/0.45 => ( setext
% 0.29/0.45 => ( emptyI
% 0.29/0.45 => ( noeltsimpempty
% 0.29/0.45 => ( setbeta
% 0.29/0.45 => ( nonemptyE1
% 0.29/0.45 => ( nonemptyI
% 0.29/0.45 => ( nonemptyI1
% 0.29/0.45 => ( setadjoinIL
% 0.29/0.45 => ( emptyinunitempty
% 0.29/0.45 => ( setadjoinIR
% 0.29/0.45 => ( setadjoinE
% 0.29/0.45 => ( setadjoinOr
% 0.29/0.45 => ( setoftrueEq
% 0.29/0.45 => ( powersetI
% 0.29/0.45 => ( emptyinPowerset
% 0.29/0.45 => ( emptyInPowerset
% 0.29/0.45 => ( powersetE
% 0.29/0.45 => ( setunionI
% 0.29/0.45 => ( setunionE
% 0.29/0.45 => ( subPowSU
% 0.29/0.45 => ( exuE2
% 0.29/0.45 => ( nonemptyImpWitness
% 0.29/0.45 => ( uniqinunit
% 0.29/0.45 => ( notinsingleton
% 0.29/0.45 => ( eqinunit
% 0.29/0.45 => ( singletonsswitch
% 0.29/0.45 => ( upairsetE
% 0.29/0.45 => ( upairsetIL
% 0.29/0.45 => ( upairsetIR
% 0.29/0.45 => ( emptyE1
% 0.29/0.45 => ( vacuousDall
% 0.29/0.45 => ( quantDeMorgan1
% 0.29/0.45 => ( quantDeMorgan2
% 0.29/0.45 => ( quantDeMorgan3
% 0.29/0.45 => ( quantDeMorgan4
% 0.29/0.45 => ( prop2setI
% 0.29/0.45 => ( prop2set2propI
% 0.29/0.45 => ( notdexE
% 0.29/0.45 => ( notdallE
% 0.29/0.45 => ( exuI1
% 0.29/0.45 => ( exuI3
% 0.29/0.45 => ( exuI2
% 0.29/0.45 => ( inCongP
% 0.29/0.45 => ( in__Cong
% 0.29/0.45 => ( exuE3u
% 0.29/0.45 => ( exu__Cong
% 0.29/0.45 => ( emptyset__Cong
% 0.29/0.45 => ( setadjoin__Cong
% 0.29/0.45 => ( powerset__Cong
% 0.29/0.45 => ( setunion__Cong
% 0.29/0.45 => ( omega__Cong
% 0.29/0.45 => ( exuEu
% 0.29/0.45 => ( descr__Cong
% 0.29/0.45 => ( dsetconstr__Cong
% 0.29/0.45 => ( subsetI1
% 0.29/0.45 => ( eqimpsubset2
% 0.29/0.45 => ( eqimpsubset1
% 0.29/0.45 => ( subsetI2
% 0.29/0.45 => ( emptysetsubset
% 0.29/0.45 => ( subsetE
% 0.29/0.45 => ( subsetE2
% 0.29/0.45 => ( notsubsetI
% 0.29/0.45 => ( notequalI1
% 0.29/0.45 => ( notequalI2
% 0.29/0.45 => ( subsetRefl
% 0.29/0.45 => ( subsetTrans
% 0.29/0.45 => ( setadjoinSub
% 0.29/0.45 => ( setadjoinSub2
% 0.29/0.45 => ( subset2powerset
% 0.29/0.45 => ( setextsub
% 0.29/0.45 => ( subsetemptysetimpeq
% 0.29/0.45 => ( powersetI1
% 0.29/0.45 => ( powersetE1
% 0.29/0.45 => ( inPowerset
% 0.29/0.45 => ( powersetsubset
% 0.29/0.45 => ( sepInPowerset
% 0.29/0.45 => ( sepSubset
% 0.29/0.45 => ( binunionIL
% 0.29/0.45 => ( upairset2IR
% 0.29/0.45 => ( binunionIR
% 0.29/0.45 => ( binunionEcases
% 0.29/0.45 => ( binunionE
% 0.29/0.45 => ( binunionLsub
% 0.29/0.45 => ( binunionRsub
% 0.29/0.45 => ( binintersectI
% 0.29/0.45 => ( binintersectSubset5
% 0.29/0.45 => ( binintersectEL
% 0.29/0.45 => ( binintersectLsub
% 0.29/0.45 => ( binintersectSubset2
% 0.29/0.45 => ( binintersectSubset3
% 0.29/0.45 => ( binintersectER
% 0.29/0.45 => ( disjointsetsI1
% 0.29/0.45 => ( binintersectRsub
% 0.29/0.45 => ( binintersectSubset4
% 0.29/0.45 => ( binintersectSubset1
% 0.29/0.45 => ( bs114d
% 0.29/0.45 => ( setminusI
% 0.29/0.45 => ( setminusEL
% 0.29/0.45 => ( setminusER
% 0.29/0.45 => ( setminusSubset2
% 0.29/0.45 => ( setminusERneg
% 0.29/0.45 => ( setminusELneg
% 0.29/0.45 => ( setminusILneg
% 0.29/0.45 => ( setminusIRneg
% 0.29/0.45 => ( setminusLsub
% 0.29/0.45 => ( setminusSubset1
% 0.29/0.45 => ( symdiffE
% 0.29/0.45 => ( symdiffI1
% 0.29/0.45 => ( symdiffI2
% 0.29/0.45 => ( symdiffIneg1
% 0.29/0.45 => ( symdiffIneg2
% 0.29/0.45 => ( secondinupair
% 0.29/0.45 => ( setukpairIL
% 0.29/0.45 => ( setukpairIR
% 0.29/0.45 => ( kpairiskpair
% 0.29/0.45 => ( kpairp
% 0.29/0.45 => ( singletonsubset
% 0.29/0.45 => ( singletoninpowerset
% 0.29/0.45 => ( singletoninpowunion
% 0.29/0.45 => ( upairset2E
% 0.29/0.45 => ( upairsubunion
% 0.29/0.45 => ( upairinpowunion
% 0.29/0.45 => ( ubforcartprodlem1
% 0.29/0.45 => ( ubforcartprodlem2
% 0.29/0.45 => ( ubforcartprodlem3
% 0.29/0.45 => ( cartprodpairin
% 0.29/0.45 => ( cartprodmempair1
% 0.29/0.45 => ( cartprodmempair
% 0.29/0.45 => ( setunionE2
% 0.29/0.45 => ( setunionsingleton1
% 0.29/0.45 => ( setunionsingleton2
% 0.29/0.45 => ( setunionsingleton
% 0.29/0.45 => ( singletonprop
% 0.29/0.45 => ( ex1E1
% 0.29/0.45 => ( ex1I
% 0.29/0.45 => ( ex1I2
% 0.29/0.45 => ( singletonsuniq
% 0.29/0.45 => ( setukpairinjL1
% 0.29/0.45 => ( kfstsingleton
% 0.29/0.45 => ( theprop
% 0.29/0.45 => ( kfstpairEq
% 0.29/0.45 => ( cartprodfstin
% 0.29/0.45 => ( setukpairinjL2
% 0.29/0.45 => ( setukpairinjL
% 0.29/0.45 => ( setukpairinjR11
% 0.29/0.45 => ( setukpairinjR12
% 0.29/0.45 => ( setukpairinjR1
% 0.29/0.45 => ( upairequniteq
% 0.29/0.45 => ( setukpairinjR2
% 0.29/0.45 => ( setukpairinjR
% 0.29/0.45 => ( ksndsingleton
% 0.29/0.45 => ( ksndpairEq
% 0.29/0.45 => ( kpairsurjEq
% 0.29/0.45 => ( cartprodsndin
% 0.29/0.45 => ( cartprodpairmemEL
% 0.29/0.45 => ( cartprodpairmemER
% 0.29/0.45 => ( cartprodmempaircEq
% 0.29/0.45 => ( cartprodfstpairEq
% 0.29/0.45 => ( cartprodsndpairEq
% 0.29/0.45 => ( cartprodpairsurjEq
% 0.29/0.45 => ( dpsetconstrI
% 0.29/0.45 => ( dpsetconstrSub
% 0.29/0.45 => ( setOfPairsIsBReln
% 0.29/0.45 => ( dpsetconstrERa
% 0.29/0.45 => ( dpsetconstrEL1
% 0.29/0.45 => ( dpsetconstrEL2
% 0.29/0.45 => ( dpsetconstrER
% 0.29/0.45 => ( funcImageSingleton
% 0.29/0.45 => ( apProp
% 0.29/0.45 => ( app
% 0.29/0.45 => ( infuncsetfunc
% 0.29/0.45 => ( ap2p
% 0.29/0.45 => ( funcinfuncset
% 0.29/0.45 => ( lamProp
% 0.29/0.45 => ( lamp
% 0.29/0.45 => ( lam2p
% 0.29/0.45 => ( brelnall1
% 0.29/0.45 => ( brelnall2
% 0.29/0.45 => ( ex1E2
% 0.29/0.45 => ( funcGraphProp1
% 0.29/0.45 => ( funcGraphProp3
% 0.29/0.45 => ( funcGraphProp2
% 0.29/0.45 => ( funcextLem
% 0.29/0.45 => ( funcGraphProp4
% 0.29/0.45 => ( subbreln
% 0.29/0.45 => ( eqbreln
% 0.29/0.45 => ( funcext
% 0.29/0.45 => ( funcext2
% 0.29/0.45 => ( ap2apEq1
% 0.29/0.45 => ( ap2apEq2
% 0.29/0.45 => ( beta1
% 0.29/0.45 => ( eta1
% 0.29/0.45 => ( lam2lamEq
% 0.29/0.45 => ( beta2
% 0.29/0.45 => ( eta2
% 0.29/0.45 => ( iffalseProp1
% 0.29/0.45 => ( iffalseProp2
% 0.29/0.45 => ( iftrueProp1
% 0.29/0.45 => ( iftrueProp2
% 0.29/0.45 => ( ifSingleton
% 0.29/0.45 => ( ifp
% 0.29/0.45 => ( theeq
% 0.29/0.45 => ( iftrue
% 0.29/0.45 => ( iffalse
% 0.29/0.45 => ( iftrueorfalse
% 0.29/0.45 => ( binintersectT_lem
% 0.29/0.45 => ( binunionT_lem
% 0.29/0.45 => ( powersetT_lem
% 0.29/0.45 => ( setminusT_lem
% 0.29/0.45 => ( complementT_lem
% 0.29/0.45 => ( setextT
% 0.29/0.45 => ( subsetTI
% 0.29/0.45 => ( powersetTI1
% 0.29/0.45 => ( powersetTE1
% 0.29/0.45 => ( complementTI1
% 0.29/0.45 => ( complementTE1
% 0.29/0.45 => ( binintersectTELcontra
% 0.29/0.45 => ( binintersectTERcontra
% 0.29/0.45 => ( contrasubsetT
% 0.29/0.45 => ( contrasubsetT1
% 0.29/0.45 => ( contrasubsetT2
% 0.29/0.45 => ( contrasubsetT3
% 0.29/0.45 => ( doubleComplementI1
% 0.29/0.45 => ( doubleComplementE1
% 0.29/0.45 => ( doubleComplementSub1
% 0.29/0.45 => ( doubleComplementSub2
% 0.29/0.45 => ( doubleComplementEq
% 0.29/0.45 => ( complementTnotintersectT
% 0.29/0.45 => ( complementImpComplementIntersect
% 0.29/0.45 => ( complementSubsetComplementIntersect
% 0.29/0.45 => ( complementInPowersetComplementIntersect
% 0.29/0.45 => ( contraSubsetComplement
% 0.29/0.45 => ( complementTcontraSubset
% 0.29/0.45 => ( binunionTILcontra
% 0.29/0.45 => ( binunionTIRcontra
% 0.29/0.45 => ( inIntersectImpInUnion
% 0.29/0.45 => ( inIntersectImpInUnion2
% 0.29/0.45 => ( inIntersectImpInIntersectUnions
% 0.29/0.45 => ( intersectInPowersetIntersectUnions
% 0.29/0.45 => ( inComplementUnionImpNotIn1
% 0.29/0.45 => ( inComplementUnionImpInComplement1
% 0.29/0.45 => ( binunionTE
% 0.29/0.45 => ( binunionTEcontra
% 0.29/0.45 => ( demorgan2a1
% 0.29/0.45 => ( complementUnionInPowersetComplement
% 0.29/0.45 => ( demorgan2a2
% 0.29/0.45 => ( demorgan1a
% 0.29/0.45 => ( demorgan1b
% 0.29/0.45 => ( demorgan1
% 0.29/0.45 => ( demorgan2a
% 0.29/0.45 => ( demorgan2b2
% 0.29/0.45 => ( demorgan2b
% 0.29/0.45 => ( demorgan2
% 0.29/0.45 => ( woz13rule0
% 0.29/0.45 => ( woz13rule1
% 0.29/0.45 => ( woz13rule2
% 0.29/0.45 => ( woz13rule3
% 0.29/0.45 => ( woz13rule4
% 0.29/0.45 => ( woz1_1
% 0.29/0.45 => ( woz1_2
% 0.29/0.45 => ( woz1_3
% 0.29/0.45 => ( woz1_4
% 0.29/0.45 => ( woz1_5
% 0.29/0.45 => ( breln1all2
% 0.29/0.45 => ( breln1SetBreln1
% 0.29/0.45 => ( choice2fnsingleton
% 0.29/0.45 => ( setOfPairsIsBReln1
% 0.29/0.45 => ( breln1all1
% 0.29/0.45 => ( subbreln1
% 0.29/0.45 => ( eqbreln1
% 0.29/0.45 => ( breln1invprop
% 0.29/0.45 => ! [A: $i,R: $i] :
% 0.29/0.45 ( ( breln1 @ A @ R )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.45 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 ------------------------
% 2.51/1.49 ---- Embedded in THF ---
% 2.51/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.51/1.49 %%% Generated on Tue Feb 24 12:34:21 EST 2026
% 2.51/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.51/1.49 %%% Logic specification used:
% 2.51/1.49 %%% thf(spec, logic, $$dhol).
% 2.51/1.49
% 2.51/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.ExEgOH0aih/DTF2DTF12985.p
% See solution above
% 2.51/1.49 ------------------------
% 2.84/1.56 ---- Cleaned THF ---
% 2.84/1.56 thf(in_type,type,
% 2.84/1.56 in: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(exu_type,type,
% 2.84/1.56 exu: ( $i > $o ) > $o ).
% 2.84/1.56
% 2.84/1.56 thf(exu,definition,
% 2.84/1.56 ( exu
% 2.84/1.56 = ( ^ [Xphi: $i > $o] :
% 2.84/1.56 ? [Xx: $i] :
% 2.84/1.56 ( ( Xphi @ Xx )
% 2.84/1.56 & ! [Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setextAx_type,type,
% 2.84/1.56 setextAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setextAx,definition,
% 2.84/1.56 ( setextAx
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 <=> ( in @ Xx @ B ) )
% 2.84/1.56 => ( A = B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptyset_type,type,
% 2.84/1.56 emptyset: $i ).
% 2.84/1.56
% 2.84/1.56 thf(emptysetAx_type,type,
% 2.84/1.56 emptysetAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptysetAx,definition,
% 2.84/1.56 ( emptysetAx
% 2.84/1.56 = ( ! [Xx: $i] :
% 2.84/1.56 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoin_type,type,
% 2.84/1.56 setadjoin: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinAx_type,type,
% 2.84/1.56 setadjoinAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinAx,definition,
% 2.84/1.56 ( setadjoinAx
% 2.84/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.56 <=> ( ( Xy = Xx )
% 2.84/1.56 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powerset_type,type,
% 2.84/1.56 powerset: $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(powersetAx_type,type,
% 2.84/1.56 powersetAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetAx,definition,
% 2.84/1.56 ( powersetAx
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.56 <=> ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setunion_type,type,
% 2.84/1.56 setunion: $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(setunionAx_type,type,
% 2.84/1.56 setunionAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setunionAx,definition,
% 2.84/1.56 ( setunionAx
% 2.84/1.56 = ( ! [A: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.56 <=> ? [B: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 & ( in @ B @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(omega_type,type,
% 2.84/1.56 omega: $i ).
% 2.84/1.56
% 2.84/1.56 thf(omega0Ax_type,type,
% 2.84/1.56 omega0Ax: $o ).
% 2.84/1.56
% 2.84/1.56 thf(omega0Ax,definition,
% 2.84/1.56 ( omega0Ax
% 2.84/1.56 = ( in @ emptyset @ omega ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(omegaSAx_type,type,
% 2.84/1.56 omegaSAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(omegaSAx,definition,
% 2.84/1.56 ( omegaSAx
% 2.84/1.56 = ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ omega )
% 2.84/1.56 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(omegaIndAx_type,type,
% 2.84/1.56 omegaIndAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(omegaIndAx,definition,
% 2.84/1.56 ( omegaIndAx
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ( ( in @ emptyset @ A )
% 2.84/1.56 & ! [Xx: $i] :
% 2.84/1.56 ( ( ( in @ Xx @ omega )
% 2.84/1.56 & ( in @ Xx @ A ) )
% 2.84/1.56 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ omega )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(replAx_type,type,
% 2.84/1.56 replAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(replAx,definition,
% 2.84/1.56 ( replAx
% 2.84/1.56 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( exu
% 2.84/1.56 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.84/1.56 => ? [B: $i] :
% 2.84/1.56 ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 <=> ? [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(foundationAx_type,type,
% 2.84/1.56 foundationAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(foundationAx,definition,
% 2.84/1.56 ( foundationAx
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.84/1.56 => ? [B: $i] :
% 2.84/1.56 ( ( in @ B @ A )
% 2.84/1.56 & ~ ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(wellorderingAx_type,type,
% 2.84/1.56 wellorderingAx: $o ).
% 2.84/1.56
% 2.84/1.56 thf(wellorderingAx,definition,
% 2.84/1.56 ( wellorderingAx
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ? [B: $i] :
% 2.84/1.56 ( ! [C: $i] :
% 2.84/1.56 ( ( in @ C @ B )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ C )
% 2.84/1.56 => ( in @ Xx @ A ) ) )
% 2.84/1.56 & ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( ( in @ Xx @ A )
% 2.84/1.56 & ( in @ Xy @ A ) )
% 2.84/1.56 => ( ! [C: $i] :
% 2.84/1.56 ( ( in @ C @ B )
% 2.84/1.56 => ( ( in @ Xx @ C )
% 2.84/1.56 <=> ( in @ Xy @ C ) ) )
% 2.84/1.56 => ( Xx = Xy ) ) )
% 2.84/1.56 & ! [C: $i,D: $i] :
% 2.84/1.56 ( ( ( in @ C @ B )
% 2.84/1.56 & ( in @ D @ B ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ C )
% 2.84/1.56 => ( in @ Xx @ D ) )
% 2.84/1.56 | ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ D )
% 2.84/1.56 => ( in @ Xx @ C ) ) ) )
% 2.84/1.56 & ! [C: $i] :
% 2.84/1.56 ( ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ C )
% 2.84/1.56 => ( in @ Xx @ A ) )
% 2.84/1.56 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.84/1.56 => ? [D: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ D @ B )
% 2.84/1.56 & ( in @ Xx @ C )
% 2.84/1.56 & ~ ? [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ D )
% 2.84/1.56 & ( in @ Xy @ C ) )
% 2.84/1.56 & ! [E: $i] :
% 2.84/1.56 ( ( in @ E @ B )
% 2.84/1.56 => ( ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ E )
% 2.84/1.56 => ( in @ Xy @ D ) )
% 2.84/1.56 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(descr_type,type,
% 2.84/1.56 descr: ( $i > $o ) > $i ).
% 2.84/1.56
% 2.84/1.56 thf(descrp_type,type,
% 2.84/1.56 descrp: $o ).
% 2.84/1.56
% 2.84/1.56 thf(descrp,definition,
% 2.84/1.56 ( descrp
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ( Xphi
% 2.84/1.56 @ ( descr
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstr_type,type,
% 2.84/1.56 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstrI_type,type,
% 2.84/1.56 dsetconstrI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstrI,definition,
% 2.84/1.56 ( dsetconstrI
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 => ( in @ Xx
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstrEL_type,type,
% 2.84/1.56 dsetconstrEL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstrEL,definition,
% 2.84/1.56 ( dsetconstrEL
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstrER_type,type,
% 2.84/1.56 dsetconstrER: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstrER,definition,
% 2.84/1.56 ( dsetconstrER
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.56 => ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exuE1_type,type,
% 2.84/1.56 exuE1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exuE1,definition,
% 2.84/1.56 ( exuE1
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ? [Xx: $i] :
% 2.84/1.56 ( ( Xphi @ Xx )
% 2.84/1.56 & ! [Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(prop2set_type,type,
% 2.84/1.56 prop2set: $o > $i ).
% 2.84/1.56
% 2.84/1.56 thf(prop2setE_type,type,
% 2.84/1.56 prop2setE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(prop2setE,definition,
% 2.84/1.56 ( prop2setE
% 2.84/1.56 = ( ! [Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.84/1.56 => Xphi ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptysetE_type,type,
% 2.84/1.56 emptysetE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptysetE,definition,
% 2.84/1.56 ( emptysetE
% 2.84/1.56 = ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ emptyset )
% 2.84/1.56 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptysetimpfalse_type,type,
% 2.84/1.56 emptysetimpfalse: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptysetimpfalse,definition,
% 2.84/1.56 ( emptysetimpfalse
% 2.84/1.56 = ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ emptyset )
% 2.84/1.56 => $false ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(notinemptyset_type,type,
% 2.84/1.56 notinemptyset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(notinemptyset,definition,
% 2.84/1.56 ( notinemptyset
% 2.84/1.56 = ( ! [Xx: $i] :
% 2.84/1.56 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exuE3e_type,type,
% 2.84/1.56 exuE3e: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exuE3e,definition,
% 2.84/1.56 ( exuE3e
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setext_type,type,
% 2.84/1.56 setext: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setext,definition,
% 2.84/1.56 ( setext
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ Xx @ B ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ A ) )
% 2.84/1.56 => ( A = B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptyI_type,type,
% 2.84/1.56 emptyI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptyI,definition,
% 2.84/1.56 ( emptyI
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ~ ( in @ Xx @ A )
% 2.84/1.56 => ( A = emptyset ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(noeltsimpempty_type,type,
% 2.84/1.56 noeltsimpempty: $o ).
% 2.84/1.56
% 2.84/1.56 thf(noeltsimpempty,definition,
% 2.84/1.56 ( noeltsimpempty
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ~ ( in @ Xx @ A )
% 2.84/1.56 => ( A = emptyset ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setbeta_type,type,
% 2.84/1.56 setbeta: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setbeta,definition,
% 2.84/1.56 ( setbeta
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.56 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(nonempty_type,type,
% 2.84/1.56 nonempty: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(nonempty,definition,
% 2.84/1.56 ( nonempty
% 2.84/1.56 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(nonemptyE1_type,type,
% 2.84/1.56 nonemptyE1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(nonemptyE1,definition,
% 2.84/1.56 ( nonemptyE1
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ( nonempty @ A )
% 2.84/1.56 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(nonemptyI_type,type,
% 2.84/1.56 nonemptyI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(nonemptyI,definition,
% 2.84/1.56 ( nonemptyI
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 => ( nonempty
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(nonemptyI1_type,type,
% 2.84/1.56 nonemptyI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(nonemptyI1,definition,
% 2.84/1.56 ( nonemptyI1
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.84/1.56 => ( nonempty @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinIL_type,type,
% 2.84/1.56 setadjoinIL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinIL,definition,
% 2.84/1.56 ( setadjoinIL
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptyinunitempty_type,type,
% 2.84/1.56 emptyinunitempty: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptyinunitempty,definition,
% 2.84/1.56 ( emptyinunitempty
% 2.84/1.56 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinIR_type,type,
% 2.84/1.56 setadjoinIR: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinIR,definition,
% 2.84/1.56 ( setadjoinIR
% 2.84/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinE_type,type,
% 2.84/1.56 setadjoinE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinE,definition,
% 2.84/1.56 ( setadjoinE
% 2.84/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.56 => ! [Xphi: $o] :
% 2.84/1.56 ( ( ( Xy = Xx )
% 2.84/1.56 => Xphi )
% 2.84/1.56 => ( ( ( in @ Xy @ A )
% 2.84/1.56 => Xphi )
% 2.84/1.56 => Xphi ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinOr_type,type,
% 2.84/1.56 setadjoinOr: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinOr,definition,
% 2.84/1.56 ( setadjoinOr
% 2.84/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.56 => ( ( Xy = Xx )
% 2.84/1.56 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setoftrueEq_type,type,
% 2.84/1.56 setoftrueEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setoftrueEq,definition,
% 2.84/1.56 ( setoftrueEq
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xx: $i] : $true )
% 2.84/1.56 = A ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powersetI_type,type,
% 2.84/1.56 powersetI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetI,definition,
% 2.84/1.56 ( powersetI
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ A ) )
% 2.84/1.56 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptyinPowerset_type,type,
% 2.84/1.56 emptyinPowerset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptyinPowerset,definition,
% 2.84/1.56 ( emptyinPowerset
% 2.84/1.56 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptyInPowerset_type,type,
% 2.84/1.56 emptyInPowerset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptyInPowerset,definition,
% 2.84/1.56 ( emptyInPowerset
% 2.84/1.56 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powersetE_type,type,
% 2.84/1.56 powersetE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetE,definition,
% 2.84/1.56 ( powersetE
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.56 => ( ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setunionI_type,type,
% 2.84/1.56 setunionI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setunionI,definition,
% 2.84/1.56 ( setunionI
% 2.84/1.56 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 => ( ( in @ B @ A )
% 2.84/1.56 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setunionE_type,type,
% 2.84/1.56 setunionE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setunionE,definition,
% 2.84/1.56 ( setunionE
% 2.84/1.56 = ( ! [A: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.56 => ! [Xphi: $o] :
% 2.84/1.56 ( ! [B: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 => ( ( in @ B @ A )
% 2.84/1.56 => Xphi ) )
% 2.84/1.56 => Xphi ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subPowSU_type,type,
% 2.84/1.56 subPowSU: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subPowSU,definition,
% 2.84/1.56 ( subPowSU
% 2.84/1.56 = ( ! [A: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exuE2_type,type,
% 2.84/1.56 exuE2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exuE2,definition,
% 2.84/1.56 ( exuE2
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ? [Xx: $i] :
% 2.84/1.56 ! [Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xy )
% 2.84/1.56 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(nonemptyImpWitness_type,type,
% 2.84/1.56 nonemptyImpWitness: $o ).
% 2.84/1.56
% 2.84/1.56 thf(nonemptyImpWitness,definition,
% 2.84/1.56 ( nonemptyImpWitness
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ( nonempty @ A )
% 2.84/1.56 => ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & $true ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(uniqinunit_type,type,
% 2.84/1.56 uniqinunit: $o ).
% 2.84/1.56
% 2.84/1.56 thf(uniqinunit,definition,
% 2.84/1.56 ( uniqinunit
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(notinsingleton_type,type,
% 2.84/1.56 notinsingleton: $o ).
% 2.84/1.56
% 2.84/1.56 thf(notinsingleton,definition,
% 2.84/1.56 ( notinsingleton
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx != Xy )
% 2.84/1.56 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(eqinunit_type,type,
% 2.84/1.56 eqinunit: $o ).
% 2.84/1.56
% 2.84/1.56 thf(eqinunit,definition,
% 2.84/1.56 ( eqinunit
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx = Xy )
% 2.84/1.56 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(singletonsswitch_type,type,
% 2.84/1.56 singletonsswitch: $o ).
% 2.84/1.56
% 2.84/1.56 thf(singletonsswitch,definition,
% 2.84/1.56 ( singletonsswitch
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.56 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(upairsetE_type,type,
% 2.84/1.56 upairsetE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairsetE,definition,
% 2.84/1.56 ( upairsetE
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.56 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.84/1.56 => ( ( Xz = Xx )
% 2.84/1.56 | ( Xz = Xy ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(upairsetIL_type,type,
% 2.84/1.56 upairsetIL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairsetIL,definition,
% 2.84/1.56 ( upairsetIL
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(upairsetIR_type,type,
% 2.84/1.56 upairsetIR: $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairsetIR,definition,
% 2.84/1.56 ( upairsetIR
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptyE1_type,type,
% 2.84/1.56 emptyE1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptyE1,definition,
% 2.84/1.56 ( emptyE1
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ( Xphi @ Xx ) )
% 2.84/1.56 => ( ( ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 = emptyset )
% 2.84/1.56 => $false ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(vacuousDall_type,type,
% 2.84/1.56 vacuousDall: $o ).
% 2.84/1.56
% 2.84/1.56 thf(vacuousDall,definition,
% 2.84/1.56 ( vacuousDall
% 2.84/1.56 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ emptyset )
% 2.84/1.56 => ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(quantDeMorgan1_type,type,
% 2.84/1.56 quantDeMorgan1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(quantDeMorgan1,definition,
% 2.84/1.56 ( quantDeMorgan1
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ~ ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( Xphi @ Xx ) )
% 2.84/1.56 => ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(quantDeMorgan2_type,type,
% 2.84/1.56 quantDeMorgan2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(quantDeMorgan2,definition,
% 2.84/1.56 ( quantDeMorgan2
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ~ ( Xphi @ Xx ) )
% 2.84/1.56 => ~ ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(quantDeMorgan3_type,type,
% 2.84/1.56 quantDeMorgan3: $o ).
% 2.84/1.56
% 2.84/1.56 thf(quantDeMorgan3,definition,
% 2.84/1.56 ( quantDeMorgan3
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ~ ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ( Xphi @ Xx ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(quantDeMorgan4_type,type,
% 2.84/1.56 quantDeMorgan4: $o ).
% 2.84/1.56
% 2.84/1.56 thf(quantDeMorgan4,definition,
% 2.84/1.56 ( quantDeMorgan4
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ~ ( Xphi @ Xx ) )
% 2.84/1.56 => ~ ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(prop2setI_type,type,
% 2.84/1.56 prop2setI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(prop2setI,definition,
% 2.84/1.56 ( prop2setI
% 2.84/1.56 = ( ! [Xphi: $o] :
% 2.84/1.56 ( Xphi
% 2.84/1.56 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(set2prop_type,type,
% 2.84/1.56 set2prop: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(prop2set2propI_type,type,
% 2.84/1.56 prop2set2propI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(prop2set2propI,definition,
% 2.84/1.56 ( prop2set2propI
% 2.84/1.56 = ( ! [Xphi: $o] :
% 2.84/1.56 ( Xphi
% 2.84/1.56 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(notdexE_type,type,
% 2.84/1.56 notdexE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(notdexE,definition,
% 2.84/1.56 ( notdexE
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ~ ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ( Xphi @ Xx ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(notdallE_type,type,
% 2.84/1.56 notdallE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(notdallE,definition,
% 2.84/1.56 ( notdallE
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ~ ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( Xphi @ Xx ) )
% 2.84/1.56 => ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exuI1_type,type,
% 2.84/1.56 exuI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exuI1,definition,
% 2.84/1.56 ( exuI1
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ? [Xx: $i] :
% 2.84/1.56 ( ( Xphi @ Xx )
% 2.84/1.56 & ! [Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) )
% 2.84/1.56 => ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exuI3_type,type,
% 2.84/1.56 exuI3: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exuI3,definition,
% 2.84/1.56 ( exuI3
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.84/1.56 => ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xx )
% 2.84/1.56 => ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) )
% 2.84/1.56 => ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exuI2_type,type,
% 2.84/1.56 exuI2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exuI2,definition,
% 2.84/1.56 ( exuI2
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ? [Xx: $i] :
% 2.84/1.56 ! [Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xy )
% 2.84/1.56 <=> ( Xy = Xx ) )
% 2.84/1.56 => ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(inCongP_type,type,
% 2.84/1.56 inCongP: $o ).
% 2.84/1.56
% 2.84/1.56 thf(inCongP,definition,
% 2.84/1.56 ( inCongP
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( A = B )
% 2.84/1.56 => ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx = Xy )
% 2.84/1.56 => ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(in__Cong_type,type,
% 2.84/1.56 in__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(in__Cong,definition,
% 2.84/1.56 ( in__Cong
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( A = B )
% 2.84/1.56 => ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx = Xy )
% 2.84/1.56 => ( ( in @ Xx @ A )
% 2.84/1.56 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exuE3u_type,type,
% 2.84/1.56 exuE3u: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exuE3u,definition,
% 2.84/1.56 ( exuE3u
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xx )
% 2.84/1.56 => ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exu__Cong_type,type,
% 2.84/1.56 exu__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exu__Cong,definition,
% 2.84/1.56 ( exu__Cong
% 2.84/1.56 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx = Xy )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 <=> ( Xpsi @ Xy ) ) )
% 2.84/1.56 => ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 <=> ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptyset__Cong_type,type,
% 2.84/1.56 emptyset__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptyset__Cong,definition,
% 2.84/1.56 ( emptyset__Cong
% 2.84/1.56 = ( emptyset = emptyset ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoin__Cong_type,type,
% 2.84/1.56 setadjoin__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoin__Cong,definition,
% 2.84/1.56 ( setadjoin__Cong
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx = Xy )
% 2.84/1.56 => ! [Xz: $i,Xu: $i] :
% 2.84/1.56 ( ( Xz = Xu )
% 2.84/1.56 => ( ( setadjoin @ Xx @ Xz )
% 2.84/1.56 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powerset__Cong_type,type,
% 2.84/1.56 powerset__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powerset__Cong,definition,
% 2.84/1.56 ( powerset__Cong
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( A = B )
% 2.84/1.56 => ( ( powerset @ A )
% 2.84/1.56 = ( powerset @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setunion__Cong_type,type,
% 2.84/1.56 setunion__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setunion__Cong,definition,
% 2.84/1.56 ( setunion__Cong
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( A = B )
% 2.84/1.56 => ( ( setunion @ A )
% 2.84/1.56 = ( setunion @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(omega__Cong_type,type,
% 2.84/1.56 omega__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(omega__Cong,definition,
% 2.84/1.56 ( omega__Cong
% 2.84/1.56 = ( omega = omega ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(exuEu_type,type,
% 2.84/1.56 exuEu: $o ).
% 2.84/1.56
% 2.84/1.56 thf(exuEu,definition,
% 2.84/1.56 ( exuEu
% 2.84/1.56 = ( ! [Xphi: $i > $o] :
% 2.84/1.56 ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xx )
% 2.84/1.56 => ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(descr__Cong_type,type,
% 2.84/1.56 descr__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(descr__Cong,definition,
% 2.84/1.56 ( descr__Cong
% 2.84/1.56 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx = Xy )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 <=> ( Xpsi @ Xy ) ) )
% 2.84/1.56 => ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ( ( exu
% 2.84/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.84/1.56 => ( ( descr
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 = ( descr
% 2.84/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstr__Cong_type,type,
% 2.84/1.56 dsetconstr__Cong: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dsetconstr__Cong,definition,
% 2.84/1.56 ( dsetconstr__Cong
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( A = B )
% 2.84/1.56 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( Xx = Xy )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.84/1.56 => ( ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 = ( dsetconstr @ B
% 2.84/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subset_type,type,
% 2.84/1.56 subset: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(disjoint_type,type,
% 2.84/1.56 disjoint: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(setsmeet_type,type,
% 2.84/1.56 setsmeet: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetI1_type,type,
% 2.84/1.56 subsetI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetI1,definition,
% 2.84/1.56 ( subsetI1
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ Xx @ B ) )
% 2.84/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(eqimpsubset2_type,type,
% 2.84/1.56 eqimpsubset2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(eqimpsubset2,definition,
% 2.84/1.56 ( eqimpsubset2
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( A = B )
% 2.84/1.56 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(eqimpsubset1_type,type,
% 2.84/1.56 eqimpsubset1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(eqimpsubset1,definition,
% 2.84/1.56 ( eqimpsubset1
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( A = B )
% 2.84/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subsetI2_type,type,
% 2.84/1.56 subsetI2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetI2,definition,
% 2.84/1.56 ( subsetI2
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ Xx @ B ) )
% 2.84/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(emptysetsubset_type,type,
% 2.84/1.56 emptysetsubset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(emptysetsubset,definition,
% 2.84/1.56 ( emptysetsubset
% 2.84/1.56 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subsetE_type,type,
% 2.84/1.56 subsetE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetE,definition,
% 2.84/1.56 ( subsetE
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subsetE2_type,type,
% 2.84/1.56 subsetE2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetE2,definition,
% 2.84/1.56 ( subsetE2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( ~ ( in @ Xx @ B )
% 2.84/1.56 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(notsubsetI_type,type,
% 2.84/1.56 notsubsetI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(notsubsetI,definition,
% 2.84/1.56 ( notsubsetI
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ B )
% 2.84/1.56 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(notequalI1_type,type,
% 2.84/1.56 notequalI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(notequalI1,definition,
% 2.84/1.56 ( notequalI1
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ~ ( subset @ A @ B )
% 2.84/1.56 => ( A != B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(notequalI2_type,type,
% 2.84/1.56 notequalI2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(notequalI2,definition,
% 2.84/1.56 ( notequalI2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ B )
% 2.84/1.56 => ( A != B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subsetRefl_type,type,
% 2.84/1.56 subsetRefl: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetRefl,definition,
% 2.84/1.56 ( subsetRefl
% 2.84/1.56 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subsetTrans_type,type,
% 2.84/1.56 subsetTrans: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetTrans,definition,
% 2.84/1.56 ( subsetTrans
% 2.84/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( ( subset @ B @ C )
% 2.84/1.56 => ( subset @ A @ C ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinSub_type,type,
% 2.84/1.56 setadjoinSub: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinSub,definition,
% 2.84/1.56 ( setadjoinSub
% 2.84/1.56 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinSub2_type,type,
% 2.84/1.56 setadjoinSub2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setadjoinSub2,definition,
% 2.84/1.56 ( setadjoinSub2
% 2.84/1.56 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subset2powerset_type,type,
% 2.84/1.56 subset2powerset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subset2powerset,definition,
% 2.84/1.56 ( subset2powerset
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setextsub_type,type,
% 2.84/1.56 setextsub: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setextsub,definition,
% 2.84/1.56 ( setextsub
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( ( subset @ B @ A )
% 2.84/1.56 => ( A = B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subsetemptysetimpeq_type,type,
% 2.84/1.56 subsetemptysetimpeq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetemptysetimpeq,definition,
% 2.84/1.56 ( subsetemptysetimpeq
% 2.84/1.56 = ( ! [A: $i] :
% 2.84/1.56 ( ( subset @ A @ emptyset )
% 2.84/1.56 => ( A = emptyset ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powersetI1_type,type,
% 2.84/1.56 powersetI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetI1,definition,
% 2.84/1.56 ( powersetI1
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( subset @ B @ A )
% 2.84/1.56 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powersetE1_type,type,
% 2.84/1.56 powersetE1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetE1,definition,
% 2.84/1.56 ( powersetE1
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.56 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(inPowerset_type,type,
% 2.84/1.56 inPowerset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(inPowerset,definition,
% 2.84/1.56 ( inPowerset
% 2.84/1.56 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powersetsubset_type,type,
% 2.84/1.56 powersetsubset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetsubset,definition,
% 2.84/1.56 ( powersetsubset
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(sepInPowerset_type,type,
% 2.84/1.56 sepInPowerset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(sepInPowerset,definition,
% 2.84/1.56 ( sepInPowerset
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( in
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 @ ( powerset @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(sepSubset_type,type,
% 2.84/1.56 sepSubset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(sepSubset,definition,
% 2.84/1.56 ( sepSubset
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( subset
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 @ A ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunion_type,type,
% 2.84/1.56 binunion: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(binunionIL_type,type,
% 2.84/1.56 binunionIL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionIL,definition,
% 2.84/1.56 ( binunionIL
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(upairset2IR_type,type,
% 2.84/1.56 upairset2IR: $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairset2IR,definition,
% 2.84/1.56 ( upairset2IR
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionIR_type,type,
% 2.84/1.56 binunionIR: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionIR,definition,
% 2.84/1.56 ( binunionIR
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionEcases_type,type,
% 2.84/1.56 binunionEcases: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionEcases,definition,
% 2.84/1.56 ( binunionEcases
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.84/1.56 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.84/1.56 => ( ( ( in @ Xx @ A )
% 2.84/1.56 => Xphi )
% 2.84/1.56 => ( ( ( in @ Xx @ B )
% 2.84/1.56 => Xphi )
% 2.84/1.56 => Xphi ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionE_type,type,
% 2.84/1.56 binunionE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionE,definition,
% 2.84/1.56 ( binunionE
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.84/1.56 => ( ( in @ Xx @ A )
% 2.84/1.56 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionLsub_type,type,
% 2.84/1.56 binunionLsub: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionLsub,definition,
% 2.84/1.56 ( binunionLsub
% 2.84/1.56 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionRsub_type,type,
% 2.84/1.56 binunionRsub: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionRsub,definition,
% 2.84/1.56 ( binunionRsub
% 2.84/1.56 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersect_type,type,
% 2.84/1.56 binintersect: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectI_type,type,
% 2.84/1.56 binintersectI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectI,definition,
% 2.84/1.56 ( binintersectI
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset5_type,type,
% 2.84/1.56 binintersectSubset5: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset5,definition,
% 2.84/1.56 ( binintersectSubset5
% 2.84/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.56 ( ( subset @ C @ A )
% 2.84/1.56 => ( ( subset @ C @ B )
% 2.84/1.56 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectEL_type,type,
% 2.84/1.56 binintersectEL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectEL,definition,
% 2.84/1.56 ( binintersectEL
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectLsub_type,type,
% 2.84/1.56 binintersectLsub: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectLsub,definition,
% 2.84/1.56 ( binintersectLsub
% 2.84/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset2_type,type,
% 2.84/1.56 binintersectSubset2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset2,definition,
% 2.84/1.56 ( binintersectSubset2
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( ( binintersect @ A @ B )
% 2.84/1.56 = A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset3_type,type,
% 2.84/1.56 binintersectSubset3: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset3,definition,
% 2.84/1.56 ( binintersectSubset3
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( ( binintersect @ A @ B )
% 2.84/1.56 = B )
% 2.84/1.56 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectER_type,type,
% 2.84/1.56 binintersectER: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectER,definition,
% 2.84/1.56 ( binintersectER
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.84/1.56 => ( in @ Xx @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(disjointsetsI1_type,type,
% 2.84/1.56 disjointsetsI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(disjointsetsI1,definition,
% 2.84/1.56 ( disjointsetsI1
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ~ ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ( in @ Xx @ B ) )
% 2.84/1.56 => ( ( binintersect @ A @ B )
% 2.84/1.56 = emptyset ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectRsub_type,type,
% 2.84/1.56 binintersectRsub: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectRsub,definition,
% 2.84/1.56 ( binintersectRsub
% 2.84/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset4_type,type,
% 2.84/1.56 binintersectSubset4: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset4,definition,
% 2.84/1.56 ( binintersectSubset4
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( subset @ B @ A )
% 2.84/1.56 => ( ( binintersect @ A @ B )
% 2.84/1.56 = B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset1_type,type,
% 2.84/1.56 binintersectSubset1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectSubset1,definition,
% 2.84/1.56 ( binintersectSubset1
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( ( binintersect @ A @ B )
% 2.84/1.56 = A )
% 2.84/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(bs114d_type,type,
% 2.84/1.56 bs114d: $o ).
% 2.84/1.56
% 2.84/1.56 thf(bs114d,definition,
% 2.84/1.56 ( bs114d
% 2.84/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.56 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.84/1.56 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(regular_type,type,
% 2.84/1.56 regular: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminus_type,type,
% 2.84/1.56 setminus: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(setminusI_type,type,
% 2.84/1.56 setminusI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusI,definition,
% 2.84/1.56 ( setminusI
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusEL_type,type,
% 2.84/1.56 setminusEL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusEL,definition,
% 2.84/1.56 ( setminusEL
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusER_type,type,
% 2.84/1.56 setminusER: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusER,definition,
% 2.84/1.56 ( setminusER
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.56 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusSubset2_type,type,
% 2.84/1.56 setminusSubset2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusSubset2,definition,
% 2.84/1.56 ( setminusSubset2
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( subset @ A @ B )
% 2.84/1.56 => ( ( setminus @ A @ B )
% 2.84/1.56 = emptyset ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusERneg_type,type,
% 2.84/1.56 setminusERneg: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusERneg,definition,
% 2.84/1.56 ( setminusERneg
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.56 => ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusELneg_type,type,
% 2.84/1.56 setminusELneg: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusELneg,definition,
% 2.84/1.56 ( setminusELneg
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.56 => ( ~ ( in @ Xx @ B )
% 2.84/1.56 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusILneg_type,type,
% 2.84/1.56 setminusILneg: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusILneg,definition,
% 2.84/1.56 ( setminusILneg
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ~ ( in @ Xx @ A )
% 2.84/1.56 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusIRneg_type,type,
% 2.84/1.56 setminusIRneg: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusIRneg,definition,
% 2.84/1.56 ( setminusIRneg
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ B )
% 2.84/1.56 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusLsub_type,type,
% 2.84/1.56 setminusLsub: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusLsub,definition,
% 2.84/1.56 ( setminusLsub
% 2.84/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusSubset1_type,type,
% 2.84/1.56 setminusSubset1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusSubset1,definition,
% 2.84/1.56 ( setminusSubset1
% 2.84/1.56 = ( ! [A: $i,B: $i] :
% 2.84/1.56 ( ( ( setminus @ A @ B )
% 2.84/1.56 = emptyset )
% 2.84/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(symdiff_type,type,
% 2.84/1.56 symdiff: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffE_type,type,
% 2.84/1.56 symdiffE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffE,definition,
% 2.84/1.56 ( symdiffE
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.84/1.56 => ! [Xphi: $o] :
% 2.84/1.56 ( ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ B )
% 2.84/1.56 => Xphi ) )
% 2.84/1.56 => ( ( ~ ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ B )
% 2.84/1.56 => Xphi ) )
% 2.84/1.56 => Xphi ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffI1_type,type,
% 2.84/1.56 symdiffI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffI1,definition,
% 2.84/1.56 ( symdiffI1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffI2_type,type,
% 2.84/1.56 symdiffI2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffI2,definition,
% 2.84/1.56 ( symdiffI2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ~ ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ B )
% 2.84/1.56 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffIneg1_type,type,
% 2.84/1.56 symdiffIneg1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffIneg1,definition,
% 2.84/1.56 ( symdiffIneg1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ B )
% 2.84/1.56 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffIneg2_type,type,
% 2.84/1.56 symdiffIneg2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(symdiffIneg2,definition,
% 2.84/1.56 ( symdiffIneg2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ~ ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ B )
% 2.84/1.56 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(iskpair_type,type,
% 2.84/1.56 iskpair: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(secondinupair_type,type,
% 2.84/1.56 secondinupair: $o ).
% 2.84/1.56
% 2.84/1.56 thf(secondinupair,definition,
% 2.84/1.56 ( secondinupair
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairIL_type,type,
% 2.84/1.56 setukpairIL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairIL,definition,
% 2.84/1.56 ( setukpairIL
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairIR_type,type,
% 2.84/1.56 setukpairIR: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairIR,definition,
% 2.84/1.56 ( setukpairIR
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(kpairiskpair_type,type,
% 2.84/1.56 kpairiskpair: $o ).
% 2.84/1.56
% 2.84/1.56 thf(kpairiskpair,definition,
% 2.84/1.56 ( kpairiskpair
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(kpair_type,type,
% 2.84/1.56 kpair: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(kpairp_type,type,
% 2.84/1.56 kpairp: $o ).
% 2.84/1.56
% 2.84/1.56 thf(kpairp,definition,
% 2.84/1.56 ( kpairp
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprod_type,type,
% 2.84/1.56 cartprod: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(singletonsubset_type,type,
% 2.84/1.56 singletonsubset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(singletonsubset,definition,
% 2.84/1.56 ( singletonsubset
% 2.84/1.56 = ( ! [A: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(singletoninpowerset_type,type,
% 2.84/1.56 singletoninpowerset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(singletoninpowerset,definition,
% 2.84/1.56 ( singletoninpowerset
% 2.84/1.56 = ( ! [A: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(singletoninpowunion_type,type,
% 2.84/1.56 singletoninpowunion: $o ).
% 2.84/1.56
% 2.84/1.56 thf(singletoninpowunion,definition,
% 2.84/1.56 ( singletoninpowunion
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(upairset2E_type,type,
% 2.84/1.56 upairset2E: $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairset2E,definition,
% 2.84/1.56 ( upairset2E
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.56 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.84/1.56 => ( ( Xz = Xx )
% 2.84/1.56 | ( Xz = Xy ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(upairsubunion_type,type,
% 2.84/1.56 upairsubunion: $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairsubunion,definition,
% 2.84/1.56 ( upairsubunion
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(upairinpowunion_type,type,
% 2.84/1.56 upairinpowunion: $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairinpowunion,definition,
% 2.84/1.56 ( upairinpowunion
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ubforcartprodlem1_type,type,
% 2.84/1.56 ubforcartprodlem1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ubforcartprodlem1,definition,
% 2.84/1.56 ( ubforcartprodlem1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ubforcartprodlem2_type,type,
% 2.84/1.56 ubforcartprodlem2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ubforcartprodlem2,definition,
% 2.84/1.56 ( ubforcartprodlem2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ubforcartprodlem3_type,type,
% 2.84/1.56 ubforcartprodlem3: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ubforcartprodlem3,definition,
% 2.84/1.56 ( ubforcartprodlem3
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodpairin_type,type,
% 2.84/1.56 cartprodpairin: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodpairin,definition,
% 2.84/1.56 ( cartprodpairin
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodmempair1_type,type,
% 2.84/1.56 cartprodmempair1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodmempair1,definition,
% 2.84/1.56 ( cartprodmempair1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.56 => ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ? [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 & ( Xu
% 2.84/1.56 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodmempair_type,type,
% 2.84/1.56 cartprodmempair: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodmempair,definition,
% 2.84/1.56 ( cartprodmempair
% 2.84/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.56 => ( iskpair @ Xu ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setunionE2_type,type,
% 2.84/1.56 setunionE2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setunionE2,definition,
% 2.84/1.56 ( setunionE2
% 2.84/1.56 = ( ! [A: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.56 => ? [X: $i] :
% 2.84/1.56 ( ( in @ X @ A )
% 2.84/1.56 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setunionsingleton1_type,type,
% 2.84/1.56 setunionsingleton1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setunionsingleton1,definition,
% 2.84/1.56 ( setunionsingleton1
% 2.84/1.56 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setunionsingleton2_type,type,
% 2.84/1.56 setunionsingleton2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setunionsingleton2,definition,
% 2.84/1.56 ( setunionsingleton2
% 2.84/1.56 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setunionsingleton_type,type,
% 2.84/1.56 setunionsingleton: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setunionsingleton,definition,
% 2.84/1.56 ( setunionsingleton
% 2.84/1.56 = ( ! [Xx: $i] :
% 2.84/1.56 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.84/1.56 = Xx ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(singleton_type,type,
% 2.84/1.56 singleton: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(singletonprop_type,type,
% 2.84/1.56 singletonprop: $o ).
% 2.84/1.56
% 2.84/1.56 thf(singletonprop,definition,
% 2.84/1.56 ( singletonprop
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 => ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) )
% 2.84/1.56 => ( ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ( Xphi @ Xx ) )
% 2.84/1.56 => ( singleton
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ex1_type,type,
% 2.84/1.56 ex1: $i > ( $i > $o ) > $o ).
% 2.84/1.56
% 2.84/1.56 thf(ex1E1_type,type,
% 2.84/1.56 ex1E1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ex1E1,definition,
% 2.84/1.56 ( ex1E1
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ( ex1 @ A
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ex1I_type,type,
% 2.84/1.56 ex1I: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ex1I,definition,
% 2.84/1.56 ( ex1I
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 => ( ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xy = Xx ) ) )
% 2.84/1.56 => ( ex1 @ A
% 2.84/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ex1I2_type,type,
% 2.84/1.56 ex1I2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ex1I2,definition,
% 2.84/1.56 ( ex1I2
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 => ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) )
% 2.84/1.56 => ( ? [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 & ( Xphi @ Xx ) )
% 2.84/1.56 => ( ex1 @ A
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(singletonsuniq_type,type,
% 2.84/1.56 singletonsuniq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(singletonsuniq,definition,
% 2.84/1.56 ( singletonsuniq
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( ( setadjoin @ Xx @ emptyset )
% 2.84/1.56 = ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(atmost1p_type,type,
% 2.84/1.56 atmost1p: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(atleast2p_type,type,
% 2.84/1.56 atleast2p: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(atmost2p_type,type,
% 2.84/1.56 atmost2p: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairsetp_type,type,
% 2.84/1.56 upairsetp: $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjL1_type,type,
% 2.84/1.56 setukpairinjL1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjL1,definition,
% 2.84/1.56 ( setukpairinjL1
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.56 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.56 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(kfstsingleton_type,type,
% 2.84/1.56 kfstsingleton: $o ).
% 2.84/1.56
% 2.84/1.56 thf(kfstsingleton,definition,
% 2.84/1.56 ( kfstsingleton
% 2.84/1.56 = ( ! [Xu: $i] :
% 2.84/1.56 ( ( iskpair @ Xu )
% 2.84/1.56 => ( singleton
% 2.84/1.56 @ ( dsetconstr @ ( setunion @ Xu )
% 2.84/1.56 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(theprop_type,type,
% 2.84/1.56 theprop: $o ).
% 2.84/1.56
% 2.84/1.56 thf(theprop,definition,
% 2.84/1.56 ( theprop
% 2.84/1.56 = ( ! [X: $i] :
% 2.84/1.56 ( ( singleton @ X )
% 2.84/1.56 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(kfst_type,type,
% 2.84/1.56 kfst: $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(kfstpairEq_type,type,
% 2.84/1.56 kfstpairEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(kfstpairEq,definition,
% 2.84/1.56 ( kfstpairEq
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.84/1.56 = Xx ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodfstin_type,type,
% 2.84/1.56 cartprodfstin: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodfstin,definition,
% 2.84/1.56 ( cartprodfstin
% 2.84/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.56 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjL2_type,type,
% 2.84/1.56 setukpairinjL2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjL2,definition,
% 2.84/1.56 ( setukpairinjL2
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.56 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjL_type,type,
% 2.84/1.56 setukpairinjL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjL,definition,
% 2.84/1.56 ( setukpairinjL
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.56 ( ( ( kpair @ Xx @ Xy )
% 2.84/1.56 = ( kpair @ Xz @ Xu ) )
% 2.84/1.56 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR11_type,type,
% 2.84/1.56 setukpairinjR11: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR11,definition,
% 2.84/1.56 ( setukpairinjR11
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx = Xy )
% 2.84/1.56 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.56 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR12_type,type,
% 2.84/1.56 setukpairinjR12: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR12,definition,
% 2.84/1.56 ( setukpairinjR12
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xx = Xy )
% 2.84/1.56 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.56 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR1_type,type,
% 2.84/1.56 setukpairinjR1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR1,definition,
% 2.84/1.56 ( setukpairinjR1
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.56 => ( ( Xz = Xu )
% 2.84/1.56 => ( Xy = Xu ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(upairequniteq_type,type,
% 2.84/1.56 upairequniteq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(upairequniteq,definition,
% 2.84/1.56 ( upairequniteq
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.56 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.56 = ( setadjoin @ Xz @ emptyset ) )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR2_type,type,
% 2.84/1.56 setukpairinjR2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR2,definition,
% 2.84/1.56 ( setukpairinjR2
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.56 => ( Xy = Xu ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR_type,type,
% 2.84/1.56 setukpairinjR: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setukpairinjR,definition,
% 2.84/1.56 ( setukpairinjR
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.56 ( ( ( kpair @ Xx @ Xy )
% 2.84/1.56 = ( kpair @ Xz @ Xu ) )
% 2.84/1.56 => ( Xy = Xu ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ksndsingleton_type,type,
% 2.84/1.56 ksndsingleton: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ksndsingleton,definition,
% 2.84/1.56 ( ksndsingleton
% 2.84/1.56 = ( ! [Xu: $i] :
% 2.84/1.56 ( ( iskpair @ Xu )
% 2.84/1.56 => ( singleton
% 2.84/1.56 @ ( dsetconstr @ ( setunion @ Xu )
% 2.84/1.56 @ ^ [Xx: $i] :
% 2.84/1.56 ( Xu
% 2.84/1.56 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ksnd_type,type,
% 2.84/1.56 ksnd: $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(ksndpairEq_type,type,
% 2.84/1.56 ksndpairEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ksndpairEq,definition,
% 2.84/1.56 ( ksndpairEq
% 2.84/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.84/1.56 = Xy ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(kpairsurjEq_type,type,
% 2.84/1.56 kpairsurjEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(kpairsurjEq,definition,
% 2.84/1.56 ( kpairsurjEq
% 2.84/1.56 = ( ! [Xu: $i] :
% 2.84/1.56 ( ( iskpair @ Xu )
% 2.84/1.56 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.84/1.56 = Xu ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodsndin_type,type,
% 2.84/1.56 cartprodsndin: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodsndin,definition,
% 2.84/1.56 ( cartprodsndin
% 2.84/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.56 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodpairmemEL_type,type,
% 2.84/1.56 cartprodpairmemEL: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodpairmemEL,definition,
% 2.84/1.56 ( cartprodpairmemEL
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodpairmemER_type,type,
% 2.84/1.56 cartprodpairmemER: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodpairmemER,definition,
% 2.84/1.56 ( cartprodpairmemER
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.84/1.56 => ( in @ Xy @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodmempaircEq_type,type,
% 2.84/1.56 cartprodmempaircEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodmempaircEq,definition,
% 2.84/1.56 ( cartprodmempaircEq
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( kpair @ Xx @ Xy )
% 2.84/1.56 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodfstpairEq_type,type,
% 2.84/1.56 cartprodfstpairEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodfstpairEq,definition,
% 2.84/1.56 ( cartprodfstpairEq
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.84/1.56 = Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodsndpairEq_type,type,
% 2.84/1.56 cartprodsndpairEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodsndpairEq,definition,
% 2.84/1.56 ( cartprodsndpairEq
% 2.84/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.84/1.56 = Xy ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodpairsurjEq_type,type,
% 2.84/1.56 cartprodpairsurjEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(cartprodpairsurjEq,definition,
% 2.84/1.56 ( cartprodpairsurjEq
% 2.84/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.56 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.84/1.56 = Xu ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(breln_type,type,
% 2.84/1.56 breln: $i > $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(breln,definition,
% 2.84/1.56 ( breln
% 2.84/1.56 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstr_type,type,
% 2.84/1.56 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrI_type,type,
% 2.84/1.56 dpsetconstrI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrI,definition,
% 2.84/1.56 ( dpsetconstrI
% 2.84/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( Xphi @ Xx @ Xy )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.56 @ ( dpsetconstr @ A @ B
% 2.84/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrSub_type,type,
% 2.84/1.56 dpsetconstrSub: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrSub,definition,
% 2.84/1.56 ( dpsetconstrSub
% 2.84/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.84/1.56 ( subset
% 2.84/1.56 @ ( dpsetconstr @ A @ B
% 2.84/1.56 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.84/1.56 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setOfPairsIsBReln_type,type,
% 2.84/1.56 setOfPairsIsBReln: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setOfPairsIsBReln,definition,
% 2.84/1.56 ( setOfPairsIsBReln
% 2.84/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.84/1.56 ( breln @ A @ B
% 2.84/1.56 @ ( dpsetconstr @ A @ B
% 2.84/1.56 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrERa_type,type,
% 2.84/1.56 dpsetconstrERa: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrERa,definition,
% 2.84/1.56 ( dpsetconstrERa
% 2.84/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.56 @ ( dpsetconstr @ A @ B
% 2.84/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.56 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrEL1_type,type,
% 2.84/1.56 dpsetconstrEL1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrEL1,definition,
% 2.84/1.56 ( dpsetconstrEL1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.56 @ ( dpsetconstr @ A @ B
% 2.84/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrEL2_type,type,
% 2.84/1.56 dpsetconstrEL2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrEL2,definition,
% 2.84/1.56 ( dpsetconstrEL2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.56 @ ( dpsetconstr @ A @ B
% 2.84/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.56 => ( in @ Xy @ B ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrER_type,type,
% 2.84/1.56 dpsetconstrER: $o ).
% 2.84/1.56
% 2.84/1.56 thf(dpsetconstrER,definition,
% 2.84/1.56 ( dpsetconstrER
% 2.84/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.56 @ ( dpsetconstr @ A @ B
% 2.84/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.56 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(func_type,type,
% 2.84/1.56 func: $i > $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcSet_type,type,
% 2.84/1.56 funcSet: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(funcImageSingleton_type,type,
% 2.84/1.56 funcImageSingleton: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcImageSingleton,definition,
% 2.84/1.56 ( funcImageSingleton
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( singleton
% 2.84/1.56 @ ( dsetconstr @ B
% 2.84/1.56 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(apProp_type,type,
% 2.84/1.56 apProp: $o ).
% 2.84/1.56
% 2.84/1.56 thf(apProp,definition,
% 2.84/1.56 ( apProp
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in
% 2.84/1.56 @ ( setunion
% 2.84/1.56 @ ( dsetconstr @ B
% 2.84/1.56 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.84/1.56 @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ap_type,type,
% 2.84/1.56 ap: $i > $i > $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(app_type,type,
% 2.84/1.56 app: $o ).
% 2.84/1.56
% 2.84/1.56 thf(app,definition,
% 2.84/1.56 ( app
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(infuncsetfunc_type,type,
% 2.84/1.56 infuncsetfunc: $o ).
% 2.84/1.56
% 2.84/1.56 thf(infuncsetfunc,definition,
% 2.84/1.56 ( infuncsetfunc
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.56 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ap2p_type,type,
% 2.84/1.56 ap2p: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ap2p,definition,
% 2.84/1.56 ( ap2p
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(funcinfuncset_type,type,
% 2.84/1.56 funcinfuncset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcinfuncset,definition,
% 2.84/1.56 ( funcinfuncset
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(lamProp_type,type,
% 2.84/1.56 lamProp: $o ).
% 2.84/1.56
% 2.84/1.56 thf(lamProp,definition,
% 2.84/1.56 ( lamProp
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.56 => ( func @ A @ B
% 2.84/1.56 @ ( dpsetconstr @ A @ B
% 2.84/1.56 @ ^ [Xx: $i,Xy: $i] :
% 2.84/1.56 ( ( Xf @ Xx )
% 2.84/1.56 = Xy ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(lam_type,type,
% 2.84/1.56 lam: $i > $i > ( $i > $i ) > $i ).
% 2.84/1.56
% 2.84/1.56 thf(lamp_type,type,
% 2.84/1.56 lamp: $o ).
% 2.84/1.56
% 2.84/1.56 thf(lamp,definition,
% 2.84/1.56 ( lamp
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.56 => ( func @ A @ B
% 2.84/1.56 @ ( lam @ A @ B
% 2.84/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(lam2p_type,type,
% 2.84/1.56 lam2p: $o ).
% 2.84/1.56
% 2.84/1.56 thf(lam2p,definition,
% 2.84/1.56 ( lam2p
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.56 => ( in
% 2.84/1.56 @ ( lam @ A @ B
% 2.84/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.84/1.56 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(brelnall1_type,type,
% 2.84/1.56 brelnall1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(brelnall1,definition,
% 2.84/1.56 ( brelnall1
% 2.84/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 2.84/1.56 ( ( breln @ A @ B @ R )
% 2.84/1.56 => ! [Xphi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.56 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ R )
% 2.84/1.56 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(brelnall2_type,type,
% 2.84/1.56 brelnall2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(brelnall2,definition,
% 2.84/1.56 ( brelnall2
% 2.84/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 2.84/1.56 ( ( breln @ A @ B @ R )
% 2.84/1.56 => ! [Xphi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.56 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ R )
% 2.84/1.56 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ex1E2_type,type,
% 2.84/1.56 ex1E2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ex1E2,definition,
% 2.84/1.56 ( ex1E2
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.56 ( ( ex1 @ A
% 2.84/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( Xphi @ Xx )
% 2.84/1.56 => ( ( Xphi @ Xy )
% 2.84/1.56 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(funcGraphProp1_type,type,
% 2.84/1.56 funcGraphProp1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcGraphProp1,definition,
% 2.84/1.56 ( funcGraphProp1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(funcGraphProp3_type,type,
% 2.84/1.56 funcGraphProp3: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcGraphProp3,definition,
% 2.84/1.56 ( funcGraphProp3
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(funcGraphProp2_type,type,
% 2.84/1.56 funcGraphProp2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcGraphProp2,definition,
% 2.84/1.56 ( funcGraphProp2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.84/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.56 = Xy ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(funcextLem_type,type,
% 2.84/1.56 funcextLem: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcextLem,definition,
% 2.84/1.56 ( funcextLem
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ! [Xg: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xg )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(funcGraphProp4_type,type,
% 2.84/1.56 funcGraphProp4: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcGraphProp4,definition,
% 2.84/1.56 ( funcGraphProp4
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.84/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.56 = Xy ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subbreln_type,type,
% 2.84/1.56 subbreln: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subbreln,definition,
% 2.84/1.56 ( subbreln
% 2.84/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 2.84/1.56 ( ( breln @ A @ B @ R )
% 2.84/1.56 => ! [S: $i] :
% 2.84/1.56 ( ( breln @ A @ B @ S )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.84/1.56 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(eqbreln_type,type,
% 2.84/1.56 eqbreln: $o ).
% 2.84/1.56
% 2.84/1.56 thf(eqbreln,definition,
% 2.84/1.56 ( eqbreln
% 2.84/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 2.84/1.56 ( ( breln @ A @ B @ R )
% 2.84/1.56 => ! [S: $i] :
% 2.84/1.56 ( ( breln @ A @ B @ S )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.84/1.56 => ( R = S ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(funcext_type,type,
% 2.84/1.56 funcext: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcext,definition,
% 2.84/1.56 ( funcext
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ! [Xg: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xg )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.84/1.56 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(funcext2_type,type,
% 2.84/1.56 funcext2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(funcext2,definition,
% 2.84/1.56 ( funcext2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.56 => ! [Xg: $i] :
% 2.84/1.56 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.84/1.56 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ap2apEq1_type,type,
% 2.84/1.56 ap2apEq1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ap2apEq1,definition,
% 2.84/1.56 ( ap2apEq1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.56 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ap2apEq2_type,type,
% 2.84/1.56 ap2apEq2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ap2apEq2,definition,
% 2.84/1.56 ( ap2apEq2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.84/1.56 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(beta1_type,type,
% 2.84/1.56 beta1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(beta1,definition,
% 2.84/1.56 ( beta1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( ap @ A @ B
% 2.84/1.56 @ ( lam @ A @ B
% 2.84/1.56 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.84/1.56 @ Xx )
% 2.84/1.56 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(eta1_type,type,
% 2.84/1.56 eta1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(eta1,definition,
% 2.84/1.56 ( eta1
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( func @ A @ B @ Xf )
% 2.84/1.56 => ( ( lam @ A @ B
% 2.84/1.56 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.84/1.56 = Xf ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(lam2lamEq_type,type,
% 2.84/1.56 lam2lamEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(lam2lamEq,definition,
% 2.84/1.56 ( lam2lamEq
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.56 => ( ( lam @ A @ B
% 2.84/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.84/1.56 = ( lam @ A @ B
% 2.84/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(beta2_type,type,
% 2.84/1.56 beta2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(beta2,definition,
% 2.84/1.56 ( beta2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( ap @ A @ B
% 2.84/1.56 @ ( lam @ A @ B
% 2.84/1.56 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.84/1.56 @ Xx )
% 2.84/1.56 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(eta2_type,type,
% 2.84/1.56 eta2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(eta2,definition,
% 2.84/1.56 ( eta2
% 2.84/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.84/1.56 => ( ( lam @ A @ B
% 2.84/1.56 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.84/1.56 = Xf ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(iffalseProp1_type,type,
% 2.84/1.56 iffalseProp1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(iffalseProp1,definition,
% 2.84/1.56 ( iffalseProp1
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ~ Xphi
% 2.84/1.56 => ( in @ Xy
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xz: $i] :
% 2.84/1.56 ( ( Xphi
% 2.84/1.56 & ( Xz = Xx ) )
% 2.84/1.56 | ( ~ Xphi
% 2.84/1.56 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(iffalseProp2_type,type,
% 2.84/1.56 iffalseProp2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(iffalseProp2,definition,
% 2.84/1.56 ( iffalseProp2
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ~ Xphi
% 2.84/1.56 => ( ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xz: $i] :
% 2.84/1.56 ( ( Xphi
% 2.84/1.56 & ( Xz = Xx ) )
% 2.84/1.56 | ( ~ Xphi
% 2.84/1.56 & ( Xz = Xy ) ) ) )
% 2.84/1.56 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(iftrueProp1_type,type,
% 2.84/1.56 iftrueProp1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(iftrueProp1,definition,
% 2.84/1.56 ( iftrueProp1
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( Xphi
% 2.84/1.56 => ( in @ Xx
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xz: $i] :
% 2.84/1.56 ( ( Xphi
% 2.84/1.56 & ( Xz = Xx ) )
% 2.84/1.56 | ( ~ Xphi
% 2.84/1.56 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(iftrueProp2_type,type,
% 2.84/1.56 iftrueProp2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(iftrueProp2,definition,
% 2.84/1.56 ( iftrueProp2
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( Xphi
% 2.84/1.56 => ( ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xz: $i] :
% 2.84/1.56 ( ( Xphi
% 2.84/1.56 & ( Xz = Xx ) )
% 2.84/1.56 | ( ~ Xphi
% 2.84/1.56 & ( Xz = Xy ) ) ) )
% 2.84/1.56 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(ifSingleton_type,type,
% 2.84/1.56 ifSingleton: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ifSingleton,definition,
% 2.84/1.56 ( ifSingleton
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( singleton
% 2.84/1.56 @ ( dsetconstr @ A
% 2.84/1.56 @ ^ [Xz: $i] :
% 2.84/1.56 ( ( Xphi
% 2.84/1.56 & ( Xz = Xx ) )
% 2.84/1.56 | ( ~ Xphi
% 2.84/1.56 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(if_type,type,
% 2.84/1.56 if: $i > $o > $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(ifp_type,type,
% 2.84/1.56 ifp: $o ).
% 2.84/1.56
% 2.84/1.56 thf(ifp,definition,
% 2.84/1.56 ( ifp
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(theeq_type,type,
% 2.84/1.56 theeq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(theeq,definition,
% 2.84/1.56 ( theeq
% 2.84/1.56 = ( ! [X: $i] :
% 2.84/1.56 ( ( singleton @ X )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ X )
% 2.84/1.56 => ( ( setunion @ X )
% 2.84/1.56 = Xx ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(iftrue_type,type,
% 2.84/1.56 iftrue: $o ).
% 2.84/1.56
% 2.84/1.56 thf(iftrue,definition,
% 2.84/1.56 ( iftrue
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( Xphi
% 2.84/1.56 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.84/1.56 = Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(iffalse_type,type,
% 2.84/1.56 iffalse: $o ).
% 2.84/1.56
% 2.84/1.56 thf(iffalse,definition,
% 2.84/1.56 ( iffalse
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ~ Xphi
% 2.84/1.56 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.84/1.56 = Xy ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(iftrueorfalse_type,type,
% 2.84/1.56 iftrueorfalse: $o ).
% 2.84/1.56
% 2.84/1.56 thf(iftrueorfalse,definition,
% 2.84/1.56 ( iftrueorfalse
% 2.84/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectT_lem_type,type,
% 2.84/1.56 binintersectT_lem: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectT_lem,definition,
% 2.84/1.56 ( binintersectT_lem
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionT_lem_type,type,
% 2.84/1.56 binunionT_lem: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionT_lem,definition,
% 2.84/1.56 ( binunionT_lem
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powersetT_lem_type,type,
% 2.84/1.56 powersetT_lem: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetT_lem,definition,
% 2.84/1.56 ( powersetT_lem
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setminusT_lem_type,type,
% 2.84/1.56 setminusT_lem: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setminusT_lem,definition,
% 2.84/1.56 ( setminusT_lem
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementT_lem_type,type,
% 2.84/1.56 complementT_lem: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementT_lem,definition,
% 2.84/1.56 ( complementT_lem
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setextT_type,type,
% 2.84/1.56 setextT: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setextT,definition,
% 2.84/1.56 ( setextT
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ X )
% 2.84/1.56 => ( in @ Xx @ Y ) ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ Y )
% 2.84/1.56 => ( in @ Xx @ X ) ) )
% 2.84/1.56 => ( X = Y ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subsetTI_type,type,
% 2.84/1.56 subsetTI: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subsetTI,definition,
% 2.84/1.56 ( subsetTI
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ X )
% 2.84/1.56 => ( in @ Xx @ Y ) ) )
% 2.84/1.56 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powersetTI1_type,type,
% 2.84/1.56 powersetTI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetTI1,definition,
% 2.84/1.56 ( powersetTI1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ X )
% 2.84/1.56 => ( in @ Xx @ Y ) ) )
% 2.84/1.56 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(powersetTE1_type,type,
% 2.84/1.56 powersetTE1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(powersetTE1,definition,
% 2.84/1.56 ( powersetTE1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ X @ ( powerset @ Y ) )
% 2.84/1.56 => ( ( in @ Xx @ X )
% 2.84/1.56 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementTI1_type,type,
% 2.84/1.56 complementTI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementTI1,definition,
% 2.84/1.56 ( complementTI1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ X )
% 2.84/1.56 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementTE1_type,type,
% 2.84/1.56 complementTE1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementTE1,definition,
% 2.84/1.56 ( complementTE1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.84/1.56 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectTELcontra_type,type,
% 2.84/1.56 binintersectTELcontra: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectTELcontra,definition,
% 2.84/1.56 ( binintersectTELcontra
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ X )
% 2.84/1.56 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectTERcontra_type,type,
% 2.84/1.56 binintersectTERcontra: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binintersectTERcontra,definition,
% 2.84/1.56 ( binintersectTERcontra
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ Y )
% 2.84/1.56 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(contrasubsetT_type,type,
% 2.84/1.56 contrasubsetT: $o ).
% 2.84/1.56
% 2.84/1.56 thf(contrasubsetT,definition,
% 2.84/1.56 ( contrasubsetT
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.84/1.56 => ( ( in @ Xx @ Y )
% 2.84/1.56 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(contrasubsetT1_type,type,
% 2.84/1.56 contrasubsetT1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(contrasubsetT1,definition,
% 2.84/1.56 ( contrasubsetT1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( subset @ X @ Y )
% 2.84/1.56 => ( ~ ( in @ Xx @ Y )
% 2.84/1.56 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(contrasubsetT2_type,type,
% 2.84/1.56 contrasubsetT2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(contrasubsetT2,definition,
% 2.84/1.56 ( contrasubsetT2
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ X @ Y )
% 2.84/1.56 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(contrasubsetT3_type,type,
% 2.84/1.56 contrasubsetT3: $o ).
% 2.84/1.56
% 2.84/1.56 thf(contrasubsetT3,definition,
% 2.84/1.56 ( contrasubsetT3
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.84/1.56 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementI1_type,type,
% 2.84/1.56 doubleComplementI1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementI1,definition,
% 2.84/1.56 ( doubleComplementI1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ X )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementE1_type,type,
% 2.84/1.56 doubleComplementE1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementE1,definition,
% 2.84/1.56 ( doubleComplementE1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.84/1.56 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementSub1_type,type,
% 2.84/1.56 doubleComplementSub1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementSub1,definition,
% 2.84/1.56 ( doubleComplementSub1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementSub2_type,type,
% 2.84/1.56 doubleComplementSub2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementSub2,definition,
% 2.84/1.56 ( doubleComplementSub2
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementEq_type,type,
% 2.84/1.56 doubleComplementEq: $o ).
% 2.84/1.56
% 2.84/1.56 thf(doubleComplementEq,definition,
% 2.84/1.56 ( doubleComplementEq
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ( X
% 2.84/1.56 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementTnotintersectT_type,type,
% 2.84/1.56 complementTnotintersectT: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementTnotintersectT,definition,
% 2.84/1.56 ( complementTnotintersectT
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.84/1.56 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementImpComplementIntersect_type,type,
% 2.84/1.56 complementImpComplementIntersect: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementImpComplementIntersect,definition,
% 2.84/1.56 ( complementImpComplementIntersect
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementSubsetComplementIntersect_type,type,
% 2.84/1.56 complementSubsetComplementIntersect: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementSubsetComplementIntersect,definition,
% 2.84/1.56 ( complementSubsetComplementIntersect
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementInPowersetComplementIntersect_type,type,
% 2.84/1.56 complementInPowersetComplementIntersect: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementInPowersetComplementIntersect,definition,
% 2.84/1.56 ( complementInPowersetComplementIntersect
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(contraSubsetComplement_type,type,
% 2.84/1.56 contraSubsetComplement: $o ).
% 2.84/1.56
% 2.84/1.56 thf(contraSubsetComplement,definition,
% 2.84/1.56 ( contraSubsetComplement
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ Y )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementTcontraSubset_type,type,
% 2.84/1.56 complementTcontraSubset: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementTcontraSubset,definition,
% 2.84/1.56 ( complementTcontraSubset
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.84/1.56 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionTILcontra_type,type,
% 2.84/1.56 binunionTILcontra: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionTILcontra,definition,
% 2.84/1.56 ( binunionTILcontra
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.84/1.56 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionTIRcontra_type,type,
% 2.84/1.56 binunionTIRcontra: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionTIRcontra,definition,
% 2.84/1.56 ( binunionTIRcontra
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.84/1.56 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(inIntersectImpInUnion_type,type,
% 2.84/1.56 inIntersectImpInUnion: $o ).
% 2.84/1.56
% 2.84/1.56 thf(inIntersectImpInUnion,definition,
% 2.84/1.56 ( inIntersectImpInUnion
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.84/1.56 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(inIntersectImpInUnion2_type,type,
% 2.84/1.56 inIntersectImpInUnion2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(inIntersectImpInUnion2,definition,
% 2.84/1.56 ( inIntersectImpInUnion2
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.84/1.56 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(inIntersectImpInIntersectUnions_type,type,
% 2.84/1.56 inIntersectImpInIntersectUnions: $o ).
% 2.84/1.56
% 2.84/1.56 thf(inIntersectImpInIntersectUnions,definition,
% 2.84/1.56 ( inIntersectImpInIntersectUnions
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.84/1.56 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(intersectInPowersetIntersectUnions_type,type,
% 2.84/1.56 intersectInPowersetIntersectUnions: $o ).
% 2.84/1.56
% 2.84/1.56 thf(intersectInPowersetIntersectUnions,definition,
% 2.84/1.56 ( intersectInPowersetIntersectUnions
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(inComplementUnionImpNotIn1_type,type,
% 2.84/1.56 inComplementUnionImpNotIn1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(inComplementUnionImpNotIn1,definition,
% 2.84/1.56 ( inComplementUnionImpNotIn1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.56 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(inComplementUnionImpInComplement1_type,type,
% 2.84/1.56 inComplementUnionImpInComplement1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(inComplementUnionImpInComplement1,definition,
% 2.84/1.56 ( inComplementUnionImpInComplement1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionTE_type,type,
% 2.84/1.56 binunionTE: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionTE,definition,
% 2.84/1.56 ( binunionTE
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xphi: $o,Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.84/1.56 => ( ( ( in @ Xx @ X )
% 2.84/1.56 => Xphi )
% 2.84/1.56 => ( ( ( in @ Xx @ Y )
% 2.84/1.56 => Xphi )
% 2.84/1.56 => Xphi ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(binunionTEcontra_type,type,
% 2.84/1.56 binunionTEcontra: $o ).
% 2.84/1.56
% 2.84/1.56 thf(binunionTEcontra,definition,
% 2.84/1.56 ( binunionTEcontra
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ~ ( in @ Xx @ X )
% 2.84/1.56 => ( ~ ( in @ Xx @ Y )
% 2.84/1.56 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2a1_type,type,
% 2.84/1.56 demorgan2a1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2a1,definition,
% 2.84/1.56 ( demorgan2a1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(complementUnionInPowersetComplement_type,type,
% 2.84/1.56 complementUnionInPowersetComplement: $o ).
% 2.84/1.56
% 2.84/1.56 thf(complementUnionInPowersetComplement,definition,
% 2.84/1.56 ( complementUnionInPowersetComplement
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2a2_type,type,
% 2.84/1.56 demorgan2a2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2a2,definition,
% 2.84/1.56 ( demorgan2a2
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan1a_type,type,
% 2.84/1.56 demorgan1a: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan1a,definition,
% 2.84/1.56 ( demorgan1a
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.84/1.56 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan1b_type,type,
% 2.84/1.56 demorgan1b: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan1b,definition,
% 2.84/1.56 ( demorgan1b
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan1_type,type,
% 2.84/1.56 demorgan1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan1,definition,
% 2.84/1.56 ( demorgan1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.84/1.56 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2a_type,type,
% 2.84/1.56 demorgan2a: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2a,definition,
% 2.84/1.56 ( demorgan2a
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.84/1.56 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2b2_type,type,
% 2.84/1.56 demorgan2b2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2b2,definition,
% 2.84/1.56 ( demorgan2b2
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.84/1.56 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2b_type,type,
% 2.84/1.56 demorgan2b: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2b,definition,
% 2.84/1.56 ( demorgan2b
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.84/1.56 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2_type,type,
% 2.84/1.56 demorgan2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(demorgan2,definition,
% 2.84/1.56 ( demorgan2
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.84/1.56 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule0_type,type,
% 2.84/1.56 woz13rule0: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule0,definition,
% 2.84/1.56 ( woz13rule0
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.84/1.56 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule1_type,type,
% 2.84/1.56 woz13rule1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule1,definition,
% 2.84/1.56 ( woz13rule1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ X @ Z )
% 2.84/1.56 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule2_type,type,
% 2.84/1.56 woz13rule2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule2,definition,
% 2.84/1.56 ( woz13rule2
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ Y @ Z )
% 2.84/1.56 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule3_type,type,
% 2.84/1.56 woz13rule3: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule3,definition,
% 2.84/1.56 ( woz13rule3
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ X @ Y )
% 2.84/1.56 => ( ( subset @ X @ Z )
% 2.84/1.56 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule4_type,type,
% 2.84/1.56 woz13rule4: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz13rule4,definition,
% 2.84/1.56 ( woz13rule4
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ! [W: $i] :
% 2.84/1.56 ( ( in @ W @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ X @ Z )
% 2.84/1.56 => ( ( subset @ Y @ W )
% 2.84/1.56 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_1_type,type,
% 2.84/1.56 woz1_1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_1,definition,
% 2.84/1.56 ( woz1_1
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_2_type,type,
% 2.84/1.56 woz1_2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_2,definition,
% 2.84/1.56 ( woz1_2
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ! [W: $i] :
% 2.84/1.56 ( ( in @ W @ ( powerset @ A ) )
% 2.84/1.56 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.84/1.56 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_3_type,type,
% 2.84/1.56 woz1_3: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_3,definition,
% 2.84/1.56 ( woz1_3
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ! [Z: $i] :
% 2.84/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_4_type,type,
% 2.84/1.56 woz1_4: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_4,definition,
% 2.84/1.56 ( woz1_4
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.84/1.56 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_5_type,type,
% 2.84/1.56 woz1_5: $o ).
% 2.84/1.56
% 2.84/1.56 thf(woz1_5,definition,
% 2.84/1.56 ( woz1_5
% 2.84/1.56 = ( ! [A: $i,X: $i] :
% 2.84/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.84/1.56 => ! [Y: $i] :
% 2.84/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.84/1.56 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(breln1_type,type,
% 2.84/1.56 breln1: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(breln1,definition,
% 2.84/1.56 ( breln1
% 2.84/1.56 = ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(breln1all2_type,type,
% 2.84/1.56 breln1all2: $o ).
% 2.84/1.56
% 2.84/1.56 thf(breln1all2,definition,
% 2.84/1.56 ( breln1all2
% 2.84/1.56 = ( ! [A: $i,R: $i] :
% 2.84/1.56 ( ( breln1 @ A @ R )
% 2.84/1.56 => ! [Xphi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.56 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ R )
% 2.84/1.56 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(breln1Set_type,type,
% 2.84/1.56 breln1Set: $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(breln1SetBreln1_type,type,
% 2.84/1.56 breln1SetBreln1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(breln1SetBreln1,definition,
% 2.84/1.56 ( breln1SetBreln1
% 2.84/1.56 = ( ! [A: $i,R: $i] :
% 2.84/1.56 ( ( in @ R @ ( breln1Set @ A ) )
% 2.84/1.56 => ( breln1 @ A @ R ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(transitive_type,type,
% 2.84/1.56 transitive: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(antisymmetric_type,type,
% 2.84/1.56 antisymmetric: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(reflexive_type,type,
% 2.84/1.56 reflexive: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(refltransitive_type,type,
% 2.84/1.56 refltransitive: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(refllinearorder_type,type,
% 2.84/1.56 refllinearorder: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(reflwellordering_type,type,
% 2.84/1.56 reflwellordering: $i > $i > $o ).
% 2.84/1.56
% 2.84/1.56 thf(choice2fnsingleton_type,type,
% 2.84/1.56 choice2fnsingleton: $o ).
% 2.84/1.56
% 2.84/1.56 thf(choice2fnsingleton,definition,
% 2.84/1.56 ( choice2fnsingleton
% 2.84/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ? [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ B )
% 2.84/1.56 & ( Xphi @ Xx @ Xy ) ) )
% 2.84/1.56 => ! [R: $i] :
% 2.84/1.56 ( ( in @ R @ ( breln1Set @ B ) )
% 2.84/1.56 => ( ( reflwellordering @ B @ R )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ( singleton
% 2.84/1.56 @ ( dsetconstr @ B
% 2.84/1.56 @ ^ [Xy: $i] :
% 2.84/1.56 ( ( Xphi @ Xx @ Xy )
% 2.84/1.56 & ! [Xz: $i] :
% 2.84/1.56 ( ( in @ Xz @ B )
% 2.84/1.56 => ( ( Xphi @ Xx @ Xz )
% 2.84/1.56 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(setOfPairsIsBReln1_type,type,
% 2.84/1.56 setOfPairsIsBReln1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(setOfPairsIsBReln1,definition,
% 2.84/1.56 ( setOfPairsIsBReln1
% 2.84/1.56 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 2.84/1.56 ( breln1 @ A
% 2.84/1.56 @ ( dpsetconstr @ A @ A
% 2.84/1.56 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(breln1all1_type,type,
% 2.84/1.56 breln1all1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(breln1all1,definition,
% 2.84/1.56 ( breln1all1
% 2.84/1.56 = ( ! [A: $i,R: $i] :
% 2.84/1.56 ( ( breln1 @ A @ R )
% 2.84/1.56 => ! [Xphi: $i > $o] :
% 2.84/1.56 ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.56 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.84/1.56 => ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ R )
% 2.84/1.56 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(subbreln1_type,type,
% 2.84/1.56 subbreln1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(subbreln1,definition,
% 2.84/1.56 ( subbreln1
% 2.84/1.56 = ( ! [A: $i,R: $i] :
% 2.84/1.56 ( ( breln1 @ A @ R )
% 2.84/1.56 => ! [S: $i] :
% 2.84/1.56 ( ( breln1 @ A @ S )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.84/1.56 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(eqbreln1_type,type,
% 2.84/1.56 eqbreln1: $o ).
% 2.84/1.56
% 2.84/1.56 thf(eqbreln1,definition,
% 2.84/1.56 ( eqbreln1
% 2.84/1.56 = ( ! [A: $i,R: $i] :
% 2.84/1.56 ( ( breln1 @ A @ R )
% 2.84/1.56 => ! [S: $i] :
% 2.84/1.56 ( ( breln1 @ A @ S )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.84/1.56 => ( ! [Xx: $i] :
% 2.84/1.56 ( ( in @ Xx @ A )
% 2.84/1.56 => ! [Xy: $i] :
% 2.84/1.56 ( ( in @ Xy @ A )
% 2.84/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.84/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.84/1.56 => ( R = S ) ) ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(breln1invset_type,type,
% 2.84/1.56 breln1invset: $i > $i > $i ).
% 2.84/1.56
% 2.84/1.56 thf(breln1invset,definition,
% 2.84/1.56 ( breln1invset
% 2.84/1.56 = ( ^ [A: $i,R: $i] :
% 2.84/1.56 ( dpsetconstr @ A @ A
% 2.84/1.56 @ ^ [Xx: $i,Xy: $i] : ( in @ ( kpair @ Xy @ Xx ) @ R ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(breln1invprop_type,type,
% 2.84/1.56 breln1invprop: $o ).
% 2.84/1.56
% 2.84/1.56 thf(breln1invprop,definition,
% 2.84/1.56 ( breln1invprop
% 2.84/1.56 = ( ! [A: $i,R: $i] :
% 2.84/1.56 ( ( breln1 @ A @ R )
% 2.84/1.56 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 2.84/1.56
% 2.84/1.56 thf(breln1invI,conjecture,
% 2.84/1.56 ( setextAx
% 2.84/1.56 => ( emptysetAx
% 2.84/1.56 => ( setadjoinAx
% 2.84/1.56 => ( powersetAx
% 2.84/1.56 => ( setunionAx
% 2.84/1.56 => ( omega0Ax
% 2.84/1.56 => ( omegaSAx
% 2.84/1.56 => ( omegaIndAx
% 2.84/1.56 => ( replAx
% 2.84/1.56 => ( foundationAx
% 2.84/1.56 => ( wellorderingAx
% 2.84/1.56 => ( descrp
% 2.84/1.57 => ( dsetconstrI
% 2.84/1.57 => ( dsetconstrEL
% 2.84/1.57 => ( dsetconstrER
% 2.84/1.57 => ( exuE1
% 2.84/1.57 => ( prop2setE
% 2.84/1.57 => ( emptysetE
% 2.84/1.57 => ( emptysetimpfalse
% 2.84/1.57 => ( notinemptyset
% 2.84/1.57 => ( exuE3e
% 2.84/1.57 => ( setext
% 2.84/1.57 => ( emptyI
% 2.84/1.57 => ( noeltsimpempty
% 2.84/1.57 => ( setbeta
% 2.84/1.57 => ( nonemptyE1
% 2.84/1.57 => ( nonemptyI
% 2.84/1.57 => ( nonemptyI1
% 2.84/1.57 => ( setadjoinIL
% 2.84/1.57 => ( emptyinunitempty
% 2.84/1.57 => ( setadjoinIR
% 2.84/1.57 => ( setadjoinE
% 2.84/1.57 => ( setadjoinOr
% 2.84/1.57 => ( setoftrueEq
% 2.84/1.57 => ( powersetI
% 2.84/1.57 => ( emptyinPowerset
% 2.84/1.57 => ( emptyInPowerset
% 2.84/1.57 => ( powersetE
% 2.84/1.57 => ( setunionI
% 2.84/1.57 => ( setunionE
% 2.84/1.57 => ( subPowSU
% 2.84/1.57 => ( exuE2
% 2.84/1.57 => ( nonemptyImpWitness
% 2.84/1.57 => ( uniqinunit
% 2.84/1.57 => ( notinsingleton
% 2.84/1.57 => ( eqinunit
% 2.84/1.57 => ( singletonsswitch
% 2.84/1.57 => ( upairsetE
% 2.84/1.57 => ( upairsetIL
% 2.84/1.57 => ( upairsetIR
% 2.84/1.57 => ( emptyE1
% 2.84/1.57 => ( vacuousDall
% 2.84/1.57 => ( quantDeMorgan1
% 2.84/1.57 => ( quantDeMorgan2
% 2.84/1.57 => ( quantDeMorgan3
% 2.84/1.57 => ( quantDeMorgan4
% 2.84/1.57 => ( prop2setI
% 2.84/1.57 => ( prop2set2propI
% 2.84/1.57 => ( notdexE
% 2.84/1.57 => ( notdallE
% 2.84/1.57 => ( exuI1
% 2.84/1.57 => ( exuI3
% 2.84/1.57 => ( exuI2
% 2.84/1.57 => ( inCongP
% 2.84/1.57 => ( in__Cong
% 2.84/1.57 => ( exuE3u
% 2.84/1.57 => ( exu__Cong
% 2.84/1.57 => ( emptyset__Cong
% 2.84/1.57 => ( setadjoin__Cong
% 2.84/1.57 => ( powerset__Cong
% 2.84/1.57 => ( setunion__Cong
% 2.84/1.57 => ( omega__Cong
% 2.84/1.57 => ( exuEu
% 2.84/1.57 => ( descr__Cong
% 2.84/1.57 => ( dsetconstr__Cong
% 2.84/1.57 => ( subsetI1
% 2.84/1.57 => ( eqimpsubset2
% 2.84/1.57 => ( eqimpsubset1
% 2.84/1.57 => ( subsetI2
% 2.84/1.57 => ( emptysetsubset
% 2.84/1.57 => ( subsetE
% 2.84/1.57 => ( subsetE2
% 2.84/1.57 => ( notsubsetI
% 2.84/1.57 => ( notequalI1
% 2.84/1.57 => ( notequalI2
% 2.84/1.57 => ( subsetRefl
% 2.84/1.57 => ( subsetTrans
% 2.84/1.57 => ( setadjoinSub
% 2.84/1.57 => ( setadjoinSub2
% 2.84/1.57 => ( subset2powerset
% 2.84/1.57 => ( setextsub
% 2.84/1.57 => ( subsetemptysetimpeq
% 2.84/1.57 => ( powersetI1
% 2.84/1.57 => ( powersetE1
% 2.84/1.57 => ( inPowerset
% 2.84/1.57 => ( powersetsubset
% 2.84/1.57 => ( sepInPowerset
% 2.84/1.57 => ( sepSubset
% 2.84/1.57 => ( binunionIL
% 2.84/1.57 => ( upairset2IR
% 2.84/1.57 => ( binunionIR
% 2.84/1.57 => ( binunionEcases
% 2.84/1.57 => ( binunionE
% 2.84/1.57 => ( binunionLsub
% 2.84/1.57 => ( binunionRsub
% 2.84/1.57 => ( binintersectI
% 2.84/1.57 => ( binintersectSubset5
% 2.84/1.57 => ( binintersectEL
% 2.84/1.57 => ( binintersectLsub
% 2.84/1.57 => ( binintersectSubset2
% 2.84/1.57 => ( binintersectSubset3
% 2.84/1.57 => ( binintersectER
% 2.84/1.57 => ( disjointsetsI1
% 2.84/1.57 => ( binintersectRsub
% 2.84/1.57 => ( binintersectSubset4
% 2.84/1.57 => ( binintersectSubset1
% 2.84/1.57 => ( bs114d
% 2.84/1.57 => ( setminusI
% 2.84/1.57 => ( setminusEL
% 2.84/1.57 => ( setminusER
% 2.84/1.57 => ( setminusSubset2
% 2.84/1.57 => ( setminusERneg
% 2.84/1.57 => ( setminusELneg
% 2.84/1.57 => ( setminusILneg
% 2.84/1.57 => ( setminusIRneg
% 2.84/1.57 => ( setminusLsub
% 2.84/1.57 => ( setminusSubset1
% 2.84/1.57 => ( symdiffE
% 2.84/1.57 => ( symdiffI1
% 2.84/1.57 => ( symdiffI2
% 2.84/1.57 => ( symdiffIneg1
% 2.84/1.57 => ( symdiffIneg2
% 2.84/1.57 => ( secondinupair
% 2.84/1.57 => ( setukpairIL
% 2.84/1.57 => ( setukpairIR
% 2.84/1.57 => ( kpairiskpair
% 2.84/1.57 => ( kpairp
% 2.84/1.57 => ( singletonsubset
% 2.84/1.57 => ( singletoninpowerset
% 2.84/1.57 => ( singletoninpowunion
% 2.84/1.57 => ( upairset2E
% 2.84/1.57 => ( upairsubunion
% 2.84/1.57 => ( upairinpowunion
% 2.84/1.57 => ( ubforcartprodlem1
% 2.84/1.57 => ( ubforcartprodlem2
% 2.84/1.57 => ( ubforcartprodlem3
% 2.84/1.57 => ( cartprodpairin
% 2.84/1.57 => ( cartprodmempair1
% 2.84/1.57 => ( cartprodmempair
% 2.84/1.57 => ( setunionE2
% 2.84/1.57 => ( setunionsingleton1
% 2.84/1.57 => ( setunionsingleton2
% 2.84/1.57 => ( setunionsingleton
% 2.84/1.57 => ( singletonprop
% 2.84/1.57 => ( ex1E1
% 2.84/1.57 => ( ex1I
% 2.84/1.57 => ( ex1I2
% 2.84/1.57 => ( singletonsuniq
% 2.84/1.57 => ( setukpairinjL1
% 2.84/1.57 => ( kfstsingleton
% 2.84/1.57 => ( theprop
% 2.84/1.57 => ( kfstpairEq
% 2.84/1.57 => ( cartprodfstin
% 2.84/1.57 => ( setukpairinjL2
% 2.84/1.57 => ( setukpairinjL
% 2.84/1.57 => ( setukpairinjR11
% 2.84/1.57 => ( setukpairinjR12
% 2.84/1.57 => ( setukpairinjR1
% 2.84/1.57 => ( upairequniteq
% 2.84/1.57 => ( setukpairinjR2
% 2.84/1.57 => ( setukpairinjR
% 2.84/1.57 => ( ksndsingleton
% 2.84/1.57 => ( ksndpairEq
% 2.84/1.57 => ( kpairsurjEq
% 2.84/1.57 => ( cartprodsndin
% 2.84/1.57 => ( cartprodpairmemEL
% 2.84/1.57 => ( cartprodpairmemER
% 2.84/1.57 => ( cartprodmempaircEq
% 2.84/1.57 => ( cartprodfstpairEq
% 2.84/1.57 => ( cartprodsndpairEq
% 2.84/1.57 => ( cartprodpairsurjEq
% 2.84/1.57 => ( dpsetconstrI
% 2.84/1.57 => ( dpsetconstrSub
% 2.84/1.57 => ( setOfPairsIsBReln
% 2.84/1.57 => ( dpsetconstrERa
% 2.84/1.57 => ( dpsetconstrEL1
% 2.84/1.57 => ( dpsetconstrEL2
% 2.84/1.57 => ( dpsetconstrER
% 2.84/1.57 => ( funcImageSingleton
% 2.84/1.57 => ( apProp
% 2.84/1.57 => ( app
% 2.84/1.57 => ( infuncsetfunc
% 2.84/1.57 => ( ap2p
% 2.84/1.57 => ( funcinfuncset
% 2.84/1.57 => ( lamProp
% 2.84/1.57 => ( lamp
% 2.84/1.57 => ( lam2p
% 2.84/1.57 => ( brelnall1
% 2.84/1.57 => ( brelnall2
% 2.84/1.57 => ( ex1E2
% 2.84/1.57 => ( funcGraphProp1
% 2.84/1.57 => ( funcGraphProp3
% 2.84/1.57 => ( funcGraphProp2
% 2.84/1.57 => ( funcextLem
% 2.84/1.57 => ( funcGraphProp4
% 2.84/1.57 => ( subbreln
% 2.84/1.57 => ( eqbreln
% 2.84/1.57 => ( funcext
% 2.84/1.57 => ( funcext2
% 2.84/1.57 => ( ap2apEq1
% 2.84/1.57 => ( ap2apEq2
% 2.84/1.57 => ( beta1
% 2.84/1.57 => ( eta1
% 2.84/1.57 => ( lam2lamEq
% 2.84/1.57 => ( beta2
% 2.84/1.57 => ( eta2
% 2.84/1.57 => ( iffalseProp1
% 2.84/1.57 => ( iffalseProp2
% 2.84/1.57 => ( iftrueProp1
% 2.84/1.57 => ( iftrueProp2
% 2.84/1.57 => ( ifSingleton
% 2.84/1.57 => ( ifp
% 2.84/1.57 => ( theeq
% 2.84/1.57 => ( iftrue
% 2.84/1.57 => ( iffalse
% 2.84/1.57 => ( iftrueorfalse
% 2.84/1.57 => ( binintersectT_lem
% 2.84/1.57 => ( binunionT_lem
% 2.84/1.57 => ( powersetT_lem
% 2.84/1.57 => ( setminusT_lem
% 2.84/1.57 => ( complementT_lem
% 2.84/1.57 => ( setextT
% 2.84/1.57 => ( subsetTI
% 2.84/1.57 => ( powersetTI1
% 2.84/1.57 => ( powersetTE1
% 2.84/1.57 => ( complementTI1
% 2.84/1.57 => ( complementTE1
% 2.84/1.57 => ( binintersectTELcontra
% 2.84/1.57 => ( binintersectTERcontra
% 2.84/1.57 => ( contrasubsetT
% 2.84/1.57 => ( contrasubsetT1
% 2.84/1.57 => ( contrasubsetT2
% 2.84/1.57 => ( contrasubsetT3
% 2.84/1.57 => ( doubleComplementI1
% 2.84/1.57 => ( doubleComplementE1
% 2.84/1.57 => ( doubleComplementSub1
% 2.84/1.57 => ( doubleComplementSub2
% 2.84/1.57 => ( doubleComplementEq
% 2.84/1.57 => ( complementTnotintersectT
% 2.84/1.57 => ( complementImpComplementIntersect
% 2.84/1.57 => ( complementSubsetComplementIntersect
% 2.84/1.57 => ( complementInPowersetComplementIntersect
% 2.84/1.57 => ( contraSubsetComplement
% 2.84/1.57 => ( complementTcontraSubset
% 2.84/1.57 => ( binunionTILcontra
% 2.84/1.57 => ( binunionTIRcontra
% 2.84/1.57 => ( inIntersectImpInUnion
% 2.84/1.57 => ( inIntersectImpInUnion2
% 2.84/1.57 => ( inIntersectImpInIntersectUnions
% 2.84/1.57 => ( intersectInPowersetIntersectUnions
% 2.84/1.57 => ( inComplementUnionImpNotIn1
% 2.84/1.57 => ( inComplementUnionImpInComplement1
% 2.84/1.57 => ( binunionTE
% 2.84/1.57 => ( binunionTEcontra
% 2.84/1.57 => ( demorgan2a1
% 2.84/1.57 => ( complementUnionInPowersetComplement
% 2.84/1.57 => ( demorgan2a2
% 2.84/1.57 => ( demorgan1a
% 2.84/1.57 => ( demorgan1b
% 2.84/1.57 => ( demorgan1
% 2.84/1.57 => ( demorgan2a
% 2.84/1.57 => ( demorgan2b2
% 2.84/1.57 => ( demorgan2b
% 2.84/1.57 => ( demorgan2
% 2.84/1.57 => ( woz13rule0
% 2.84/1.57 => ( woz13rule1
% 2.84/1.57 => ( woz13rule2
% 2.84/1.57 => ( woz13rule3
% 2.84/1.57 => ( woz13rule4
% 2.84/1.57 => ( woz1_1
% 2.84/1.57 => ( woz1_2
% 2.84/1.57 => ( woz1_3
% 2.84/1.57 => ( woz1_4
% 2.84/1.57 => ( woz1_5
% 2.84/1.57 => ( breln1all2
% 2.84/1.57 => ( breln1SetBreln1
% 2.84/1.57 => ( choice2fnsingleton
% 2.84/1.57 => ( setOfPairsIsBReln1
% 2.84/1.57 => ( breln1all1
% 2.84/1.57 => ( subbreln1
% 2.84/1.57 => ( eqbreln1
% 2.84/1.57 => ( breln1invprop
% 2.84/1.57 => ! [A: $i,R: $i] :
% 2.84/1.57 ( ( breln1 @ A @ R )
% 2.84/1.57 => ! [Xx: $i] :
% 2.84/1.57 ( ( in @ Xx @ A )
% 2.84/1.57 => ! [Xy: $i] :
% 2.84/1.57 ( ( in @ Xy @ A )
% 2.84/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.84/1.57 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.57 ------------------------
% 2.84/1.60 % (13255)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_12985 for (2999ds/27Mi)
% 2.84/1.60 % (13253)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/183Mi)
% 2.90/1.60 % (13254)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_12985 for (2999ds/4Mi)
% 2.90/1.60 % (13256)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/2Mi)
% 2.90/1.60 % (13257)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_12985 for (2999ds/2Mi)
% 2.90/1.60 % (13255)Instruction limit reached!
% 2.90/1.60 % (13255)------------------------------
% 2.90/1.60 % (13255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.60 % (13258)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_12985 for (2999ds/275Mi)
% 2.90/1.60 % (13255)Termination reason: Unknown
% 2.90/1.60 % (13255)Termination phase: shuffling
% 2.90/1.60
% 2.90/1.60 % (13255)Memory used [KB]: 2046
% 2.90/1.60 % (13255)Time elapsed: 0.010 s
% 2.90/1.60 % (13255)Instructions burned: 29 (million)
% 2.90/1.60 % (13255)------------------------------
% 2.90/1.60 % (13255)------------------------------
% 2.90/1.60 % (13259)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/18Mi)
% 2.90/1.60 % (13256)Instruction limit reached!
% 2.90/1.60 % (13256)------------------------------
% 2.90/1.60 % (13256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.60 % (13256)Termination reason: Unknown
% 2.90/1.60 % (13256)Termination phase: shuffling
% 2.90/1.60
% 2.90/1.60 % (13256)Memory used [KB]: 1535
% 2.90/1.60 % (13256)Time elapsed: 0.003 s
% 2.90/1.60 % (13256)Instructions burned: 2 (million)
% 2.90/1.60 % (13256)------------------------------
% 2.90/1.60 % (13256)------------------------------
% 2.90/1.60 % (13257)Instruction limit reached!
% 2.90/1.60 % (13257)------------------------------
% 2.90/1.60 % (13257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.60 % (13257)Termination reason: Unknown
% 2.90/1.60 % (13257)Termination phase: shuffling
% 2.90/1.60
% 2.90/1.60 % (13257)Memory used [KB]: 1535
% 2.90/1.60 % (13257)Time elapsed: 0.003 s
% 2.90/1.60 % (13257)Instructions burned: 2 (million)
% 2.90/1.60 % (13257)------------------------------
% 2.90/1.60 % (13257)------------------------------
% 2.90/1.61 % (13254)Instruction limit reached!
% 2.90/1.61 % (13254)------------------------------
% 2.90/1.61 % (13254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (13254)Termination reason: Unknown
% 2.90/1.61 % (13254)Termination phase: shuffling
% 2.90/1.61
% 2.90/1.61 % (13254)Memory used [KB]: 1535
% 2.90/1.61 % (13254)Time elapsed: 0.004 s
% 2.90/1.61 % (13254)Instructions burned: 5 (million)
% 2.90/1.61 % (13254)------------------------------
% 2.90/1.61 % (13254)------------------------------
% 2.90/1.61 % (13266)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/3Mi)
% 2.90/1.61 % (13259)Instruction limit reached!
% 2.90/1.61 % (13259)------------------------------
% 2.90/1.61 % (13259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (13259)Termination reason: Unknown
% 2.90/1.61 % (13259)Termination phase: shuffling
% 2.90/1.61
% 2.90/1.61 % (13259)Memory used [KB]: 1791
% 2.90/1.61 % (13259)Time elapsed: 0.013 s
% 2.90/1.61 % (13259)Instructions burned: 19 (million)
% 2.90/1.61 % (13259)------------------------------
% 2.90/1.61 % (13259)------------------------------
% 2.90/1.62 % (13266)Instruction limit reached!
% 2.90/1.62 % (13266)------------------------------
% 2.90/1.62 % (13266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.62 % (13266)Termination reason: Unknown
% 2.90/1.62 % (13266)Termination phase: shuffling
% 2.90/1.62
% 2.90/1.62 % (13266)Memory used [KB]: 1535
% 2.90/1.62 % (13266)Time elapsed: 0.003 s
% 2.90/1.62 % (13266)Instructions burned: 5 (million)
% 2.90/1.62 % (13266)------------------------------
% 2.90/1.62 % (13266)------------------------------
% 2.90/1.62 % (13267)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_12985 for (2999ds/37Mi)
% 2.90/1.62 % (13268)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_12985 for (2999ds/15Mi)
% 2.90/1.62 % (13269)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_12985 for (2999ds/3Mi)
% 2.90/1.62 % (13278)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_12985 for (2999ds/1041Mi)
% 2.90/1.63 % (13269)Instruction limit reached!
% 2.90/1.63 % (13269)------------------------------
% 2.90/1.63 % (13269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.63 % (13269)Termination reason: Unknown
% 2.90/1.63 % (13269)Termination phase: shuffling
% 2.90/1.63
% 2.90/1.63 % (13269)Memory used [KB]: 1535
% 2.90/1.63 % (13269)Time elapsed: 0.004 s
% 2.90/1.63 % (13269)Instructions burned: 3 (million)
% 2.90/1.63 % (13269)------------------------------
% 2.90/1.63 % (13269)------------------------------
% 2.90/1.63 % (13268)Instruction limit reached!
% 2.90/1.63 % (13268)------------------------------
% 2.90/1.63 % (13268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.63 % (13268)Termination reason: Unknown
% 2.90/1.63 % (13268)Termination phase: shuffling
% 2.90/1.63
% 2.90/1.63 % (13268)Memory used [KB]: 1791
% 2.90/1.63 % (13268)Time elapsed: 0.012 s
% 2.90/1.63 % (13268)Instructions burned: 16 (million)
% 2.90/1.63 % (13268)------------------------------
% 2.90/1.63 % (13268)------------------------------
% 2.90/1.63 % (13279)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/7Mi)
% 2.90/1.64 % (13279)Instruction limit reached!
% 2.90/1.64 % (13279)------------------------------
% 2.90/1.64 % (13279)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.64 % (13279)Termination reason: Unknown
% 2.90/1.64 % (13279)Termination phase: shuffling
% 2.90/1.64
% 2.90/1.64 % (13279)Memory used [KB]: 1663
% 2.90/1.64 % (13279)Time elapsed: 0.006 s
% 2.90/1.64 % (13279)Instructions burned: 7 (million)
% 2.90/1.64 % (13279)------------------------------
% 2.90/1.64 % (13279)------------------------------
% 2.90/1.64 % (13267)Instruction limit reached!
% 2.90/1.64 % (13267)------------------------------
% 2.90/1.64 % (13267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.64 % (13267)Termination reason: Unknown
% 2.90/1.64 % (13267)Termination phase: shuffling
% 2.90/1.64
% 2.90/1.64 % (13267)Memory used [KB]: 2302
% 2.90/1.64 % (13267)Time elapsed: 0.021 s
% 2.90/1.64 % (13267)Instructions burned: 37 (million)
% 2.90/1.64 % (13267)------------------------------
% 2.90/1.64 % (13267)------------------------------
% 2.90/1.64 % (13253)Instruction limit reached!
% 2.90/1.64 % (13253)------------------------------
% 2.90/1.64 % (13253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.64 % (13253)Termination reason: Unknown
% 2.90/1.64 % (13253)Termination phase: Saturation
% 2.90/1.64
% 2.90/1.64 % (13253)Memory used [KB]: 7419
% 2.90/1.64 % (13253)Time elapsed: 0.048 s
% 2.90/1.64 % (13253)Instructions burned: 183 (million)
% 2.90/1.64 % (13253)------------------------------
% 2.90/1.64 % (13253)------------------------------
% 2.90/1.64 % (13285)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/16Mi)
% 2.90/1.65 % (13285)Instruction limit reached!
% 2.90/1.65 % (13285)------------------------------
% 2.90/1.65 % (13285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.65 % (13285)Termination reason: Unknown
% 2.90/1.65 % (13285)Termination phase: shuffling
% 2.90/1.65
% 2.90/1.65 % (13285)Memory used [KB]: 1791
% 2.90/1.65 % (13285)Time elapsed: 0.009 s
% 2.90/1.65 % (13285)Instructions burned: 16 (million)
% 2.90/1.65 % (13285)------------------------------
% 2.90/1.65 % (13285)------------------------------
% 2.90/1.65 % (13291)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_12985 for (2999ds/3Mi)
% 2.90/1.65 % (13291)Instruction limit reached!
% 2.90/1.65 % (13291)------------------------------
% 2.90/1.65 % (13291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.65 % (13291)Termination reason: Unknown
% 2.90/1.65 % (13291)Termination phase: shuffling
% 2.90/1.65
% 2.90/1.65 % (13291)Memory used [KB]: 1535
% 2.90/1.65 % (13291)Time elapsed: 0.004 s
% 2.90/1.65 % (13291)Instructions burned: 4 (million)
% 2.90/1.65 % (13291)------------------------------
% 2.90/1.65 % (13291)------------------------------
% 2.90/1.65 % (13296)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/3Mi)
% 2.90/1.65 % (13296)Instruction limit reached!
% 2.90/1.65 % (13296)------------------------------
% 2.90/1.65 % (13296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.65 % (13296)Termination reason: Unknown
% 2.90/1.65 % (13296)Termination phase: shuffling
% 2.90/1.65
% 2.90/1.65 % (13296)Memory used [KB]: 1535
% 2.90/1.65 % (13296)Time elapsed: 0.003 s
% 2.90/1.65 % (13296)Instructions burned: 6 (million)
% 2.90/1.65 % (13296)------------------------------
% 2.90/1.65 % (13296)------------------------------
% 2.90/1.65 % (13294)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_12985 for (2999ds/3Mi)
% 2.90/1.66 % (13294)Instruction limit reached!
% 2.90/1.66 % (13294)------------------------------
% 2.90/1.66 % (13294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.66 % (13294)Termination reason: Unknown
% 2.90/1.66 % (13294)Termination phase: shuffling
% 2.90/1.66
% 2.90/1.66 % (13294)Memory used [KB]: 1535
% 2.90/1.66 % (13294)Time elapsed: 0.004 s
% 2.90/1.66 % (13294)Instructions burned: 3 (million)
% 2.90/1.66 % (13294)------------------------------
% 2.90/1.66 % (13294)------------------------------
% 2.90/1.66 % (13295)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/7Mi)
% 2.90/1.66 % (13297)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/4Mi)
% 2.90/1.66 % (13295)Instruction limit reached!
% 2.90/1.66 % (13295)------------------------------
% 2.90/1.66 % (13295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.66 % (13295)Termination reason: Unknown
% 2.90/1.66 % (13295)Termination phase: shuffling
% 2.90/1.66
% 2.90/1.66 % (13295)Memory used [KB]: 1663
% 2.90/1.66 % (13295)Time elapsed: 0.006 s
% 2.90/1.66 % (13295)Instructions burned: 8 (million)
% 2.90/1.66 % (13295)------------------------------
% 2.90/1.66 % (13295)------------------------------
% 2.90/1.66 % (13297)Instruction limit reached!
% 2.90/1.66 % (13297)------------------------------
% 2.90/1.66 % (13297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.66 % (13297)Termination reason: Unknown
% 2.90/1.66 % (13297)Termination phase: shuffling
% 2.90/1.66
% 2.90/1.66 % (13297)Memory used [KB]: 1535
% 2.90/1.66 % (13297)Time elapsed: 0.003 s
% 2.90/1.66 % (13297)Instructions burned: 6 (million)
% 2.90/1.66 % (13297)------------------------------
% 2.90/1.66 % (13297)------------------------------
% 2.90/1.66 % (13304)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/710Mi)
% 2.90/1.67 % (13299)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/18Mi)
% 2.90/1.67 % (13299)Instruction limit reached!
% 2.90/1.67 % (13299)------------------------------
% 2.90/1.67 % (13299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (13299)Termination reason: Unknown
% 2.90/1.67 % (13299)Termination phase: shuffling
% 2.90/1.67
% 2.90/1.67 % (13299)Memory used [KB]: 1791
% 2.90/1.67 % (13299)Time elapsed: 0.007 s
% 2.90/1.67 % (13299)Instructions burned: 18 (million)
% 2.90/1.67 % (13299)------------------------------
% 2.90/1.67 % (13299)------------------------------
% 2.90/1.67 % (13307)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/6Mi)
% 2.90/1.67 % (13313)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_12985 for (2999ds/902Mi)
% 2.90/1.67 % (13307)Instruction limit reached!
% 2.90/1.67 % (13307)------------------------------
% 2.90/1.67 % (13307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (13307)Termination reason: Unknown
% 2.90/1.67 % (13307)Termination phase: shuffling
% 2.90/1.67
% 2.90/1.67 % (13307)Memory used [KB]: 1535
% 2.90/1.67 % (13307)Time elapsed: 0.004 s
% 2.90/1.67 % (13307)Instructions burned: 6 (million)
% 2.90/1.67 % (13307)------------------------------
% 2.90/1.67 % (13307)------------------------------
% 2.90/1.68 % (13314)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_12985 for (2999ds/21Mi)
% 2.90/1.68 % (13319)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_12985 for (2999ds/5Mi)
% 2.90/1.68 % (13319)Instruction limit reached!
% 2.90/1.68 % (13319)------------------------------
% 2.90/1.68 % (13319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.68 % (13319)Termination reason: Unknown
% 2.90/1.68 % (13319)Termination phase: shuffling
% 2.90/1.68
% 2.90/1.68 % (13319)Memory used [KB]: 1535
% 2.90/1.68 % (13319)Time elapsed: 0.003 s
% 2.90/1.68 % (13319)Instructions burned: 6 (million)
% 2.90/1.68 % (13319)------------------------------
% 2.90/1.68 % (13319)------------------------------
% 2.90/1.68 % (13314)Instruction limit reached!
% 2.90/1.68 % (13314)------------------------------
% 2.90/1.68 % (13314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.68 % (13314)Termination reason: Unknown
% 2.90/1.68 % (13314)Termination phase: shuffling
% 2.90/1.68
% 2.90/1.68 % (13314)Memory used [KB]: 1918
% 2.90/1.68 % (13314)Time elapsed: 0.009 s
% 2.90/1.68 % (13314)Instructions burned: 22 (million)
% 2.90/1.68 % (13314)------------------------------
% 2.90/1.68 % (13314)------------------------------
% 2.90/1.69 % (13323)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_12985 for (2999ds/6Mi)
% 2.90/1.69 % (13323)Instruction limit reached!
% 2.90/1.69 % (13323)------------------------------
% 2.90/1.69 % (13323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (13323)Termination reason: Unknown
% 2.90/1.69 % (13323)Termination phase: shuffling
% 2.90/1.69
% 2.90/1.69 % (13323)Memory used [KB]: 1535
% 2.90/1.69 % (13323)Time elapsed: 0.003 s
% 2.90/1.69 % (13323)Instructions burned: 6 (million)
% 2.90/1.69 % (13323)------------------------------
% 2.90/1.69 % (13323)------------------------------
% 2.90/1.69 % (13329)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_12985 for (2998ds/377Mi)
% 2.90/1.69 % (13330)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_12985 for (2998ds/779Mi)
% 2.90/1.70 % (13331)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_12985 for (2998ds/19Mi)
% 2.90/1.70 % (13331)Instruction limit reached!
% 2.90/1.70 % (13331)------------------------------
% 2.90/1.70 % (13331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.70 % (13331)Termination reason: Unknown
% 2.90/1.70 % (13331)Termination phase: shuffling
% 2.90/1.70
% 2.90/1.70 % (13331)Memory used [KB]: 1918
% 2.90/1.70 % (13331)Time elapsed: 0.007 s
% 2.90/1.70 % (13331)Instructions burned: 21 (million)
% 2.90/1.70 % (13331)------------------------------
% 2.90/1.70 % (13331)------------------------------
% 2.90/1.71 % (13340)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_12985 for (2998ds/879Mi)
% 2.90/1.73 % (13258)Instruction limit reached!
% 2.90/1.73 % (13258)------------------------------
% 2.90/1.73 % (13258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.73 % (13258)Termination reason: Unknown
% 2.90/1.73 % (13258)Termination phase: Saturation
% 2.90/1.73
% 2.90/1.73 % (13258)Memory used [KB]: 10234
% 2.90/1.73 % (13258)Time elapsed: 0.126 s
% 2.90/1.73 % (13258)Instructions burned: 277 (million)
% 2.90/1.73 % (13258)------------------------------
% 2.90/1.73 % (13258)------------------------------
% 2.90/1.74 % (13351)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_12985 for (2998ds/17Mi)
% 2.90/1.74 % (13351)Instruction limit reached!
% 2.90/1.74 % (13351)------------------------------
% 2.90/1.74 % (13351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.74 % (13351)Termination reason: Unknown
% 2.90/1.74 % (13351)Termination phase: shuffling
% 2.90/1.74
% 2.90/1.74 % (13351)Memory used [KB]: 1791
% 2.90/1.74 % (13351)Time elapsed: 0.007 s
% 2.90/1.74 % (13351)Instructions burned: 20 (million)
% 2.90/1.74 % (13351)------------------------------
% 2.90/1.74 % (13351)------------------------------
% 2.90/1.76 % (13360)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_12985 for (2998ds/3Mi)
% 2.90/1.76 % (13360)Instruction limit reached!
% 2.90/1.76 % (13360)------------------------------
% 2.90/1.76 % (13360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.76 % (13360)Termination reason: Unknown
% 2.90/1.76 % (13360)Termination phase: shuffling
% 2.90/1.76
% 2.90/1.76 % (13360)Memory used [KB]: 1535
% 2.90/1.76 % (13360)Time elapsed: 0.002 s
% 2.90/1.76 % (13360)Instructions burned: 3 (million)
% 2.90/1.76 % (13360)------------------------------
% 2.90/1.76 % (13360)------------------------------
% 2.90/1.77 % (13366)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_12985 for (2998ds/30Mi)
% 2.90/1.78 % (13366)Instruction limit reached!
% 2.90/1.78 % (13366)------------------------------
% 2.90/1.78 % (13366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.78 % (13366)Termination reason: Unknown
% 2.90/1.78 % (13366)Termination phase: shuffling
% 2.90/1.78
% 2.90/1.78 % (13366)Memory used [KB]: 2046
% 2.90/1.78 % (13366)Time elapsed: 0.010 s
% 2.90/1.78 % (13366)Instructions burned: 30 (million)
% 2.90/1.78 % (13366)------------------------------
% 2.90/1.78 % (13366)------------------------------
% 2.90/1.79 % (13375)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_12985 for (2998ds/127Mi)
% 2.90/1.80 % (13329)Instruction limit reached!
% 2.90/1.80 % (13329)------------------------------
% 2.90/1.80 % (13329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.80 % (13329)Termination reason: Unknown
% 2.90/1.80 % (13329)Termination phase: Saturation
% 2.90/1.80
% 2.90/1.80 % (13329)Memory used [KB]: 9466
% 2.90/1.80 % (13329)Time elapsed: 0.111 s
% 2.90/1.80 % (13329)Instructions burned: 378 (million)
% 2.90/1.80 % (13329)------------------------------
% 2.90/1.80 % (13329)------------------------------
% 2.90/1.81 % (13385)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_12985 for (2997ds/100Mi)
% 2.90/1.82 % (13375)Instruction limit reached!
% 2.90/1.82 % (13375)------------------------------
% 2.90/1.82 % (13375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.82 % (13375)Termination reason: Unknown
% 2.90/1.82 % (13375)Termination phase: Property scanning
% 2.90/1.82
% 2.90/1.82 % (13375)Memory used [KB]: 2814
% 2.90/1.82 % (13375)Time elapsed: 0.037 s
% 2.90/1.82 % (13375)Instructions burned: 130 (million)
% 2.90/1.82 % (13375)------------------------------
% 2.90/1.82 % (13375)------------------------------
% 2.90/1.83 % (13395)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_12985 for (2997ds/3Mi)
% 2.90/1.83 % (13395)Instruction limit reached!
% 2.90/1.83 % (13395)------------------------------
% 2.90/1.83 % (13395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.83 % (13395)Termination reason: Unknown
% 2.90/1.83 % (13395)Termination phase: shuffling
% 2.90/1.83
% 2.90/1.83 % (13395)Memory used [KB]: 1535
% 2.90/1.83 % (13395)Time elapsed: 0.002 s
% 2.90/1.83 % (13395)Instructions burned: 3 (million)
% 2.90/1.83 % (13395)------------------------------
% 2.90/1.83 % (13395)------------------------------
% 2.90/1.84 % (13385)Instruction limit reached!
% 2.90/1.84 % (13385)------------------------------
% 2.90/1.84 % (13385)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.84 % (13385)Termination reason: Unknown
% 2.90/1.84 % (13385)Termination phase: Property scanning
% 2.90/1.84
% 2.90/1.84 % (13385)Memory used [KB]: 2686
% 2.90/1.84 % (13385)Time elapsed: 0.030 s
% 2.90/1.84 % (13385)Instructions burned: 103 (million)
% 2.90/1.84 % (13385)------------------------------
% 2.90/1.84 % (13385)------------------------------
% 2.90/1.84 % (13400)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_12985 for (2997ds/20Mi)
% 2.90/1.85 % (13400)Instruction limit reached!
% 2.90/1.85 % (13400)------------------------------
% 2.90/1.85 % (13400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.85 % (13400)Termination reason: Unknown
% 2.90/1.85 % (13400)Termination phase: shuffling
% 2.90/1.85
% 2.90/1.85 % (13400)Memory used [KB]: 1791
% 2.90/1.85 % (13400)Time elapsed: 0.008 s
% 2.90/1.85 % (13400)Instructions burned: 21 (million)
% 2.90/1.85 % (13400)------------------------------
% 2.90/1.85 % (13400)------------------------------
% 2.90/1.85 % (13402)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_12985 for (2997ds/86Mi)
% 2.90/1.86 % (13406)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_12985 for (2997ds/107Mi)
% 2.90/1.88 % (13402)Instruction limit reached!
% 2.90/1.88 % (13402)------------------------------
% 2.90/1.88 % (13402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.88 % (13402)Termination reason: Unknown
% 2.90/1.88 % (13402)Termination phase: Property scanning
% 2.90/1.88
% 2.90/1.88 % (13402)Memory used [KB]: 2558
% 2.90/1.88 % (13402)Time elapsed: 0.025 s
% 2.90/1.88 % (13402)Instructions burned: 86 (million)
% 2.90/1.88 % (13402)------------------------------
% 2.90/1.88 % (13402)------------------------------
% 2.90/1.88 % (13304)Instruction limit reached!
% 2.90/1.88 % (13304)------------------------------
% 2.90/1.88 % (13304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.88 % (13304)Termination reason: Unknown
% 2.90/1.88 % (13304)Termination phase: Saturation
% 2.90/1.88
% 2.90/1.88 % (13304)Memory used [KB]: 11513
% 2.90/1.88 % (13304)Time elapsed: 0.221 s
% 2.90/1.88 % (13304)Instructions burned: 711 (million)
% 2.90/1.88 % (13304)------------------------------
% 2.90/1.88 % (13304)------------------------------
% 2.90/1.89 % (13330)Instruction limit reached!
% 2.90/1.89 % (13330)------------------------------
% 2.90/1.89 % (13330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.89 % (13330)Termination reason: Unknown
% 2.90/1.89 % (13330)Termination phase: Saturation
% 2.90/1.89
% 2.90/1.89 % (13330)Memory used [KB]: 7675
% 2.90/1.89 % (13330)Time elapsed: 0.195 s
% 2.90/1.89 % (13330)Instructions burned: 780 (million)
% 2.90/1.89 % (13330)------------------------------
% 2.90/1.89 % (13330)------------------------------
% 2.90/1.89 % (13415)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_12985 for (2997ds/39Mi)
% 2.90/1.89 % (13278)Instruction limit reached!
% 2.90/1.89 % (13278)------------------------------
% 2.90/1.89 % (13278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.89 % (13278)Termination reason: Unknown
% 2.90/1.89 % (13278)Termination phase: Saturation
% 2.90/1.89
% 2.90/1.89 % (13278)Memory used [KB]: 16630
% 2.90/1.89 % (13278)Time elapsed: 0.267 s
% 2.90/1.89 % (13278)Instructions burned: 1041 (million)
% 2.90/1.89 % (13278)------------------------------
% 2.90/1.89 % (13278)------------------------------
% 2.90/1.89 % (13406)Instruction limit reached!
% 2.90/1.89 % (13406)------------------------------
% 2.90/1.89 % (13406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.89 % (13406)Termination reason: Unknown
% 2.90/1.89 % (13406)Termination phase: SInE selection
% 2.90/1.89
% 2.90/1.89 % (13406)Memory used [KB]: 2558
% 2.90/1.89 % (13406)Time elapsed: 0.031 s
% 2.90/1.89 % (13406)Instructions burned: 109 (million)
% 2.90/1.89 % (13406)------------------------------
% 2.90/1.89 % (13406)------------------------------
% 2.90/1.89 % (13421)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_12985 for (2997ds/448Mi)
% 2.90/1.90 % (13423)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_12985 for (2996ds/46Mi)
% 2.90/1.90 % (13415)Instruction limit reached!
% 2.90/1.90 % (13415)------------------------------
% 2.90/1.90 % (13415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.90 % (13415)Termination reason: Unknown
% 2.90/1.90 % (13415)Termination phase: shuffling
% 2.90/1.90
% 2.90/1.90 % (13415)Memory used [KB]: 2302
% 2.90/1.90 % (13415)Time elapsed: 0.013 s
% 2.90/1.90 % (13415)Instructions burned: 42 (million)
% 2.90/1.90 % (13415)------------------------------
% 2.90/1.90 % (13415)------------------------------
% 2.90/1.90 % (13424)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_12985 for (2996ds/98Mi)
% 2.90/1.90 % (13425)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_12985 for (2996ds/507Mi)
% 2.90/1.91 % (13423)Instruction limit reached!
% 2.90/1.91 % (13423)------------------------------
% 2.90/1.91 % (13423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.91 % (13423)Termination reason: Unknown
% 2.90/1.91 % (13423)Termination phase: shuffling
% 2.90/1.91
% 2.90/1.91 % (13423)Memory used [KB]: 2430
% 2.90/1.91 % (13423)Time elapsed: 0.014 s
% 2.90/1.91 % (13423)Instructions burned: 47 (million)
% 2.90/1.91 % (13423)------------------------------
% 2.90/1.91 % (13423)------------------------------
% 2.90/1.91 % (13428)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_12985 for (2996ds/149Mi)
% 2.90/1.92 % (13431)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_12985 for (2996ds/39Mi)
% 2.90/1.93 % (13424)Instruction limit reached!
% 2.90/1.93 % (13424)------------------------------
% 2.90/1.93 % (13424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.93 % (13424)Termination reason: Unknown
% 2.90/1.93 % (13424)Termination phase: Property scanning
% 2.90/1.93
% 2.90/1.93 % (13424)Memory used [KB]: 2686
% 2.90/1.93 % (13424)Time elapsed: 0.027 s
% 2.90/1.93 % (13424)Instructions burned: 101 (million)
% 2.90/1.93 % (13424)------------------------------
% 2.90/1.93 % (13424)------------------------------
% 2.90/1.93 % (13431)Instruction limit reached!
% 2.90/1.93 % (13431)------------------------------
% 2.90/1.93 % (13431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.93 % (13431)Termination reason: Unknown
% 2.90/1.93 % (13431)Termination phase: shuffling
% 2.90/1.93
% 2.90/1.93 % (13431)Memory used [KB]: 2302
% 2.90/1.93 % (13431)Time elapsed: 0.012 s
% 2.90/1.93 % (13431)Instructions burned: 39 (million)
% 2.90/1.93 % (13431)------------------------------
% 2.90/1.93 % (13431)------------------------------
% 2.90/1.94 % (13444)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_12985 for (2996ds/6Mi)
% 2.90/1.94 % (13444)Instruction limit reached!
% 2.90/1.94 % (13444)------------------------------
% 2.90/1.94 % (13444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.94 % (13444)Termination reason: Unknown
% 2.90/1.94 % (13444)Termination phase: shuffling
% 2.90/1.94
% 2.90/1.94 % (13444)Memory used [KB]: 1535
% 2.90/1.94 % (13444)Time elapsed: 0.003 s
% 2.90/1.94 % (13444)Instructions burned: 7 (million)
% 2.90/1.94 % (13444)------------------------------
% 2.90/1.94 % (13444)------------------------------
% 2.90/1.94 % (13448)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_12985 for (2996ds/1087Mi)
% 2.90/1.95 % (13454)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_12985 for (2996ds/47Mi)
% 2.90/1.95 % (13428)Instruction limit reached!
% 2.90/1.95 % (13428)------------------------------
% 2.90/1.95 % (13428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.95 % (13428)Termination reason: Unknown
% 2.90/1.95 % (13428)Termination phase: Twee Goal Transformation
% 2.90/1.95
% 2.90/1.95 % (13428)Memory used [KB]: 3070
% 2.90/1.95 % (13428)Time elapsed: 0.043 s
% 2.90/1.95 % (13428)Instructions burned: 150 (million)
% 2.90/1.95 % (13428)------------------------------
% 2.90/1.95 % (13428)------------------------------
% 2.90/1.96 % (13454)Instruction limit reached!
% 2.90/1.96 % (13454)------------------------------
% 2.90/1.96 % (13454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.96 % (13454)Termination reason: Unknown
% 2.90/1.96 % (13454)Termination phase: shuffling
% 2.90/1.96
% 2.90/1.96 % (13454)Memory used [KB]: 2430
% 2.90/1.96 % (13454)Time elapsed: 0.014 s
% 2.90/1.96 % (13454)Instructions burned: 47 (million)
% 2.90/1.96 % (13454)------------------------------
% 2.90/1.96 % (13454)------------------------------
% 2.90/1.96 % (13313)Instruction limit reached!
% 2.90/1.96 % (13313)------------------------------
% 2.90/1.96 % (13313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.96 % (13313)Termination reason: Unknown
% 2.90/1.96 % (13313)Termination phase: Saturation
% 2.90/1.96
% 2.90/1.96 % (13313)Memory used [KB]: 12025
% 2.90/1.96 % (13313)Time elapsed: 0.290 s
% 2.90/1.96 % (13313)Instructions burned: 903 (million)
% 2.90/1.96 % (13313)------------------------------
% 2.90/1.96 % (13313)------------------------------
% 2.90/1.96 % (13463)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_12985 for (2996ds/44Mi)
% 2.90/1.97 % (13468)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_12985 for (2996ds/432Mi)
% 2.90/1.97 % (13470)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_12985 for (2996ds/62Mi)
% 2.90/1.98 % (13463)Instruction limit reached!
% 2.90/1.98 % (13463)------------------------------
% 2.90/1.98 % (13463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.98 % (13463)Termination reason: Unknown
% 2.90/1.98 % (13463)Termination phase: shuffling
% 2.90/1.98
% 2.90/1.98 % (13463)Memory used [KB]: 2430
% 2.90/1.98 % (13463)Time elapsed: 0.014 s
% 2.90/1.98 % (13463)Instructions burned: 45 (million)
% 2.90/1.98 % (13463)------------------------------
% 2.90/1.98 % (13463)------------------------------
% 2.90/1.99 % (13473)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_12985 for (2996ds/31Mi)
% 2.90/1.99 % (13470)Instruction limit reached!
% 2.90/1.99 % (13470)------------------------------
% 2.90/1.99 % (13470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.99 % (13470)Termination reason: Unknown
% 2.90/1.99 % (13470)Termination phase: Property scanning
% 2.90/1.99
% 2.90/1.99 % (13470)Memory used [KB]: 2558
% 2.90/1.99 % (13470)Time elapsed: 0.017 s
% 2.90/1.99 % (13470)Instructions burned: 62 (million)
% 2.90/1.99 % (13470)------------------------------
% 2.90/1.99 % (13470)------------------------------
% 2.90/2.00 % (13473)Instruction limit reached!
% 2.90/2.00 % (13473)------------------------------
% 2.90/2.00 % (13473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/2.00 % (13473)Termination reason: Unknown
% 2.90/2.00 % (13473)Termination phase: shuffling
% 2.90/2.00
% 2.90/2.00 % (13473)Memory used [KB]: 2174
% 2.90/2.00 % (13473)Time elapsed: 0.010 s
% 2.90/2.00 % (13473)Instructions burned: 31 (million)
% 2.90/2.00 % (13473)------------------------------
% 2.90/2.00 % (13473)------------------------------
% 2.90/2.00 % (13340)Instruction limit reached!
% 2.90/2.00 % (13340)------------------------------
% 2.90/2.00 % (13340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/2.00 % (13340)Termination reason: Unknown
% 2.90/2.00 % (13340)Termination phase: Saturation
% 2.90/2.00
% 2.90/2.00 % (13340)Memory used [KB]: 10234
% 2.90/2.00 % (13340)Time elapsed: 0.283 s
% 2.90/2.00 % (13340)Instructions burned: 880 (million)
% 2.90/2.00 % (13340)------------------------------
% 2.90/2.00 % (13340)------------------------------
% 2.90/2.00 % (13478)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_12985 for (2995ds/96Mi)
% 2.90/2.00 % (13421)Instruction limit reached!
% 2.90/2.00 % (13421)------------------------------
% 2.90/2.00 % (13421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/2.00 % (13421)Termination reason: Unknown
% 2.90/2.00 % (13421)Termination phase: Saturation
% 2.90/2.00
% 2.90/2.00 % (13421)Memory used [KB]: 7675
% 2.90/2.00 % (13421)Time elapsed: 0.112 s
% 2.90/2.00 % (13421)Instructions burned: 450 (million)
% 2.90/2.00 % (13421)------------------------------
% 2.90/2.00 % (13421)------------------------------
% 2.90/2.00 % (13480)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_12985 for (2995ds/453Mi)
% 2.90/2.01 % (13481)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_12985 for (2995ds/71Mi)
% 2.90/2.01 % (13482)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_12985 for (2995ds/83Mi)
% 2.90/2.02 % (13478)Instruction limit reached!
% 2.90/2.02 % (13478)------------------------------
% 2.90/2.02 % (13478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/2.02 % (13478)Termination reason: Unknown
% 2.90/2.02 % (13478)Termination phase: Property scanning
% 2.90/2.02
% 2.90/2.02 % (13478)Memory used [KB]: 2814
% 2.90/2.02 % (13478)Time elapsed: 0.026 s
% 2.90/2.02 % (13478)Instructions burned: 96 (million)
% 2.90/2.02 % (13478)------------------------------
% 2.90/2.02 % (13478)------------------------------
% 5.87/2.03 % (13481)Instruction limit reached!
% 5.87/2.03 % (13481)------------------------------
% 5.87/2.03 % (13481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.03 % (13481)Termination reason: Unknown
% 5.87/2.03 % (13481)Termination phase: Property scanning
% 5.87/2.03
% 5.87/2.03 % (13481)Memory used [KB]: 2558
% 5.87/2.03 % (13481)Time elapsed: 0.021 s
% 5.87/2.03 % (13481)Instructions burned: 72 (million)
% 5.87/2.03 % (13481)------------------------------
% 5.87/2.03 % (13481)------------------------------
% 5.87/2.03 % (13483)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_12985 for (2995ds/11Mi)
% 5.87/2.04 % (13482)Instruction limit reached!
% 5.87/2.04 % (13482)------------------------------
% 5.87/2.04 % (13482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.04 % (13482)Termination reason: Unknown
% 5.87/2.04 % (13482)Termination phase: Property scanning
% 5.87/2.04
% 5.87/2.04 % (13482)Memory used [KB]: 2558
% 5.87/2.04 % (13482)Time elapsed: 0.023 s
% 5.87/2.04 % (13482)Instructions burned: 83 (million)
% 5.87/2.04 % (13482)------------------------------
% 5.87/2.04 % (13482)------------------------------
% 5.87/2.04 % (13484)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_12985 for (2995ds/41Mi)
% 5.87/2.04 % (13483)Instruction limit reached!
% 5.87/2.04 % (13483)------------------------------
% 5.87/2.04 % (13483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.04 % (13483)Termination reason: Unknown
% 5.87/2.04 % (13483)Termination phase: shuffling
% 5.87/2.04
% 5.87/2.04 % (13483)Memory used [KB]: 1791
% 5.87/2.04 % (13483)Time elapsed: 0.005 s
% 5.87/2.04 % (13483)Instructions burned: 14 (million)
% 5.87/2.04 % (13483)------------------------------
% 5.87/2.04 % (13483)------------------------------
% 5.87/2.05 % (13485)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12985 for (2995ds/6Mi)
% 5.87/2.05 % (13486)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_12985 for (2995ds/150Mi)
% 5.87/2.05 % (13485)Instruction limit reached!
% 5.87/2.05 % (13485)------------------------------
% 5.87/2.05 % (13485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.05 % (13485)Termination reason: Unknown
% 5.87/2.05 % (13485)Termination phase: shuffling
% 5.87/2.05
% 5.87/2.05 % (13485)Memory used [KB]: 1535
% 5.87/2.05 % (13485)Time elapsed: 0.003 s
% 5.87/2.05 % (13485)Instructions burned: 6 (million)
% 5.87/2.05 % (13485)------------------------------
% 5.87/2.05 % (13485)------------------------------
% 5.87/2.05 % (13484)Instruction limit reached!
% 5.87/2.05 % (13484)------------------------------
% 5.87/2.05 % (13484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.05 % (13484)Termination reason: Unknown
% 5.87/2.05 % (13484)Termination phase: shuffling
% 5.87/2.05
% 5.87/2.05 % (13484)Memory used [KB]: 2302
% 5.87/2.05 % (13484)Time elapsed: 0.012 s
% 5.87/2.05 % (13484)Instructions burned: 41 (million)
% 5.87/2.05 % (13484)------------------------------
% 5.87/2.05 % (13484)------------------------------
% 5.87/2.06 % (13487)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_12985 for (2995ds/343Mi)
% 5.87/2.06 % (13488)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_12985 for (2995ds/109Mi)
% 5.87/2.06 % (13425)Instruction limit reached!
% 5.87/2.06 % (13425)------------------------------
% 5.87/2.06 % (13425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.06 % (13425)Termination reason: Unknown
% 5.87/2.06 % (13425)Termination phase: Saturation
% 5.87/2.06
% 5.87/2.06 % (13425)Memory used [KB]: 8187
% 5.87/2.06 % (13425)Time elapsed: 0.156 s
% 5.87/2.06 % (13425)Instructions burned: 508 (million)
% 5.87/2.06 % (13425)------------------------------
% 5.87/2.06 % (13425)------------------------------
% 5.87/2.07 % (13489)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_12985 for (2995ds/253Mi)
% 5.87/2.08 % (13468)Instruction limit reached!
% 5.87/2.08 % (13468)------------------------------
% 5.87/2.08 % (13468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.08 % (13468)Termination reason: Unknown
% 5.87/2.08 % (13468)Termination phase: Saturation
% 5.87/2.08
% 5.87/2.08 % (13468)Memory used [KB]: 3070
% 5.87/2.08 % (13468)Time elapsed: 0.108 s
% 5.87/2.08 % (13468)Instructions burned: 435 (million)
% 5.87/2.08 % (13468)------------------------------
% 5.87/2.08 % (13468)------------------------------
% 5.87/2.09 % (13488)Instruction limit reached!
% 5.87/2.09 % (13488)------------------------------
% 5.87/2.09 % (13488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.09 % (13488)Termination reason: Unknown
% 5.87/2.09 % (13488)Termination phase: SInE selection
% 5.87/2.09
% 5.87/2.09 % (13488)Memory used [KB]: 2558
% 5.87/2.09 % (13488)Time elapsed: 0.029 s
% 5.87/2.09 % (13488)Instructions burned: 111 (million)
% 5.87/2.09 % (13488)------------------------------
% 5.87/2.09 % (13488)------------------------------
% 5.87/2.09 % (13490)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_12985 for (2994ds/140Mi)
% 5.87/2.09 % (13486)Instruction limit reached!
% 5.87/2.09 % (13486)------------------------------
% 5.87/2.09 % (13486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.09 % (13486)Termination reason: Unknown
% 5.87/2.09 % (13486)Termination phase: Property scanning
% 5.87/2.09
% 5.87/2.09 % (13486)Memory used [KB]: 5117
% 5.87/2.09 % (13486)Time elapsed: 0.045 s
% 5.87/2.09 % (13486)Instructions burned: 153 (million)
% 5.87/2.09 % (13486)------------------------------
% 5.87/2.09 % (13486)------------------------------
% 5.87/2.10 % (13491)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_12985 for (2994ds/614Mi)
% 5.87/2.10 % (13492)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_12985 for (2994ds/202Mi)
% 5.87/2.12 % (13480)Instruction limit reached!
% 5.87/2.12 % (13480)------------------------------
% 5.87/2.12 % (13480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.12 % (13480)Termination reason: Unknown
% 5.87/2.12 % (13480)Termination phase: Saturation
% 5.87/2.12
% 5.87/2.12 % (13480)Memory used [KB]: 7675
% 5.87/2.12 % (13480)Time elapsed: 0.112 s
% 5.87/2.12 % (13480)Instructions burned: 453 (million)
% 5.87/2.12 % (13480)------------------------------
% 5.87/2.12 % (13480)------------------------------
% 5.87/2.12 % (13490)Instruction limit reached!
% 5.87/2.12 % (13490)------------------------------
% 5.87/2.12 % (13490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.87/2.12 % (13490)Termination reason: Unknown
% 5.87/2.12 % (13490)Termination phase: Function definition elimination
% 5.87/2.12
% 5.87/2.12 % (13490)Memory used [KB]: 2814
% 5.87/2.12 % (13490)Time elapsed: 0.037 s
% 5.87/2.12 % (13490)Instructions burned: 143 (million)
% 5.87/2.12 % (13490)------------------------------
% 5.87/2.12 % (13490)------------------------------
% 5.87/2.13 % (13493)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_12985 for (2994ds/339Mi)
% 6.44/2.13 % (13494)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_12985 for (2994ds/283Mi)
% 6.44/2.14 % (13489)Instruction limit reached!
% 6.44/2.14 % (13489)------------------------------
% 6.44/2.14 % (13489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.44/2.14 % (13489)Termination reason: Unknown
% 6.44/2.14 % (13489)Termination phase: Saturation
% 6.44/2.14
% 6.44/2.14 % (13489)Memory used [KB]: 5628
% 6.44/2.14 % (13489)Time elapsed: 0.071 s
% 6.44/2.14 % (13489)Instructions burned: 255 (million)
% 6.44/2.14 % (13489)------------------------------
% 6.44/2.14 % (13489)------------------------------
% 6.44/2.14 % (13487)Instruction limit reached!
% 6.44/2.14 % (13487)------------------------------
% 6.44/2.14 % (13487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.44/2.14 % (13487)Termination reason: Unknown
% 6.44/2.14 % (13487)Termination phase: Saturation
% 6.44/2.14
% 6.44/2.14 % (13487)Memory used [KB]: 8315
% 6.44/2.14 % (13487)Time elapsed: 0.089 s
% 6.44/2.14 % (13487)Instructions burned: 343 (million)
% 6.44/2.14 % (13487)------------------------------
% 6.44/2.14 % (13487)------------------------------
% 6.44/2.15 % (13495)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_12985 for (2994ds/134Mi)
% 6.44/2.16 % (13492)Instruction limit reached!
% 6.44/2.16 % (13492)------------------------------
% 6.44/2.16 % (13492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.44/2.16 % (13492)Termination reason: Unknown
% 6.44/2.16 % (13492)Termination phase: Property scanning
% 6.44/2.16
% 6.44/2.16 % (13492)Memory used [KB]: 5245
% 6.44/2.16 % (13492)Time elapsed: 0.058 s
% 6.44/2.16 % (13492)Instructions burned: 206 (million)
% 6.44/2.16 % (13492)------------------------------
% 6.44/2.16 % (13492)------------------------------
% 6.44/2.16 % (13496)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_12985 for (2994ds/298Mi)
% 6.44/2.17 % (13497)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_12985 for (2994ds/192Mi)
% 6.44/2.18 % (13495)Instruction limit reached!
% 6.44/2.18 % (13495)------------------------------
% 6.44/2.18 % (13495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.44/2.18 % (13495)Termination reason: Unknown
% 6.44/2.18 % (13495)Termination phase: Property scanning
% 6.44/2.18
% 6.44/2.18 % (13495)Memory used [KB]: 2942
% 6.44/2.18 % (13495)Time elapsed: 0.036 s
% 6.44/2.18 % (13495)Instructions burned: 137 (million)
% 6.44/2.18 % (13495)------------------------------
% 6.44/2.18 % (13495)------------------------------
% 6.44/2.19 % (13498)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_12985 for (2993ds/258Mi)
% 6.79/2.20 % (13494)Instruction limit reached!
% 6.79/2.20 % (13494)------------------------------
% 6.79/2.20 % (13494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.20 % (13494)Termination reason: Unknown
% 6.79/2.20 % (13494)Termination phase: Saturation
% 6.79/2.20
% 6.79/2.20 % (13494)Memory used [KB]: 7547
% 6.79/2.20 % (13494)Time elapsed: 0.072 s
% 6.79/2.20 % (13494)Instructions burned: 283 (million)
% 6.79/2.20 % (13494)------------------------------
% 6.79/2.20 % (13494)------------------------------
% 6.79/2.21 % (13499)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_12985 for (2993ds/90Mi)
% 6.79/2.22 % (13493)Instruction limit reached!
% 6.79/2.22 % (13493)------------------------------
% 6.79/2.22 % (13493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.22 % (13493)Termination reason: Unknown
% 6.79/2.22 % (13493)Termination phase: Saturation
% 6.79/2.22
% 6.79/2.22 % (13493)Memory used [KB]: 7931
% 6.79/2.22 % (13493)Time elapsed: 0.092 s
% 6.79/2.22 % (13493)Instructions burned: 339 (million)
% 6.79/2.22 % (13493)------------------------------
% 6.79/2.22 % (13493)------------------------------
% 6.79/2.22 % (13497)Instruction limit reached!
% 6.79/2.22 % (13497)------------------------------
% 6.79/2.22 % (13497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.22 % (13497)Termination reason: Unknown
% 6.79/2.22 % (13497)Termination phase: Saturation
% 6.79/2.22
% 6.79/2.22 % (13497)Memory used [KB]: 7547
% 6.79/2.22 % (13497)Time elapsed: 0.051 s
% 6.79/2.22 % (13497)Instructions burned: 193 (million)
% 6.79/2.22 % (13497)------------------------------
% 6.79/2.22 % (13497)------------------------------
% 6.79/2.23 % (13500)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_12985 for (2993ds/231Mi)
% 6.79/2.23 % (13501)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_12985 for (2993ds/458Mi)
% 6.79/2.24 % (13499)Instruction limit reached!
% 6.79/2.24 % (13499)------------------------------
% 6.79/2.24 % (13499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.24 % (13499)Termination reason: Unknown
% 6.79/2.24 % (13499)Termination phase: shuffling
% 6.79/2.24
% 6.79/2.24 % (13499)Memory used [KB]: 2558
% 6.79/2.24 % (13499)Time elapsed: 0.026 s
% 6.79/2.24 % (13499)Instructions burned: 91 (million)
% 6.79/2.24 % (13499)------------------------------
% 6.79/2.24 % (13499)------------------------------
% 6.79/2.25 % (13502)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_12985 for (2993ds/31Mi)
% 6.79/2.26 % (13502)Instruction limit reached!
% 6.79/2.26 % (13502)------------------------------
% 6.79/2.26 % (13502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.26 % (13502)Termination reason: Unknown
% 6.79/2.26 % (13502)Termination phase: shuffling
% 6.79/2.26
% 6.79/2.26 % (13502)Memory used [KB]: 2174
% 6.79/2.26 % (13502)Time elapsed: 0.010 s
% 6.79/2.26 % (13502)Instructions burned: 33 (million)
% 6.79/2.26 % (13502)------------------------------
% 6.79/2.26 % (13502)------------------------------
% 6.79/2.26 % (13498)Instruction limit reached!
% 6.79/2.26 % (13498)------------------------------
% 6.79/2.26 % (13498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.26 % (13498)Termination reason: Unknown
% 6.79/2.26 % (13498)Termination phase: Saturation
% 6.79/2.26
% 6.79/2.26 % (13498)Memory used [KB]: 7931
% 6.79/2.26 % (13498)Time elapsed: 0.070 s
% 6.79/2.26 % (13498)Instructions burned: 261 (million)
% 6.79/2.26 % (13498)------------------------------
% 6.79/2.26 % (13498)------------------------------
% 6.79/2.27 % (13503)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_12985 for (2993ds/297Mi)
% 6.79/2.27 % (13504)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_12985 for (2993ds/987Mi)
% 6.79/2.27 % (13491)Instruction limit reached!
% 6.79/2.27 % (13491)------------------------------
% 6.79/2.27 % (13491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.27 % (13491)Termination reason: Unknown
% 6.79/2.27 % (13491)Termination phase: Saturation
% 6.79/2.27
% 6.79/2.27 % (13491)Memory used [KB]: 10234
% 6.79/2.27 % (13491)Time elapsed: 0.178 s
% 6.79/2.27 % (13491)Instructions burned: 615 (million)
% 6.79/2.27 % (13491)------------------------------
% 6.79/2.27 % (13491)------------------------------
% 6.79/2.28 % (13496)Instruction limit reached!
% 6.79/2.28 % (13496)------------------------------
% 6.79/2.28 % (13496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.28 % (13496)Termination reason: Unknown
% 6.79/2.28 % (13496)Termination phase: Saturation
% 6.79/2.28
% 6.79/2.28 % (13496)Memory used [KB]: 7547
% 6.79/2.28 % (13496)Time elapsed: 0.117 s
% 6.79/2.28 % (13496)Instructions burned: 298 (million)
% 6.79/2.28 % (13496)------------------------------
% 6.79/2.28 % (13496)------------------------------
% 6.79/2.28 % (13448)Instruction limit reached!
% 6.79/2.28 % (13448)------------------------------
% 6.79/2.28 % (13448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.28 % (13448)Termination reason: Unknown
% 6.79/2.28 % (13448)Termination phase: Saturation
% 6.79/2.28
% 6.79/2.28 % (13448)Memory used [KB]: 12537
% 6.79/2.28 % (13448)Time elapsed: 0.363 s
% 6.79/2.28 % (13448)Instructions burned: 1089 (million)
% 6.79/2.28 % (13448)------------------------------
% 6.79/2.28 % (13448)------------------------------
% 6.79/2.28 % (13505)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_12985 for (2993ds/182Mi)
% 6.79/2.29 % (13500)Instruction limit reached!
% 6.79/2.29 % (13500)------------------------------
% 6.79/2.29 % (13500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.79/2.29 % (13500)Termination reason: Unknown
% 6.79/2.29 % (13500)Termination phase: Saturation
% 6.79/2.29
% 6.79/2.29 % (13500)Memory used [KB]: 8443
% 6.79/2.29 % (13500)Time elapsed: 0.061 s
% 6.79/2.29 % (13500)Instructions burned: 233 (million)
% 6.79/2.29 % (13500)------------------------------
% 6.79/2.29 % (13500)------------------------------
% 6.79/2.29 % (13506)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_12985 for (2993ds/182Mi)
% 6.79/2.29 % (13507)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_12985 for (2993ds/154Mi)
% 6.79/2.30 % (13508)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_12985 for (2993ds/79Mi)
% 7.67/2.32 % (13508)Instruction limit reached!
% 7.67/2.32 % (13508)------------------------------
% 7.67/2.32 % (13508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.32 % (13508)Termination reason: Unknown
% 7.67/2.32 % (13508)Termination phase: Property scanning
% 7.67/2.32
% 7.67/2.32 % (13508)Memory used [KB]: 2558
% 7.67/2.32 % (13508)Time elapsed: 0.022 s
% 7.67/2.32 % (13508)Instructions burned: 81 (million)
% 7.67/2.32 % (13508)------------------------------
% 7.67/2.32 % (13508)------------------------------
% 7.67/2.33 % (13509)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_12985 for (2992ds/216Mi)
% 7.67/2.33 % (13505)Instruction limit reached!
% 7.67/2.33 % (13505)------------------------------
% 7.67/2.33 % (13505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.33 % (13505)Termination reason: Unknown
% 7.67/2.33 % (13505)Termination phase: Saturation
% 7.67/2.33
% 7.67/2.33 % (13505)Memory used [KB]: 7419
% 7.67/2.33 % (13505)Time elapsed: 0.048 s
% 7.67/2.33 % (13505)Instructions burned: 185 (million)
% 7.67/2.33 % (13505)------------------------------
% 7.67/2.33 % (13505)------------------------------
% 7.67/2.33 % (13507)Instruction limit reached!
% 7.67/2.33 % (13507)------------------------------
% 7.67/2.33 % (13507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.33 % (13507)Termination reason: Unknown
% 7.67/2.33 % (13507)Termination phase: Property scanning
% 7.67/2.33
% 7.67/2.33 % (13507)Memory used [KB]: 2686
% 7.67/2.33 % (13507)Time elapsed: 0.040 s
% 7.67/2.33 % (13507)Instructions burned: 157 (million)
% 7.67/2.33 % (13507)------------------------------
% 7.67/2.33 % (13507)------------------------------
% 7.67/2.34 % (13510)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_12985 for (2992ds/114Mi)
% 7.67/2.34 % (13503)Instruction limit reached!
% 7.67/2.34 % (13503)------------------------------
% 7.67/2.34 % (13503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.34 % (13503)Termination reason: Unknown
% 7.67/2.34 % (13503)Termination phase: Saturation
% 7.67/2.34
% 7.67/2.34 % (13503)Memory used [KB]: 7547
% 7.67/2.34 % (13503)Time elapsed: 0.075 s
% 7.67/2.34 % (13503)Instructions burned: 299 (million)
% 7.67/2.34 % (13503)------------------------------
% 7.67/2.34 % (13503)------------------------------
% 7.67/2.34 % (13511)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_12985 for (2992ds/922Mi)
% 7.67/2.34 % (13506)Instruction limit reached!
% 7.67/2.34 % (13506)------------------------------
% 7.67/2.34 % (13506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.34 % (13506)Termination reason: Unknown
% 7.67/2.34 % (13506)Termination phase: Property scanning
% 7.67/2.34
% 7.67/2.34 % (13506)Memory used [KB]: 5245
% 7.67/2.34 % (13506)Time elapsed: 0.057 s
% 7.67/2.34 % (13506)Instructions burned: 185 (million)
% 7.67/2.34 % (13506)------------------------------
% 7.67/2.34 % (13506)------------------------------
% 7.67/2.35 % (13512)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_12985 for (2992ds/182Mi)
% 7.67/2.35 % (13501)Instruction limit reached!
% 7.67/2.35 % (13501)------------------------------
% 7.67/2.35 % (13501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.35 % (13501)Termination reason: Unknown
% 7.67/2.35 % (13501)Termination phase: Saturation
% 7.67/2.35
% 7.67/2.35 % (13501)Memory used [KB]: 9722
% 7.67/2.35 % (13501)Time elapsed: 0.124 s
% 7.67/2.35 % (13501)Instructions burned: 458 (million)
% 7.67/2.35 % (13501)------------------------------
% 7.67/2.35 % (13501)------------------------------
% 7.67/2.36 % (13513)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_12985 for (2992ds/164Mi)
% 7.67/2.36 % (13514)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_12985 for (2992ds/50Mi)
% 7.67/2.37 % (13510)Instruction limit reached!
% 7.67/2.37 % (13510)------------------------------
% 7.67/2.37 % (13510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.37 % (13510)Termination reason: Unknown
% 7.67/2.37 % (13510)Termination phase: Property scanning
% 7.67/2.37
% 7.67/2.37 % (13510)Memory used [KB]: 2814
% 7.67/2.37 % (13510)Time elapsed: 0.031 s
% 7.67/2.37 % (13510)Instructions burned: 117 (million)
% 7.67/2.37 % (13510)------------------------------
% 7.67/2.37 % (13510)------------------------------
% 7.67/2.37 % (13514)Instruction limit reached!
% 7.67/2.37 % (13514)------------------------------
% 7.67/2.37 % (13514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.37 % (13514)Termination reason: Unknown
% 7.67/2.37 % (13514)Termination phase: shuffling
% 7.67/2.37
% 7.67/2.37 % (13514)Memory used [KB]: 2430
% 7.67/2.37 % (13514)Time elapsed: 0.015 s
% 7.67/2.37 % (13514)Instructions burned: 53 (million)
% 7.67/2.37 % (13514)------------------------------
% 7.67/2.37 % (13514)------------------------------
% 7.67/2.38 % (13515)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_12985 for (2992ds/368Mi)
% 7.67/2.38 % (13509)Instruction limit reached!
% 7.67/2.38 % (13509)------------------------------
% 7.67/2.38 % (13509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.38 % (13509)Termination reason: Unknown
% 7.67/2.38 % (13509)Termination phase: Property scanning
% 7.67/2.38
% 7.67/2.38 % (13509)Memory used [KB]: 2814
% 7.67/2.38 % (13509)Time elapsed: 0.054 s
% 7.67/2.38 % (13509)Instructions burned: 217 (million)
% 7.67/2.38 % (13509)------------------------------
% 7.67/2.38 % (13509)------------------------------
% 7.67/2.38 % (13516)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_12985 for (2991ds/1021Mi)
% 7.67/2.39 % (13517)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_12985 for (2991ds/267Mi)
% 7.67/2.39 % (13512)Instruction limit reached!
% 7.67/2.39 % (13512)------------------------------
% 7.67/2.39 % (13512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.67/2.39 % (13512)Termination reason: Unknown
% 7.67/2.39 % (13512)Termination phase: Saturation
% 7.67/2.39
% 7.67/2.39 % (13512)Memory used [KB]: 7291
% 7.67/2.39 % (13512)Time elapsed: 0.046 s
% 7.67/2.39 % (13512)Instructions burned: 182 (million)
% 7.67/2.39 % (13512)------------------------------
% 7.67/2.39 % (13512)------------------------------
% 8.26/2.40 % (13513)Instruction limit reached!
% 8.26/2.40 % (13513)------------------------------
% 8.26/2.40 % (13513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.40 % (13513)Termination reason: Unknown
% 8.26/2.40 % (13513)Termination phase: Function definition elimination
% 8.26/2.40
% 8.26/2.40 % (13513)Memory used [KB]: 2814
% 8.26/2.40 % (13513)Time elapsed: 0.045 s
% 8.26/2.40 % (13513)Instructions burned: 164 (million)
% 8.26/2.40 % (13513)------------------------------
% 8.26/2.40 % (13513)------------------------------
% 8.26/2.40 % (13518)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_12985 for (2991ds/204Mi)
% 8.26/2.41 % (13519)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_12985 for (2991ds/26Mi)
% 8.26/2.42 % (13519)Instruction limit reached!
% 8.26/2.42 % (13519)------------------------------
% 8.26/2.42 % (13519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.42 % (13519)Termination reason: Unknown
% 8.26/2.42 % (13519)Termination phase: shuffling
% 8.26/2.42
% 8.26/2.42 % (13519)Memory used [KB]: 2174
% 8.26/2.42 % (13519)Time elapsed: 0.009 s
% 8.26/2.42 % (13519)Instructions burned: 28 (million)
% 8.26/2.42 % (13519)------------------------------
% 8.26/2.42 % (13519)------------------------------
% 8.26/2.43 % (13520)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_12985 for (2991ds/53Mi)
% 8.26/2.44 % (13520)Instruction limit reached!
% 8.26/2.44 % (13520)------------------------------
% 8.26/2.44 % (13520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.44 % (13520)Termination reason: Unknown
% 8.26/2.44 % (13520)Termination phase: shuffling
% 8.26/2.44
% 8.26/2.44 % (13520)Memory used [KB]: 2558
% 8.26/2.44 % (13520)Time elapsed: 0.016 s
% 8.26/2.44 % (13520)Instructions burned: 53 (million)
% 8.26/2.44 % (13520)------------------------------
% 8.26/2.44 % (13520)------------------------------
% 8.26/2.45 % (13521)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_12985 for (2991ds/4436Mi)
% 8.26/2.45 % (13518)Instruction limit reached!
% 8.26/2.45 % (13518)------------------------------
% 8.26/2.45 % (13518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.45 % (13518)Termination reason: Unknown
% 8.26/2.45 % (13518)Termination phase: Property scanning
% 8.26/2.45
% 8.26/2.45 % (13518)Memory used [KB]: 2942
% 8.26/2.45 % (13518)Time elapsed: 0.052 s
% 8.26/2.45 % (13518)Instructions burned: 205 (million)
% 8.26/2.45 % (13518)------------------------------
% 8.26/2.45 % (13518)------------------------------
% 8.26/2.46 % (13517)Instruction limit reached!
% 8.26/2.46 % (13517)------------------------------
% 8.26/2.46 % (13517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.46 % (13517)Termination reason: Unknown
% 8.26/2.46 % (13517)Termination phase: Saturation
% 8.26/2.46
% 8.26/2.46 % (13517)Memory used [KB]: 7547
% 8.26/2.46 % (13517)Time elapsed: 0.068 s
% 8.26/2.46 % (13517)Instructions burned: 270 (million)
% 8.26/2.46 % (13517)------------------------------
% 8.26/2.46 % (13517)------------------------------
% 8.26/2.46 % (13522)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_12985 for (2991ds/1539Mi)
% 8.26/2.47 % (13523)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_12985 for (2991ds/408Mi)
% 8.26/2.48 % (13515)Instruction limit reached!
% 8.26/2.48 % (13515)------------------------------
% 8.26/2.48 % (13515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.48 % (13515)Termination reason: Unknown
% 8.26/2.48 % (13515)Termination phase: Saturation
% 8.26/2.48
% 8.26/2.48 % (13515)Memory used [KB]: 9338
% 8.26/2.48 % (13515)Time elapsed: 0.099 s
% 8.26/2.48 % (13515)Instructions burned: 371 (million)
% 8.26/2.48 % (13515)------------------------------
% 8.26/2.48 % (13515)------------------------------
% 8.26/2.49 % (13524)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_12985 for (2990ds/5205Mi)
% 8.62/2.52 % (13504)Instruction limit reached!
% 8.62/2.52 % (13504)------------------------------
% 8.62/2.52 % (13504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.52 % (13504)Termination reason: Unknown
% 8.62/2.52 % (13504)Termination phase: Saturation
% 8.62/2.52
% 8.62/2.52 % (13504)Memory used [KB]: 16375
% 8.62/2.52 % (13504)Time elapsed: 0.250 s
% 8.62/2.52 % (13504)Instructions burned: 988 (million)
% 8.62/2.52 % (13504)------------------------------
% 8.62/2.52 % (13504)------------------------------
% 8.62/2.53 % (13525)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_12985 for (2990ds/1913Mi)
% 8.62/2.59 % (13523)Instruction limit reached!
% 8.62/2.59 % (13523)------------------------------
% 8.62/2.59 % (13523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.62/2.59 % (13523)Termination reason: Unknown
% 8.62/2.59 % (13523)Termination phase: Saturation
% 8.62/2.59
% 8.62/2.59 % (13523)Memory used [KB]: 11513
% 8.62/2.59 % (13523)Time elapsed: 0.121 s
% 8.62/2.59 % (13523)Instructions burned: 410 (million)
% 8.62/2.59 % (13523)------------------------------
% 8.62/2.59 % (13523)------------------------------
% 8.62/2.60 % (13526)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_12985 for (2989ds/4350Mi)
% 9.00/2.62 % (13511)Instruction limit reached!
% 9.00/2.62 % (13511)------------------------------
% 9.00/2.62 % (13511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.00/2.62 % (13511)Termination reason: Unknown
% 9.00/2.62 % (13511)Termination phase: Saturation
% 9.00/2.62
% 9.00/2.62 % (13511)Memory used [KB]: 12409
% 9.00/2.62 % (13511)Time elapsed: 0.307 s
% 9.00/2.62 % (13511)Instructions burned: 924 (million)
% 9.00/2.62 % (13511)------------------------------
% 9.00/2.62 % (13511)------------------------------
% 9.00/2.64 % (13527)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=68:si=on:rtra=on_0 on DTF2THF_12985 for (2989ds/68Mi)
% 9.00/2.65 % (13527)Instruction limit reached!
% 9.00/2.65 % (13527)------------------------------
% 9.00/2.65 % (13527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.00/2.65 % (13527)Termination reason: Unknown
% 9.00/2.65 % (13527)Termination phase: Property scanning
% 9.00/2.65
% 9.00/2.65 % (13527)Memory used [KB]: 2558
% 9.00/2.65 % (13527)Time elapsed: 0.021 s
% 9.00/2.65 % (13527)Instructions burned: 72 (million)
% 9.00/2.65 % (13527)------------------------------
% 9.00/2.65 % (13527)------------------------------
% 9.00/2.66 % (13528)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_12985 for (2989ds/358Mi)
% 9.33/2.70 % (13516)Instruction limit reached!
% 9.33/2.70 % (13516)------------------------------
% 9.33/2.70 % (13516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.33/2.70 % (13516)Termination reason: Unknown
% 9.33/2.70 % (13516)Termination phase: Saturation
% 9.33/2.70
% 9.33/2.70 % (13516)Memory used [KB]: 12792
% 9.33/2.70 % (13516)Time elapsed: 0.317 s
% 9.33/2.70 % (13516)Instructions burned: 1021 (million)
% 9.33/2.70 % (13516)------------------------------
% 9.33/2.70 % (13516)------------------------------
% 9.33/2.71 % (13529)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_12985 for (2988ds/31Mi)
% 9.33/2.72 % (13529)Instruction limit reached!
% 9.33/2.72 % (13529)------------------------------
% 9.33/2.72 % (13529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.33/2.72 % (13529)Termination reason: Unknown
% 9.33/2.72 % (13529)Termination phase: shuffling
% 9.33/2.72
% 9.33/2.72 % (13529)Memory used [KB]: 2046
% 9.33/2.72 % (13529)Time elapsed: 0.010 s
% 9.33/2.72 % (13529)Instructions burned: 31 (million)
% 9.33/2.72 % (13529)------------------------------
% 9.33/2.72 % (13529)------------------------------
% 9.33/2.74 % (13530)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_12985 for (2988ds/279Mi)
% 9.33/2.76 % (13528)Instruction limit reached!
% 9.33/2.76 % (13528)------------------------------
% 9.33/2.76 % (13528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.33/2.76 % (13528)Termination reason: Unknown
% 9.33/2.76 % (13528)Termination phase: Saturation
% 9.33/2.76
% 9.33/2.76 % (13528)Memory used [KB]: 8315
% 9.33/2.76 % (13528)Time elapsed: 0.093 s
% 9.33/2.76 % (13528)Instructions burned: 361 (million)
% 9.33/2.76 % (13528)------------------------------
% 9.33/2.76 % (13528)------------------------------
% 9.33/2.77 % (13531)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_12985 for (2988ds/258Mi)
% 9.78/2.84 % (13531)Instruction limit reached!
% 9.78/2.84 % (13531)------------------------------
% 9.78/2.84 % (13531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.78/2.84 % (13531)Termination reason: Unknown
% 9.78/2.84 % (13531)Termination phase: Saturation
% 9.78/2.84
% 9.78/2.84 % (13531)Memory used [KB]: 8315
% 9.78/2.84 % (13531)Time elapsed: 0.072 s
% 9.78/2.84 % (13531)Instructions burned: 259 (million)
% 9.78/2.84 % (13531)------------------------------
% 9.78/2.84 % (13531)------------------------------
% 9.78/2.84 % (13530)Instruction limit reached!
% 9.78/2.84 % (13530)------------------------------
% 9.78/2.84 % (13530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.78/2.84 % (13530)Termination reason: Unknown
% 9.78/2.84 % (13530)Termination phase: Saturation
% 9.78/2.84
% 9.78/2.84 % (13530)Memory used [KB]: 7547
% 9.78/2.84 % (13530)Time elapsed: 0.108 s
% 9.78/2.84 % (13530)Instructions burned: 282 (million)
% 9.78/2.84 % (13530)------------------------------
% 9.78/2.84 % (13530)------------------------------
% 9.78/2.85 % (13532)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_12985 for (2987ds/26Mi)
% 9.78/2.85 % (13533)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_12985 for (2987ds/995Mi)
% 9.78/2.85 % (13532)Instruction limit reached!
% 9.78/2.85 % (13532)------------------------------
% 9.78/2.85 % (13532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.78/2.85 % (13532)Termination reason: Unknown
% 9.78/2.85 % (13532)Termination phase: shuffling
% 9.78/2.85
% 9.78/2.85 % (13532)Memory used [KB]: 2174
% 9.78/2.85 % (13532)Time elapsed: 0.009 s
% 9.78/2.85 % (13532)Instructions burned: 27 (million)
% 9.78/2.85 % (13532)------------------------------
% 9.78/2.85 % (13532)------------------------------
% 9.78/2.86 % (13534)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_12985 for (2987ds/373Mi)
% 10.00/2.94 % (13522)Instruction limit reached!
% 10.00/2.94 % (13522)------------------------------
% 10.00/2.94 % (13522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.00/2.94 % (13522)Termination reason: Unknown
% 10.00/2.94 % (13522)Termination phase: Saturation
% 10.00/2.94
% 10.00/2.94 % (13522)Memory used [KB]: 11001
% 10.00/2.94 % (13522)Time elapsed: 0.474 s
% 10.00/2.94 % (13522)Instructions burned: 1541 (million)
% 10.00/2.94 % (13522)------------------------------
% 10.00/2.94 % (13522)------------------------------
% 10.00/2.95 % (13535)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_12985 for (2986ds/253Mi)
% 10.00/2.96 % (13534)Instruction limit reached!
% 10.00/2.96 % (13534)------------------------------
% 10.00/2.96 % (13534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.00/2.96 % (13534)Termination reason: Unknown
% 10.00/2.96 % (13534)Termination phase: Saturation
% 10.00/2.96
% 10.00/2.96 % (13534)Memory used [KB]: 9338
% 10.00/2.96 % (13534)Time elapsed: 0.099 s
% 10.00/2.96 % (13534)Instructions burned: 373 (million)
% 10.00/2.96 % (13534)------------------------------
% 10.00/2.96 % (13534)------------------------------
% 10.00/2.97 % (13536)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_12985 for (2986ds/832Mi)
% 10.00/2.98 % (13521)First to succeed.
% 10.00/2.98 % (13521)Refutation found. Thanks to Tanya!
% 10.00/2.98 % SZS status Theorem for DTF2THF_12985
% 10.00/2.98 % SZS output start Proof for DTF2THF_12985
% 10.00/2.98 thf(func_def_0, type, in: $i > $i > $o).
% 10.00/2.98 thf(func_def_1, type, exu: ($i > $o) > $o).
% 10.00/2.98 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 10.00/2.98 thf(func_def_8, type, powerset: $i > $i).
% 10.00/2.98 thf(func_def_10, type, setunion: $i > $i).
% 10.00/2.98 thf(func_def_19, type, descr: ($i > $o) > $i).
% 10.00/2.98 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_26, type, prop2set: $o > $i).
% 10.00/2.98 thf(func_def_36, type, nonempty: $i > $o).
% 10.00/2.98 thf(func_def_69, type, set2prop: $i > $o).
% 10.00/2.98 thf(func_def_88, type, subset: $i > $i > $o).
% 10.00/2.98 thf(func_def_89, type, disjoint: $i > $i > $o).
% 10.00/2.98 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 10.00/2.98 thf(func_def_114, type, binunion: $i > $i > $i).
% 10.00/2.98 thf(func_def_122, type, binintersect: $i > $i > $i).
% 10.00/2.98 thf(func_def_135, type, regular: $i > $o).
% 10.00/2.98 thf(func_def_136, type, setminus: $i > $i > $i).
% 10.00/2.98 thf(func_def_147, type, symdiff: $i > $i > $i).
% 10.00/2.98 thf(func_def_153, type, iskpair: $i > $o).
% 10.00/2.98 thf(func_def_158, type, kpair: $i > $i > $i).
% 10.00/2.98 thf(func_def_160, type, cartprod: $i > $i > $i).
% 10.00/2.98 thf(func_def_177, type, singleton: $i > $o).
% 10.00/2.98 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_184, type, atmost1p: $i > $o).
% 10.00/2.98 thf(func_def_185, type, atleast2p: $i > $o).
% 10.00/2.98 thf(func_def_186, type, atmost2p: $i > $o).
% 10.00/2.98 thf(func_def_187, type, upairsetp: $i > $o).
% 10.00/2.98 thf(func_def_191, type, kfst: $i > $i).
% 10.00/2.98 thf(func_def_203, type, ksnd: $i > $i).
% 10.00/2.98 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 10.00/2.98 thf(func_def_222, type, func: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_223, type, funcSet: $i > $i > $i).
% 10.00/2.98 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 10.00/2.98 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 10.00/2.98 thf(func_def_323, type, breln1: $i > $i > $o).
% 10.00/2.98 thf(func_def_325, type, breln1Set: $i > $i).
% 10.00/2.98 thf(func_def_327, type, transitive: $i > $i > $o).
% 10.00/2.98 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 10.00/2.98 thf(func_def_329, type, reflexive: $i > $i > $o).
% 10.00/2.98 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 10.00/2.98 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 10.00/2.98 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 10.00/2.98 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 10.00/2.98 thf(func_def_354, type, sP1: $i > $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_356, type, sP3: ($i > $o) > $o).
% 10.00/2.98 thf(func_def_357, type, sP4: $i > $o).
% 10.00/2.98 thf(func_def_358, type, sP5: ($i > $i > $o) > $i > $i > $o).
% 10.00/2.98 thf(func_def_359, type, sP6: ($i > $i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_361, type, sP8: ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_363, type, sP10: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_364, type, sP11: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_365, type, sP12: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_366, type, sP13: $i > $i > $o).
% 10.00/2.98 thf(func_def_367, type, sP14: $i > $i > $o).
% 10.00/2.98 thf(func_def_368, type, sP15: $i > $i > $o).
% 10.00/2.98 thf(func_def_369, type, sP16: $i > $i > $o).
% 10.00/2.98 thf(func_def_370, type, sP17: $i > $o).
% 10.00/2.98 thf(func_def_371, type, sP18: $i > $i > $o).
% 10.00/2.98 thf(func_def_373, type, sP20: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_374, type, sP21: $i > $o).
% 10.00/2.98 thf(func_def_375, type, sP22: $i > $o).
% 10.00/2.98 thf(func_def_377, type, sP24: ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_379, type, sP26: $i > $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_380, type, sP27: $i > $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_381, type, sP28: ($i > $i > $o) > $i > $i > $o).
% 10.00/2.98 thf(func_def_382, type, sP29: $i > ($i > $i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_383, type, sP30: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_384, type, sP31: $i > $i > ($i > $i) > $o).
% 10.00/2.98 thf(func_def_386, type, sP33: $i > $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_387, type, sP34: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_388, type, sP35: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_389, type, sP36: $i > ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_390, type, sP37: ($i > $o) > $o).
% 10.00/2.98 thf(func_def_391, type, sP38: $i > $i > ($i > $i) > $o).
% 10.00/2.98 thf(func_def_393, type, sP40: $i > $i > $o).
% 10.00/2.98 thf(func_def_395, type, sP42: $o > $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_396, type, sP43: $i > $i > $i > $o > $o).
% 10.00/2.98 thf(func_def_397, type, sP44: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_398, type, sP45: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_399, type, sP46: $i > $i > $o > $o).
% 10.00/2.98 thf(func_def_403, type, sP50: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_404, type, sP51: $i > ($i > $o) > $i > $i > $o).
% 10.00/2.98 thf(func_def_405, type, sP52: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_406, type, sP53: $i > ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_408, type, sP55: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_410, type, sP57: $i > ($i > $o) > ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_411, type, sP58: $o > $i > $i > $o).
% 10.00/2.98 thf(func_def_412, type, sP59: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_413, type, sP60: $i > $o > $i > $o).
% 10.00/2.98 thf(func_def_415, type, sP62: $i > $i > $o).
% 10.00/2.98 thf(func_def_417, type, sP64: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_418, type, sP65: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_419, type, sP66: ($i > $o) > $o).
% 10.00/2.98 thf(func_def_420, type, sP67: ($i > $o) > $o).
% 10.00/2.98 thf(func_def_421, type, sP68: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_422, type, sP69: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_425, type, sP72: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_426, type, sP73: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_427, type, sP74: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_429, type, sP76: ($i > $o) > $o).
% 10.00/2.98 thf(func_def_431, type, sP78: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_434, type, sP81: $o > $i > $i > $o).
% 10.00/2.98 thf(func_def_435, type, sP82: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_436, type, sP83: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_437, type, sP84: $i > ($i > $o) > $o).
% 10.00/2.98 thf(func_def_438, type, sP85: ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_441, type, sP88: ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_443, type, sP90: ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_444, type, sP91: $i > $i > ($i > $o) > $i > $o).
% 10.00/2.98 thf(func_def_445, type, sP92: $i > $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_453, type, sK100: $i > $i).
% 10.00/2.98 thf(func_def_456, type, sK103: $i > $i > ($i > $i) > $i).
% 10.00/2.98 thf(func_def_461, type, sK108: $i > $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_463, type, sK110: $i > $o).
% 10.00/2.98 thf(func_def_469, type, sK116: $i > $i).
% 10.00/2.98 thf(func_def_472, type, sK119: $i > $i > ($i > $i) > $i).
% 10.00/2.98 thf(func_def_474, type, sK121: $i > $o).
% 10.00/2.98 thf(func_def_493, type, sK140: $i > $i > $o).
% 10.00/2.98 thf(func_def_494, type, sK141: $i > $i > $o).
% 10.00/2.98 thf(func_def_499, type, sK146: $i > $o).
% 10.00/2.98 thf(func_def_520, type, sK167: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_521, type, sK168: ($i > $o) > $i).
% 10.00/2.98 thf(func_def_522, type, sK169: $i > $o).
% 10.00/2.98 thf(func_def_531, type, sK178: $i > $i > $i).
% 10.00/2.98 thf(func_def_532, type, sK179: $i > $i).
% 10.00/2.98 thf(func_def_536, type, sK183: $i > $i > $o).
% 10.00/2.98 thf(func_def_538, type, sK185: $i > ($i > $i > $o) > $i).
% 10.00/2.98 thf(func_def_539, type, sK186: $i > ($i > $i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_540, type, sK187: ($i > $i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_541, type, sK188: ($i > $i > $o) > $i > $i > $i).
% 10.00/2.98 thf(func_def_546, type, sK193: $i > $o).
% 10.00/2.98 thf(func_def_548, type, sK195: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_549, type, sK196: ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_552, type, sK199: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_553, type, sK200: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_557, type, sK204: $i > $i).
% 10.00/2.98 thf(func_def_558, type, sK205: $i > $i).
% 10.00/2.98 thf(func_def_559, type, sK206: $i > $i).
% 10.00/2.98 thf(func_def_560, type, sK207: $i > $i).
% 10.00/2.98 thf(func_def_561, type, sK208: $i > $i > $i).
% 10.00/2.98 thf(func_def_562, type, sK209: $i > $i > $i).
% 10.00/2.98 thf(func_def_563, type, sK210: $i > $i > $i).
% 10.00/2.98 thf(func_def_564, type, sK211: $i > $i > $i).
% 10.00/2.98 thf(func_def_565, type, sK212: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_566, type, sK213: $i > $i > $i).
% 10.00/2.98 thf(func_def_567, type, sK214: $i > $i > $i).
% 10.00/2.98 thf(func_def_568, type, sK215: $i > $i > $i).
% 10.00/2.98 thf(func_def_569, type, sK216: $i > $i > $i).
% 10.00/2.98 thf(func_def_570, type, sK217: $i > $i > $i).
% 10.00/2.98 thf(func_def_571, type, sK218: $i > $i > $i).
% 10.00/2.98 thf(func_def_572, type, sK219: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_573, type, sK220: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_574, type, sK221: $i > $i).
% 10.00/2.98 thf(func_def_587, type, sK234: $i > $i > $i).
% 10.00/2.98 thf(func_def_588, type, sK235: $i > $i > $i).
% 10.00/2.98 thf(func_def_592, type, sK239: $i > $o).
% 10.00/2.98 thf(func_def_595, type, sK242: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_596, type, sK243: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_624, type, sK271: $i > $i).
% 10.00/2.98 thf(func_def_625, type, sK272: $i > $i).
% 10.00/2.98 thf(func_def_631, type, sK278: $i > $o).
% 10.00/2.98 thf(func_def_632, type, sK279: ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_633, type, sK280: $i > $o).
% 10.00/2.98 thf(func_def_635, type, sK282: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_636, type, sK283: ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_637, type, sK284: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_638, type, sK285: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_639, type, sK286: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_640, type, sK287: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_645, type, sK292: $i > $o).
% 10.00/2.98 thf(func_def_660, type, sK307: $i > $o).
% 10.00/2.98 thf(func_def_666, type, sK313: $i > ($i > $i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_667, type, sK314: $i > ($i > $i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_668, type, sK315: ($i > $i > $o) > $i > $i > $i).
% 10.00/2.98 thf(func_def_669, type, sK316: ($i > $i > $o) > $i > $i > $i).
% 10.00/2.98 thf(func_def_671, type, sK318: $i > $i > $o).
% 10.00/2.98 thf(func_def_683, type, sK330: $i > $o).
% 10.00/2.98 thf(func_def_706, type, sK353: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_707, type, sK354: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_736, type, sK383: $i > $i).
% 10.00/2.98 thf(func_def_738, type, sK385: $i > ($i > $i) > $i > $i).
% 10.00/2.98 thf(func_def_739, type, sK386: $i > $i > ($i > $i) > $i).
% 10.00/2.98 thf(func_def_755, type, sK402: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_756, type, sK403: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_757, type, sK404: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_758, type, sK405: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_765, type, sK412: $i > ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_766, type, sK413: $i > ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_767, type, sK414: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_770, type, sK417: $i > $o).
% 10.00/2.98 thf(func_def_775, type, sK422: ($i > $o) > $i).
% 10.00/2.98 thf(func_def_776, type, sK423: ($i > $o) > $i).
% 10.00/2.98 thf(func_def_777, type, sK424: $i > $o).
% 10.00/2.98 thf(func_def_782, type, sK429: $i > $i).
% 10.00/2.98 thf(func_def_785, type, sK432: $i > $i > ($i > $i) > $i).
% 10.00/2.98 thf(func_def_786, type, sK433: $i > $i > ($i > $i) > $i).
% 10.00/2.98 thf(func_def_789, type, sK436: $i > $i > $i).
% 10.00/2.98 thf(func_def_798, type, sK445: $i > $i > $i).
% 10.00/2.98 thf(func_def_828, type, sK475: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_829, type, sK476: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_830, type, sK477: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_831, type, sK478: $i > $o).
% 10.00/2.98 thf(func_def_892, type, sK539: $i > $i > $i).
% 10.00/2.98 thf(func_def_897, type, sK544: $i > $i > $i).
% 10.00/2.98 thf(func_def_898, type, sK545: $i > ($i > $o) > $i > $i > $i).
% 10.00/2.98 thf(func_def_899, type, sK546: $i > ($i > $o) > $i > $i > $i).
% 10.00/2.98 thf(func_def_900, type, sK547: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_904, type, sK551: $i > $o).
% 10.00/2.98 thf(func_def_925, type, sK572: $i > ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_926, type, sK573: $i > ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_927, type, sK574: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_930, type, sK577: $i > $o).
% 10.00/2.98 thf(func_def_935, type, sK582: $i > $i).
% 10.00/2.98 thf(func_def_938, type, sK585: $i > $i > ($i > $i) > $i).
% 10.00/2.98 thf(func_def_941, type, sK588: $i > $o).
% 10.00/2.98 thf(func_def_943, type, sK590: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_944, type, sK591: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_949, type, sK596: $i > ($i > $o) > ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_950, type, sK597: $i > ($i > $o) > ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_953, type, sK600: $i > $o).
% 10.00/2.98 thf(func_def_954, type, sK601: $i > $o).
% 10.00/2.98 thf(func_def_957, type, sK604: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_968, type, sK615: $i > $i > $o).
% 10.00/2.98 thf(func_def_973, type, sK620: $i > $o > $i > $i).
% 10.00/2.98 thf(func_def_982, type, sK629: $i > $i > $i).
% 10.00/2.98 thf(func_def_1027, type, sK674: $i > $i > $o).
% 10.00/2.98 thf(func_def_1029, type, sK676: $i > $o).
% 10.00/2.98 thf(func_def_1035, type, sK682: $i > $i > $i).
% 10.00/2.98 thf(func_def_1040, type, sK687: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1041, type, sK688: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1082, type, sK729: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1089, type, sK736: ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1090, type, sK737: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1091, type, sK738: $i > $o).
% 10.00/2.98 thf(func_def_1097, type, sK744: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1098, type, sK745: ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1099, type, sK746: $i > $o).
% 10.00/2.98 thf(func_def_1101, type, sK748: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1102, type, sK749: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1134, type, sK781: $i > $o).
% 10.00/2.98 thf(func_def_1151, type, sK798: $i > $i > $i).
% 10.00/2.98 thf(func_def_1156, type, sK803: $i > $o).
% 10.00/2.98 thf(func_def_1157, type, sK804: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1161, type, sK808: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1162, type, sK809: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1163, type, sK810: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1164, type, sK811: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1180, type, sK827: $i > $o).
% 10.00/2.98 thf(func_def_1184, type, sK831: $i > $o).
% 10.00/2.98 thf(func_def_1185, type, sK832: $i > $o).
% 10.00/2.98 thf(func_def_1186, type, sK833: ($i > $o) > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1187, type, sK834: ($i > $o) > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1191, type, sK838: ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1192, type, sK839: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1193, type, sK840: $i > $o).
% 10.00/2.98 thf(func_def_1196, type, sK843: $i > $i > $o).
% 10.00/2.98 thf(func_def_1200, type, sK847: $i > $i > $i).
% 10.00/2.98 thf(func_def_1202, type, sK849: $i > $o).
% 10.00/2.98 thf(func_def_1203, type, sK850: ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_1204, type, sK851: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1208, type, sK855: $i > $i).
% 10.00/2.98 thf(func_def_1211, type, sK858: $i > $i > ($i > $i) > $i).
% 10.00/2.98 thf(func_def_1218, type, sK865: $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1219, type, sK866: $i > $i > $i > $o).
% 10.00/2.98 thf(func_def_1225, type, sK872: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1226, type, sK873: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1227, type, sK874: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1229, type, sK876: $i > $o).
% 10.00/2.98 thf(func_def_1233, type, sK880: $i > $o).
% 10.00/2.98 thf(func_def_1236, type, sK883: $i > $o).
% 10.00/2.98 thf(func_def_1237, type, sK884: ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1263, type, sK910: $i > $o).
% 10.00/2.98 thf(func_def_1264, type, sK911: ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_1265, type, sK912: ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_1266, type, sK913: $i > $o).
% 10.00/2.98 thf(func_def_1267, type, sK914: $i > $o).
% 10.00/2.98 thf(func_def_1268, type, sK915: ($i > $o) > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1269, type, sK916: ($i > $o) > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1285, type, sK932: $i > $i).
% 10.00/2.98 thf(func_def_1293, type, sK940: $i > $o).
% 10.00/2.98 thf(func_def_1295, type, sK942: $i > ($i > $o) > $i).
% 10.00/2.98 thf(func_def_1296, type, sK943: ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_1307, type, sK954: $i > $o).
% 10.00/2.98 thf(func_def_1308, type, sK955: $i > $i).
% 10.00/2.98 thf(func_def_1314, type, sK961: $i > $i > $o).
% 10.00/2.98 thf(func_def_1322, type, sK969: $i > $i).
% 10.00/2.98 thf(func_def_1327, type, sK974: $i > $o).
% 10.00/2.98 thf(func_def_1329, type, sK976: $i > $i > ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_1330, type, sK977: $i > $i > ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_1331, type, sK978: ($i > $o) > $i > $i).
% 10.00/2.98 thf(func_def_1335, type, sK982: $i > $o).
% 10.00/2.98 thf(func_def_1347, type, sK994: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1348, type, sK995: $i > $i > $i > $i > $i).
% 10.00/2.98 thf(func_def_1359, type, sK1006: $i > $i > $o).
% 10.00/2.98 thf(func_def_1372, type, sK1019: $i > $i).
% 10.00/2.98 thf(func_def_1375, type, sK1022: $i > $i > $o).
% 10.00/2.98 thf(func_def_1395, type, ph1042: !>[X0: $tType]:(X0)).
% 10.00/2.98 thf(f10357,plain,(
% 10.00/2.98 $false),
% 10.00/2.98 inference(subsumption_resolution,[status(thm)],[f10356,f4133])).
% 10.00/2.98 thf(f4133,plain,(
% 10.00/2.98 ($true = (in @ sK640 @ sK639))),
% 10.00/2.98 inference(cnf_transformation,[status(thm)],[f2414])).
% 10.00/2.98 thf(f2414,plain,(
% 10.00/2.98 (cartprodfstpairEq = $true) & (singletonprop = $true) & (prop2setE = $true) & (eqbreln = $true) & (quantDeMorgan4 = $true) & (nonemptyE1 = $true) & (eta1 = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (binintersectSubset1 = $true) & (sepSubset = $true) & (binintersectER = $true) & (iffalseProp1 = $true) & (singletonsuniq = $true) & (cartprodsndpairEq = $true) & (setminusIRneg = $true) & (notsubsetI = $true) & (setukpairIR = $true) & (binintersectT_lem = $true) & (apProp = $true) & (complementSubsetComplementIntersect = $true) & (binintersectRsub = $true) & (eqinunit = $true) & (kfstsingleton = $true) & (iftrueorfalse = $true) & (sepInPowerset = $true) & (upairsubunion = $true) & (notequalI2 = $true) & (setminusEL = $true) & (complementInPowersetComplementIntersect = $true) & (setext = $true) & (powerset__Cong = $true) & (contrasubsetT1 = $true) & (binintersectEL = $true) & (doubleComplementE1 = $true) & (exuE3e = $true) & (doubleComplementI1 = $true) & (setadjoinE = $true) & (singletoninpowerset = $true) & (intersectInPowersetIntersectUnions = $true) & (demorgan2a = $true) & (lamProp = $true) & (exuI1 = $true) & (ifp = $true) & (emptyinunitempty = $true) & (woz1_5 = $true) & (symdiffI1 = $true) & (notdexE = $true) & (nonemptyImpWitness = $true) & (emptysetsubset = $true) & (exu__Cong = $true) & (complementTnotintersectT = $true) & (inIntersectImpInIntersectUnions = $true) & (upairinpowunion = $true) & (setadjoinIR = $true) & (emptyE1 = $true) & (doubleComplementSub1 = $true) & (setbeta = $true) & (cartprodpairsurjEq = $true) & (symdiffE = $true) & (binunionIR = $true) & (woz1_3 = $true) & (contrasubsetT2 = $true) & (iffalse = $true) & (cartprodpairmemEL = $true) & (subsetI1 = $true) & (doubleComplementEq = $true) & (setminusSubset2 = $true) & (setminusI = $true) & (ap2apEq2 = $true) & (notinsingleton = $true) & (demorgan2a2 = $true) & (breln1all2 = $true) & (binunionIL = $true) & (complementUnionInPowersetComplement = $true) & (setminusERneg = $true) & (woz1_1 = $true) & (setadjoinSub = $true) & (wellorderingAx = $true) & (demorgan2a1 = $true) & (singletonsubset = $true) & (setukpairinjR1 = $true) & (contrasubsetT3 = $true) & (doubleComplementSub2 = $true) & (setunion__Cong = $true) & (breln1invprop = $true) & (binunionRsub = $true) & (setadjoinOr = $true) & (ex1I = $true) & (setminusLsub = $true) & (powersetI = $true) & (brelnall2 = $true) & (demorgan1a = $true) & (foundationAx = $true) & (powersetsubset = $true) & (upairsetIL = $true) & (dpsetconstrEL2 = $true) & (exuE3u = $true) & (dpsetconstrEL1 = $true) & (noeltsimpempty = $true) & (setadjoin__Cong = $true) & (brelnall1 = $true) & (app = $true) & (binunionTEcontra = $true) & (exuEu = $true) & (dpsetconstrSub = $true) & (subsetemptysetimpeq = $true) & (woz1_2 = $true) & (dsetconstrER = $true) & (iftrueProp1 = $true) & (((breln1 @ sK639 @ sK638) = $true) & (($true = (in @ sK640 @ sK639)) & (($true = (in @ sK641 @ sK639)) & ($true != (in @ (kpair @ sK641 @ sK640) @ (breln1invset @ sK639 @ sK638))) & ($true = (in @ (kpair @ sK640 @ sK641) @ sK638))))) & (cartprodmempair = $true) & (notequalI1 = $true) & (binintersectTERcontra = $true) & (cartprodpairmemER = $true) & (emptysetimpfalse = $true) & (quantDeMorgan1 = $true) & (nonemptyI1 = $true) & (demorgan2b2 = $true) & (setunionE2 = $true) & (emptysetAx = $true) & (demorgan1 = $true) & (infuncsetfunc = $true) & (upairset2E = $true) & (funcextLem = $true) & (funcext2 = $true) & (complementT_lem = $true) & (setextsub = $true) & (demorgan2 = $true) & (cartprodmempair1 = $true) & (iftrueProp2 = $true) & (inIntersectImpInUnion = $true) & (binintersectLsub = $true) & (inIntersectImpInUnion2 = $true) & (singletoninpowunion = $true) & (contraSubsetComplement = $true) & (binintersectSubset5 = $true) & (kpairp = $true) & (funcGraphProp2 = $true) & (subPowSU = $true) & (binintersectSubset3 = $true) & (binintersectTELcontra = $true) & (funcGraphProp4 = $true) & (powersetE1 = $true) & (binintersectI = $true) & (cartprodfstin = $true) & (eqbreln1 = $true) & (singletonsswitch = $true) & (ap2p = $true) & (omegaIndAx = $true) & (subsetI2 = $true) & (complementTI1 = $true) & (setukpairinjL = $true) & (setOfPairsIsBReln = $true) & (binunionEcases = $true) & (exuE1 = $true) & (funcinfuncset = $true) & (complementTE1 = $true) & (binunionTIRcontra = $true) & (kpairiskpair = $true) & (ubforcartprodlem2 = $true) & (beta1 = $true) & (dsetconstrI = $true) & (setminusILneg = $true) & (subsetTrans = $true) & (vacuousDall = $true) & (powersetE = $true) & (powersetTE1 = $true) & (woz13rule0 = $true) & (replAx = $true) & (powersetT_lem = $true) & (eqimpsubset2 = $true) & (setminusSubset1 = $true) & (notinemptyset = $true) & (cartprodsndin = $true) & (eqimpsubset1 = $true) & (setunionE = $true) & (inPowerset = $true) & (breln1all1 = $true) & (setminusELneg = $true) & (subsetRefl = $true) & (emptyset__Cong = $true) & (contrasubsetT = $true) & (dsetconstrEL = $true) & (funcext = $true) & (ifSingleton = $true) & (exuE2 = $true) & (quantDeMorgan3 = $true) & (omegaSAx = $true) & (binunionE = $true) & (ex1E1 = $true) & (powersetTI1 = $true) & (bs114d = $true) & (setextT = $true) & (woz13rule3 = $true) & (setadjoinSub2 = $true) & (funcGraphProp3 = $true) & (subbreln = $true) & (breln1SetBreln1 = $true) & (descrp = $true) & (inComplementUnionImpInComplement1 = $true) & (binunionTE = $true) & (ksndsingleton = $true) & (demorgan1b = $true) & (complementImpComplementIntersect = $true) & (emptyI = $true) & (binintersectSubset2 = $true) & (theprop = $true) & (kfstpairEq = $true) & (binunionTILcontra = $true) & (descr__Cong = $true) & (prop2set2propI = $true) & (symdiffIneg2 = $true) & (subsetE = $true) & (setminusT_lem = $true) & (theeq = $true) & (iftrue = $true) & (exuI3 = $true) & (kpairsurjEq = $true) & (upairset2IR = $true) & (iffalseProp2 = $true) & (dsetconstr__Cong = $true) & (ubforcartprodlem3 = $true) & (lam2lamEq = $true) & (woz13rule4 = $true) & (setOfPairsIsBReln1 = $true) & (notdallE = $true) & (ubforcartprodlem1 = $true) & (funcGraphProp1 = $true) & (upairequniteq = $true) & (disjointsetsI1 = $true) & (cartprodmempaircEq = $true) & (emptyinPowerset = $true) & (setunionsingleton = $true) & (binunionT_lem = $true) & (omega__Cong = $true) & (symdiffI2 = $true) & (cartprodpairin = $true) & (upairsetIR = $true) & (setminusER = $true) & (beta2 = $true) & (emptyInPowerset = $true) & (setukpairinjR12 = $true) & (lamp = $true) & (powersetI1 = $true) & (inComplementUnionImpNotIn1 = $true) & (setunionI = $true) & (in__Cong = $true) & (ap2apEq1 = $true) & (symdiffIneg1 = $true) & (emptysetE = $true) & (setadjoinAx = $true) & (nonemptyI = $true) & (choice2fnsingleton = $true) & (uniqinunit = $true) & (setukpairinjL2 = $true) & (ex1E2 = $true) & (demorgan2b = $true) & (binintersectSubset4 = $true) & (prop2setI = $true) & (subsetE2 = $true) & (subset2powerset = $true) & (lam2p = $true) & (setadjoinIL = $true) & (dpsetconstrERa = $true) & (setunionsingleton1 = $true) & (setunionAx = $true) & (funcImageSingleton = $true) & (subbreln1 = $true) & (ex1I2 = $true) & (ksndpairEq = $true) & (dpsetconstrER = $true) & (upairsetE = $true) & (setukpairinjR = $true) & (setukpairinjR2 = $true) & (woz13rule2 = $true) & (secondinupair = $true) & (dpsetconstrI = $true) & (woz13rule1 = $true) & (complementTcontraSubset = $true) & (setukpairIL = $true) & (woz1_4 = $true) & (binunionLsub = $true) & (omega0Ax = $true) & (setukpairinjR11 = $true) & (setukpairinjL1 = $true) & (quantDeMorgan2 = $true) & (inCongP = $true) & (eta2 = $true) & (exuI2 = $true) & (setextAx = $true) & (subsetTI = $true) & (setunionsingleton2 = $true)),
% 10.00/2.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK638,sK639,sK640,sK641])],[f1079,f2413,f2412,f2411])).
% 10.00/2.98 thf(f2411,plain,(
% 10.00/2.98 ? [X0 : $i,X1 : $i] : (($true = (breln1 @ X1 @ X0)) & ? [X2 : $i] : (((in @ X2 @ X1) = $true) & ? [X3 : $i] : (((in @ X3 @ X1) = $true) & ($true != (in @ (kpair @ X3 @ X2) @ (breln1invset @ X1 @ X0))) & ($true = (in @ (kpair @ X2 @ X3) @ X0))))) => (((breln1 @ sK639 @ sK638) = $true) & ? [X2 : $i] : (((in @ X2 @ sK639) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ sK639)) & ($true != (in @ (kpair @ X3 @ X2) @ (breln1invset @ sK639 @ sK638))) & ((in @ (kpair @ X2 @ X3) @ sK638) = $true))))),
% 10.00/2.98 introduced(definition,[],[choice_axiom])).
% 10.00/2.98 thf(f2412,plain,(
% 10.00/2.98 ? [X2 : $i] : (((in @ X2 @ sK639) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ sK639)) & ($true != (in @ (kpair @ X3 @ X2) @ (breln1invset @ sK639 @ sK638))) & ((in @ (kpair @ X2 @ X3) @ sK638) = $true))) => (($true = (in @ sK640 @ sK639)) & ? [X3 : $i] : (($true = (in @ X3 @ sK639)) & ($true != (in @ (kpair @ X3 @ sK640) @ (breln1invset @ sK639 @ sK638))) & ($true = (in @ (kpair @ sK640 @ X3) @ sK638))))),
% 10.00/2.98 introduced(definition,[],[choice_axiom])).
% 10.00/2.98 thf(f2413,plain,(
% 10.00/2.98 ? [X3 : $i] : (($true = (in @ X3 @ sK639)) & ($true != (in @ (kpair @ X3 @ sK640) @ (breln1invset @ sK639 @ sK638))) & ($true = (in @ (kpair @ sK640 @ X3) @ sK638))) => (($true = (in @ sK641 @ sK639)) & ($true != (in @ (kpair @ sK641 @ sK640) @ (breln1invset @ sK639 @ sK638))) & ($true = (in @ (kpair @ sK640 @ sK641) @ sK638)))),
% 10.00/2.98 introduced(definition,[],[choice_axiom])).
% 10.00/2.98 thf(f1079,plain,(
% 10.00/2.98 (cartprodfstpairEq = $true) & (singletonprop = $true) & (prop2setE = $true) & (eqbreln = $true) & (quantDeMorgan4 = $true) & (nonemptyE1 = $true) & (eta1 = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (binintersectSubset1 = $true) & (sepSubset = $true) & (binintersectER = $true) & (iffalseProp1 = $true) & (singletonsuniq = $true) & (cartprodsndpairEq = $true) & (setminusIRneg = $true) & (notsubsetI = $true) & (setukpairIR = $true) & (binintersectT_lem = $true) & (apProp = $true) & (complementSubsetComplementIntersect = $true) & (binintersectRsub = $true) & (eqinunit = $true) & (kfstsingleton = $true) & (iftrueorfalse = $true) & (sepInPowerset = $true) & (upairsubunion = $true) & (notequalI2 = $true) & (setminusEL = $true) & (complementInPowersetComplementIntersect = $true) & (setext = $true) & (powerset__Cong = $true) & (contrasubsetT1 = $true) & (binintersectEL = $true) & (doubleComplementE1 = $true) & (exuE3e = $true) & (doubleComplementI1 = $true) & (setadjoinE = $true) & (singletoninpowerset = $true) & (intersectInPowersetIntersectUnions = $true) & (demorgan2a = $true) & (lamProp = $true) & (exuI1 = $true) & (ifp = $true) & (emptyinunitempty = $true) & (woz1_5 = $true) & (symdiffI1 = $true) & (notdexE = $true) & (nonemptyImpWitness = $true) & (emptysetsubset = $true) & (exu__Cong = $true) & (complementTnotintersectT = $true) & (inIntersectImpInIntersectUnions = $true) & (upairinpowunion = $true) & (setadjoinIR = $true) & (emptyE1 = $true) & (doubleComplementSub1 = $true) & (setbeta = $true) & (cartprodpairsurjEq = $true) & (symdiffE = $true) & (binunionIR = $true) & (woz1_3 = $true) & (contrasubsetT2 = $true) & (iffalse = $true) & (cartprodpairmemEL = $true) & (subsetI1 = $true) & (doubleComplementEq = $true) & (setminusSubset2 = $true) & (setminusI = $true) & (ap2apEq2 = $true) & (notinsingleton = $true) & (demorgan2a2 = $true) & (breln1all2 = $true) & (binunionIL = $true) & (complementUnionInPowersetComplement = $true) & (setminusERneg = $true) & (woz1_1 = $true) & (setadjoinSub = $true) & (wellorderingAx = $true) & (demorgan2a1 = $true) & (singletonsubset = $true) & (setukpairinjR1 = $true) & (contrasubsetT3 = $true) & (doubleComplementSub2 = $true) & (setunion__Cong = $true) & (breln1invprop = $true) & (binunionRsub = $true) & (setadjoinOr = $true) & (ex1I = $true) & (setminusLsub = $true) & (powersetI = $true) & (brelnall2 = $true) & (demorgan1a = $true) & (foundationAx = $true) & (powersetsubset = $true) & (upairsetIL = $true) & (dpsetconstrEL2 = $true) & (exuE3u = $true) & (dpsetconstrEL1 = $true) & (noeltsimpempty = $true) & (setadjoin__Cong = $true) & (brelnall1 = $true) & (app = $true) & (binunionTEcontra = $true) & (exuEu = $true) & (dpsetconstrSub = $true) & (subsetemptysetimpeq = $true) & (woz1_2 = $true) & (dsetconstrER = $true) & (iftrueProp1 = $true) & ? [X0 : $i,X1 : $i] : (($true = (breln1 @ X1 @ X0)) & ? [X2 : $i] : (((in @ X2 @ X1) = $true) & ? [X3 : $i] : (((in @ X3 @ X1) = $true) & ($true != (in @ (kpair @ X3 @ X2) @ (breln1invset @ X1 @ X0))) & ($true = (in @ (kpair @ X2 @ X3) @ X0))))) & (cartprodmempair = $true) & (notequalI1 = $true) & (binintersectTERcontra = $true) & (cartprodpairmemER = $true) & (emptysetimpfalse = $true) & (quantDeMorgan1 = $true) & (nonemptyI1 = $true) & (demorgan2b2 = $true) & (setunionE2 = $true) & (emptysetAx = $true) & (demorgan1 = $true) & (infuncsetfunc = $true) & (upairset2E = $true) & (funcextLem = $true) & (funcext2 = $true) & (complementT_lem = $true) & (setextsub = $true) & (demorgan2 = $true) & (cartprodmempair1 = $true) & (iftrueProp2 = $true) & (inIntersectImpInUnion = $true) & (binintersectLsub = $true) & (inIntersectImpInUnion2 = $true) & (singletoninpowunion = $true) & (contraSubsetComplement = $true) & (binintersectSubset5 = $true) & (kpairp = $true) & (funcGraphProp2 = $true) & (subPowSU = $true) & (binintersectSubset3 = $true) & (binintersectTELcontra = $true) & (funcGraphProp4 = $true) & (powersetE1 = $true) & (binintersectI = $true) & (cartprodfstin = $true) & (eqbreln1 = $true) & (singletonsswitch = $true) & (ap2p = $true) & (omegaIndAx = $true) & (subsetI2 = $true) & (complementTI1 = $true) & (setukpairinjL = $true) & (setOfPairsIsBReln = $true) & (binunionEcases = $true) & (exuE1 = $true) & (funcinfuncset = $true) & (complementTE1 = $true) & (binunionTIRcontra = $true) & (kpairiskpair = $true) & (ubforcartprodlem2 = $true) & (beta1 = $true) & (dsetconstrI = $true) & (setminusILneg = $true) & (subsetTrans = $true) & (vacuousDall = $true) & (powersetE = $true) & (powersetTE1 = $true) & (woz13rule0 = $true) & (replAx = $true) & (powersetT_lem = $true) & (eqimpsubset2 = $true) & (setminusSubset1 = $true) & (notinemptyset = $true) & (cartprodsndin = $true) & (eqimpsubset1 = $true) & (setunionE = $true) & (inPowerset = $true) & (breln1all1 = $true) & (setminusELneg = $true) & (subsetRefl = $true) & (emptyset__Cong = $true) & (contrasubsetT = $true) & (dsetconstrEL = $true) & (funcext = $true) & (ifSingleton = $true) & (exuE2 = $true) & (quantDeMorgan3 = $true) & (omegaSAx = $true) & (binunionE = $true) & (ex1E1 = $true) & (powersetTI1 = $true) & (bs114d = $true) & (setextT = $true) & (woz13rule3 = $true) & (setadjoinSub2 = $true) & (funcGraphProp3 = $true) & (subbreln = $true) & (breln1SetBreln1 = $true) & (descrp = $true) & (inComplementUnionImpInComplement1 = $true) & (binunionTE = $true) & (ksndsingleton = $true) & (demorgan1b = $true) & (complementImpComplementIntersect = $true) & (emptyI = $true) & (binintersectSubset2 = $true) & (theprop = $true) & (kfstpairEq = $true) & (binunionTILcontra = $true) & (descr__Cong = $true) & (prop2set2propI = $true) & (symdiffIneg2 = $true) & (subsetE = $true) & (setminusT_lem = $true) & (theeq = $true) & (iftrue = $true) & (exuI3 = $true) & (kpairsurjEq = $true) & (upairset2IR = $true) & (iffalseProp2 = $true) & (dsetconstr__Cong = $true) & (ubforcartprodlem3 = $true) & (lam2lamEq = $true) & (woz13rule4 = $true) & (setOfPairsIsBReln1 = $true) & (notdallE = $true) & (ubforcartprodlem1 = $true) & (funcGraphProp1 = $true) & (upairequniteq = $true) & (disjointsetsI1 = $true) & (cartprodmempaircEq = $true) & (emptyinPowerset = $true) & (setunionsingleton = $true) & (binunionT_lem = $true) & (omega__Cong = $true) & (symdiffI2 = $true) & (cartprodpairin = $true) & (upairsetIR = $true) & (setminusER = $true) & (beta2 = $true) & (emptyInPowerset = $true) & (setukpairinjR12 = $true) & (lamp = $true) & (powersetI1 = $true) & (inComplementUnionImpNotIn1 = $true) & (setunionI = $true) & (in__Cong = $true) & (ap2apEq1 = $true) & (symdiffIneg1 = $true) & (emptysetE = $true) & (setadjoinAx = $true) & (nonemptyI = $true) & (choice2fnsingleton = $true) & (uniqinunit = $true) & (setukpairinjL2 = $true) & (ex1E2 = $true) & (demorgan2b = $true) & (binintersectSubset4 = $true) & (prop2setI = $true) & (subsetE2 = $true) & (subset2powerset = $true) & (lam2p = $true) & (setadjoinIL = $true) & (dpsetconstrERa = $true) & (setunionsingleton1 = $true) & (setunionAx = $true) & (funcImageSingleton = $true) & (subbreln1 = $true) & (ex1I2 = $true) & (ksndpairEq = $true) & (dpsetconstrER = $true) & (upairsetE = $true) & (setukpairinjR = $true) & (setukpairinjR2 = $true) & (woz13rule2 = $true) & (secondinupair = $true) & (dpsetconstrI = $true) & (woz13rule1 = $true) & (complementTcontraSubset = $true) & (setukpairIL = $true) & (woz1_4 = $true) & (binunionLsub = $true) & (omega0Ax = $true) & (setukpairinjR11 = $true) & (setukpairinjL1 = $true) & (quantDeMorgan2 = $true) & (inCongP = $true) & (eta2 = $true) & (exuI2 = $true) & (setextAx = $true) & (subsetTI = $true) & (setunionsingleton2 = $true)),
% 10.00/2.98 inference(flattening,[status(thm)],[f1078])).
% 10.00/2.98 thf(f1078,plain,(
% 10.00/2.98 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : ((($true != (in @ (kpair @ X3 @ X2) @ (breln1invset @ X1 @ X0))) & ($true = (in @ (kpair @ X2 @ X3) @ X0))) & ((in @ X3 @ X1) = $true)) & ((in @ X2 @ X1) = $true)) & ($true = (breln1 @ X1 @ X0))) & (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)),
% 10.00/2.98 inference(ennf_transformation,[status(thm)],[f464])).
% 10.00/2.98 thf(f464,plain,(
% 10.00/2.98 ~((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) => ! [X0 : $i,X1 : $i] : (($true = (breln1 @ X1 @ X0)) => ! [X2 : $i] : (((in @ X2 @ X1) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => (($true = (in @ (kpair @ X2 @ X3) @ X0)) => ($true = (in @ (kpair @ X3 @ X2) @ (breln1invset @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 10.00/2.98 inference(fool_elimination,[status(thm)],[f463])).
% 10.00/2.98 thf(f463,plain,(
% 10.00/2.98 ~(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 => ! [X0 : $i,X1 : $i] : ((breln1 @ X1 @ X0) => ! [X2 : $i] : ((in @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X1) => ((in @ (kpair @ X2 @ X3) @ X0) => (in @ (kpair @ X3 @ X2) @ (breln1invset @ X1 @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 10.00/2.98 inference(rectify,[status(thm)],[f299])).
% 10.00/2.98 thf(f299,negated_conjecture,(
% 10.00/2.98 ~(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 => ! [X13 : $i,X3 : $i] : ((breln1 @ X3 @ X13) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => ((in @ (kpair @ X1 @ X2) @ X13) => (in @ (kpair @ X2 @ X1) @ (breln1invset @ X3 @ X13))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 10.00/2.98 inference(negated_conjecture,[status(cth)],[f298])).
% 10.00/2.98 thf(f298,conjecture,(
% 10.00/2.98 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 => ! [X13 : $i,X3 : $i] : ((breln1 @ X3 @ X13) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => ((in @ (kpair @ X1 @ X2) @ X13) => (in @ (kpair @ X2 @ X1) @ (breln1invset @ X3 @ X13)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 10.00/2.98 file('/export/starexec/sandbox/tmp/tmp.ExEgOH0aih/DTF2THF_12985.p',breln1invI)).
% 10.00/2.98 thf(f10356,plain,(
% 10.00/2.98 ($true != (in @ sK640 @ sK639))),
% 10.00/2.98 inference(subsumption_resolution,[status(thm)],[f10355,f4130])).
% 10.00/2.98 thf(f4130,plain,(
% 10.00/2.98 ($true = (in @ (kpair @ sK640 @ sK641) @ sK638))),
% 10.00/2.98 inference(cnf_transformation,[status(thm)],[f2414])).
% 10.00/2.98 thf(f10355,plain,(
% 10.00/2.98 ($true != (in @ (kpair @ sK640 @ sK641) @ sK638)) | ($true != (in @ sK640 @ sK639))),
% 10.00/2.98 inference(subsumption_resolution,[status(thm)],[f10350,f4132])).
% 10.00/2.98 thf(f4132,plain,(
% 10.00/2.98 ($true = (in @ sK641 @ sK639))),
% 10.00/2.98 inference(cnf_transformation,[status(thm)],[f2414])).
% 10.00/2.98 thf(f10350,plain,(
% 10.00/2.98 ($true != (in @ sK641 @ sK639)) | ($true != (in @ sK640 @ sK639)) | ($true != (in @ (kpair @ sK640 @ sK641) @ sK638))),
% 10.00/2.98 inference(beta_eta_normalization,[status(thm)],[f10349])).
% 10.00/2.98 thf(f10349,plain,(
% 10.00/2.98 ($true != (in @ sK640 @ sK639)) | ($true != (in @ sK641 @ sK639)) | ($true != ((^[Y0 : $i]: ((^[Y1 : $i]: (in @ (kpair @ Y1 @ Y0) @ sK638)))) @ sK641 @ sK640))),
% 10.00/2.98 inference(trivial_inequality_removal,[status(thm)],[f10340])).
% 10.00/2.98 thf(f10340,plain,(
% 10.00/2.98 ($true != ((^[Y0 : $i]: ((^[Y1 : $i]: (in @ (kpair @ Y1 @ Y0) @ sK638)))) @ sK641 @ sK640)) | ($true != (in @ sK640 @ sK639)) | ($true != (in @ sK641 @ sK639)) | ($true != $true)),
% 10.00/2.98 inference(superposition,[status(thm)],[f6237,f6008])).
% 10.00/2.98 thf(f6008,plain,(
% 10.00/2.98 ( ! [X2 : $i,X3 : $i,X0 : $i > $i > $o,X1 : $i,X4 : $i] : (($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ X0))) | ($true != (X0 @ X2 @ X4)) | ((in @ X2 @ X3) != $true) | ($true != (in @ X4 @ X1))) )),
% 10.00/2.98 inference(beta_eta_normalization,[status(thm)],[f6007])).
% 10.00/2.98 thf(f6007,plain,(
% 10.00/2.98 ( ! [X2 : $i,X3 : $i,X0 : $i > $i > $o,X1 : $i,X4 : $i] : (($true != (X0 @ X2 @ X4)) | ((in @ X2 @ X3) != $true) | ($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) | ($true != (in @ X4 @ X1))) )),
% 10.00/2.98 inference(trivial_inequality_removal,[status(thm)],[f5854])).
% 10.00/2.98 thf(f5854,plain,(
% 10.00/2.98 ( ! [X2 : $i,X3 : $i,X0 : $i > $i > $o,X1 : $i,X4 : $i] : (($true != (X0 @ X2 @ X4)) | ((in @ X2 @ X3) != $true) | ($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) | ($true != $true) | ($true != (in @ X4 @ X1))) )),
% 10.00/2.98 inference(definition_unfolding,[status(thm)],[f4805,f3963])).
% 10.00/2.98 thf(f3963,plain,(
% 10.00/2.98 (dpsetconstrI = $true)),
% 10.00/2.98 inference(cnf_transformation,[status(thm)],[f2414])).
% 10.00/2.98 thf(f4805,plain,(
% 10.00/2.98 ( ! [X2 : $i,X3 : $i,X0 : $i > $i > $o,X1 : $i,X4 : $i] : (($true != (in @ X4 @ X1)) | ($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) | ($true != (X0 @ X2 @ X4)) | ((in @ X2 @ X3) != $true) | (dpsetconstrI != $true)) )),
% 10.00/2.98 inference(cnf_transformation,[status(thm)],[f3062])).
% 10.00/2.98 thf(f3062,plain,(
% 10.00/2.98 (! [X0 : $i > $i > $o,X1 : $i,X2 : $i,X3 : $i] : (! [X4 : $i] : (($true != (in @ X4 @ X1)) | ($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) | ($true != (X0 @ X2 @ X4))) | ((in @ X2 @ X3) != $true)) | (dpsetconstrI != $true)) & ((dpsetconstrI = $true) | ((($true = (in @ sK1010 @ sK1007)) & ($true != (in @ (kpair @ sK1008 @ sK1010) @ (dpsetconstr @ sK1009 @ sK1007 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK1006 @ Y0 @ Y1))))))) & ($true = (sK1006 @ sK1008 @ sK1010))) & ((in @ sK1008 @ sK1009) = $true)))),
% 10.00/2.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1006,sK1007,sK1008,sK1009,sK1010])],[f3059,f3061,f3060])).
% 10.00/2.98 thf(f3060,plain,(
% 10.00/2.98 ? [X5 : $i > $i > $o,X6 : $i,X7 : $i,X8 : $i] : (? [X9 : $i] : (($true = (in @ X9 @ X6)) & ($true != (in @ (kpair @ X7 @ X9) @ (dpsetconstr @ X8 @ X6 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X5 @ Y0 @ Y1))))))) & ($true = (X5 @ X7 @ X9))) & ((in @ X7 @ X8) = $true)) => (? [X9 : $i] : (($true = (in @ X9 @ sK1007)) & ($true != (in @ (kpair @ sK1008 @ X9) @ (dpsetconstr @ sK1009 @ sK1007 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK1006 @ Y0 @ Y1))))))) & ($true = (sK1006 @ sK1008 @ X9))) & ((in @ sK1008 @ sK1009) = $true))),
% 10.00/2.98 introduced(definition,[],[choice_axiom])).
% 10.00/2.98 thf(f3061,plain,(
% 10.00/2.98 ? [X9 : $i] : (($true = (in @ X9 @ sK1007)) & ($true != (in @ (kpair @ sK1008 @ X9) @ (dpsetconstr @ sK1009 @ sK1007 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK1006 @ Y0 @ Y1))))))) & ($true = (sK1006 @ sK1008 @ X9))) => (($true = (in @ sK1010 @ sK1007)) & ($true != (in @ (kpair @ sK1008 @ sK1010) @ (dpsetconstr @ sK1009 @ sK1007 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK1006 @ Y0 @ Y1))))))) & ($true = (sK1006 @ sK1008 @ sK1010)))),
% 10.00/2.98 introduced(definition,[],[choice_axiom])).
% 10.00/2.98 thf(f3059,plain,(
% 10.00/2.98 (! [X0 : $i > $i > $o,X1 : $i,X2 : $i,X3 : $i] : (! [X4 : $i] : (($true != (in @ X4 @ X1)) | ($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) | ($true != (X0 @ X2 @ X4))) | ((in @ X2 @ X3) != $true)) | (dpsetconstrI != $true)) & ((dpsetconstrI = $true) | ? [X5 : $i > $i > $o,X6 : $i,X7 : $i,X8 : $i] : (? [X9 : $i] : (($true = (in @ X9 @ X6)) & ($true != (in @ (kpair @ X7 @ X9) @ (dpsetconstr @ X8 @ X6 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X5 @ Y0 @ Y1))))))) & ($true = (X5 @ X7 @ X9))) & ((in @ X7 @ X8) = $true)))),
% 10.00/2.98 inference(rectify,[status(thm)],[f3058])).
% 10.00/2.98 thf(f3058,plain,(
% 10.00/2.98 (! [X0 : $i > $i > $o,X1 : $i,X2 : $i,X3 : $i] : (! [X4 : $i] : (($true != (in @ X4 @ X1)) | ($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) | ($true != (X0 @ X2 @ X4))) | ((in @ X2 @ X3) != $true)) | (dpsetconstrI != $true)) & ((dpsetconstrI = $true) | ? [X0 : $i > $i > $o,X1 : $i,X2 : $i,X3 : $i] : (? [X4 : $i] : (($true = (in @ X4 @ X1)) & ($true != (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) & ($true = (X0 @ X2 @ X4))) & ((in @ X2 @ X3) = $true)))),
% 10.00/2.98 inference(nnf_transformation,[status(thm)],[f1114])).
% 10.00/2.98 thf(f1114,plain,(
% 10.00/2.98 ! [X0 : $i > $i > $o,X1 : $i,X2 : $i,X3 : $i] : (! [X4 : $i] : (($true != (in @ X4 @ X1)) | ($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) | ($true != (X0 @ X2 @ X4))) | ((in @ X2 @ X3) != $true)) <=> (dpsetconstrI = $true)),
% 10.00/2.98 inference(flattening,[status(thm)],[f1113])).
% 10.00/2.98 thf(f1113,plain,(
% 10.00/2.98 ! [X0 : $i > $i > $o,X1 : $i,X2 : $i,X3 : $i] : (! [X4 : $i] : ((($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))))) | ($true != (X0 @ X2 @ X4))) | ($true != (in @ X4 @ X1))) | ((in @ X2 @ X3) != $true)) <=> (dpsetconstrI = $true)),
% 10.00/2.98 inference(ennf_transformation,[status(thm)],[f631])).
% 10.00/2.98 thf(f631,plain,(
% 10.00/2.98 ! [X0 : $i > $i > $o,X1 : $i,X2 : $i,X3 : $i] : (((in @ X2 @ X3) = $true) => ! [X4 : $i] : (($true = (in @ X4 @ X1)) => (($true = (X0 @ X2 @ X4)) => ($true = (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1)))))))))) <=> (dpsetconstrI = $true)),
% 10.00/2.98 inference(fool_elimination,[status(thm)],[f630])).
% 10.00/2.98 thf(f630,plain,(
% 10.00/2.98 (! [X0 : $i > $i > $o,X1 : $i,X2 : $i,X3 : $i] : ((in @ X2 @ X3) => ! [X4 : $i] : ((in @ X4 @ X1) => ((X0 @ X2 @ X4) => (in @ (kpair @ X2 @ X4) @ (dpsetconstr @ X3 @ X1 @ (^[X5 : $i, X6 : $i] : (X0 @ X5 @ X6))))))) = dpsetconstrI)),
% 10.00/2.98 inference(rectify,[status(thm)],[f185])).
% 10.00/2.98 thf(f185,axiom,(
% 10.00/2.98 (! [X0 : $i > $i > $o,X4 : $i,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => ((X0 @ X1 @ X2) => (in @ (kpair @ X1 @ X2) @ (dpsetconstr @ X3 @ X4 @ (^[X8 : $i, X10 : $i] : (X0 @ X8 @ X10))))))) = dpsetconstrI)),
% 10.00/2.98 file('/export/starexec/sandbox/tmp/tmp.ExEgOH0aih/DTF2THF_12985.p',dpsetconstrI)).
% 10.00/2.98 thf(f6237,plain,(
% 10.00/2.98 ($true != (in @ (kpair @ sK641 @ sK640) @ (dpsetconstr @ sK639 @ sK639 @ (^[Y0 : $i]: ((^[Y1 : $i]: (in @ (kpair @ Y1 @ Y0) @ sK638)))))))),
% 10.00/2.98 inference(beta_eta_normalization,[status(thm)],[f5426])).
% 10.00/2.98 thf(f5426,plain,(
% 10.00/2.98 ((in @ (kpair @ sK641 @ sK640) @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dpsetconstr @ Y0 @ Y0 @ (^[Y2 : $i]: ((^[Y3 : $i]: (in @ (kpair @ Y3 @ Y2) @ Y1)))))))) @ sK639 @ sK638)) != $true)),
% 10.00/2.98 inference(definition_unfolding,[status(thm)],[f4131,f3198])).
% 10.00/2.98 thf(f3198,plain,(
% 10.00/2.98 (breln1invset = (^[Y0 : $i]: ((^[Y1 : $i]: (dpsetconstr @ Y0 @ Y0 @ (^[Y2 : $i]: ((^[Y3 : $i]: (in @ (kpair @ Y3 @ Y2) @ Y1)))))))))),
% 10.00/2.98 inference(cnf_transformation,[status(thm)],[f823])).
% 10.00/2.98 thf(f823,plain,(
% 10.00/2.98 (breln1invset = (^[Y0 : $i]: ((^[Y1 : $i]: (dpsetconstr @ Y0 @ Y0 @ (^[Y2 : $i]: ((^[Y3 : $i]: (in @ (kpair @ Y3 @ Y2) @ Y1)))))))))),
% 10.00/2.98 inference(fool_elimination,[status(thm)],[f822])).
% 10.00/2.98 thf(f822,plain,(
% 10.00/2.98 ((^[X0 : $i, X1 : $i] : (dpsetconstr @ X0 @ X0 @ (^[X2 : $i, X3 : $i] : (in @ (kpair @ X3 @ X2) @ X1)))) = breln1invset)),
% 10.00/2.98 inference(rectify,[status(thm)],[f296])).
% 10.00/2.98 thf(f296,axiom,(
% 10.00/2.98 ((^[X3 : $i, X13 : $i] : (dpsetconstr @ X3 @ X3 @ (^[X1 : $i, X2 : $i] : (in @ (kpair @ X2 @ X1) @ X13)))) = breln1invset)),
% 10.00/2.98 file('/export/starexec/sandbox/tmp/tmp.ExEgOH0aih/DTF2THF_12985.p',breln1invset)).
% 10.00/2.98 thf(f4131,plain,(
% 10.00/2.98 ($true != (in @ (kpair @ sK641 @ sK640) @ (breln1invset @ sK639 @ sK638)))),
% 10.00/2.98 inference(cnf_transformation,[status(thm)],[f2414])).
% 10.00/2.98 % SZS output end Proof for DTF2THF_12985
% 10.00/2.98 % (13521)------------------------------
% 10.00/2.98 % (13521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.00/2.98 % (13521)Termination reason: Refutation
% 10.00/2.98
% 10.00/2.98 % (13521)Memory used [KB]: 13432
% 10.00/2.98 % (13521)Time elapsed: 0.528 s
% 10.00/2.98 % (13521)Instructions burned: 1613 (million)
% 10.00/2.98 % (13521)------------------------------
% 10.00/2.98 % (13521)------------------------------
% 10.00/2.98 % (13247)Success in time 1.405 s
%------------------------------------------------------------------------------