%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU801^1 : 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 : n026.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 09:37:19 AM UTC 2026
% Result : Theorem 7.07s 2.24s
% Output : Refutation 7.07s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 5
% Syntax : Number of formulae : 32 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 4996 (1332 equ; 0 cnn)
% Maximal formula atoms : 319 ( 156 avg)
% Number of connectives : 2534 ( 28 ~; 17 |; 964 &; 246 @)
% ( 2 <=>;1277 =>; 0 <=; 0 <~>)
% Maximal formula depth : 326 ( 73 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 651 ( 648 usr; 324 con; 0-6 aty)
% Number of variables : 93 ( 30 ^; 45 !; 18 ?; 93 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_360,type,
image1: $i > ( $i > $i ) > $i ).
thf(func_def_364,type,
injective: $i > $i > $i > $o ).
thf(func_def_365,type,
injFuncSet: $i > $i > $i ).
thf(func_def_369,type,
surjective: $i > $i > $i > $o ).
thf(func_def_370,type,
surjFuncSet: $i > $i > $i ).
thf(func_def_387,type,
sP0: $i > $i > $i > $i > $o ).
thf(func_def_388,type,
sP1: $i > $i > $i > $o ).
thf(func_def_389,type,
sP2: $i > $i > $i > $o ).
thf(func_def_390,type,
sP3: ( $i > $o ) > $o ).
thf(func_def_391,type,
sP4: ( $i > $o ) > $o ).
thf(func_def_392,type,
sP5: $i > ( $i > $o ) > $o ).
thf(func_def_393,type,
sP6: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_395,type,
sP8: ( $i > $o ) > $i > $o ).
thf(func_def_396,type,
sP9: $i > ( $i > $o ) > $i > $o ).
thf(func_def_397,type,
sP10: $i > $i > $o ).
thf(func_def_399,type,
sP12: ( $i > $o ) > $i > $o ).
thf(func_def_401,type,
sP14: $i > ( $i > $o ) > $o ).
thf(func_def_403,type,
sP16: ( $i > $o ) > $o ).
thf(func_def_404,type,
sP17: $i > ( $i > $o ) > $o ).
thf(func_def_407,type,
sP20: $i > $i > ( $i > $i > $o ) > $o ).
thf(func_def_408,type,
sP21: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_409,type,
sP22: $i > ( $i > $i ) > $i > $o ).
thf(func_def_411,type,
sP24: $i > $o > $i > $o ).
thf(func_def_413,type,
sP26: $i > $i > $i > $i > $o ).
thf(func_def_414,type,
sP27: $i > ( $i > $o ) > $o ).
thf(func_def_415,type,
sP28: $i > ( $i > $o ) > $i > $o ).
thf(func_def_416,type,
sP29: ( $i > $o ) > $o ).
thf(func_def_417,type,
sP30: $i > $o ).
thf(func_def_418,type,
sP31: $i > $o ).
thf(func_def_419,type,
sP32: $o > $i > $i > $i > $o ).
thf(func_def_420,type,
sP33: $i > $i > $i > $i > $o ).
thf(func_def_421,type,
sP34: $i > $i > $i > $o ).
thf(func_def_422,type,
sP35: $i > $i > $i > $o ).
thf(func_def_423,type,
sP36: $i > $i > $o > $o ).
thf(func_def_424,type,
sP37: $i > $i > $i > $o ).
thf(func_def_425,type,
sP38: $i > $i > ( $i > $i ) > $o ).
thf(func_def_427,type,
sP40: $i > $i > $i > $i > $o ).
thf(func_def_428,type,
sP41: $i > $i > $i > $i > $o ).
thf(func_def_429,type,
sP42: $i > $i > $i > $o ).
thf(func_def_430,type,
sP43: $i > $i > $i > $o ).
thf(func_def_435,type,
sP48: $o > $i > $i > $o ).
thf(func_def_436,type,
sP49: $i > $i > $i > $o ).
thf(func_def_438,type,
sP51: $i > $o > $i > $i > $o ).
thf(func_def_439,type,
sP52: $o > $i > $i > $i > $o ).
thf(func_def_440,type,
sP53: $i > $i > $i > $i > $o ).
thf(func_def_441,type,
sP54: $i > $i > $i > $o ).
thf(func_def_442,type,
sP55: $i > $i > $i > $o ).
thf(func_def_443,type,
sP56: $i > $i > $i > $o ).
thf(func_def_444,type,
sP57: ( $i > $o ) > $i > $o ).
thf(func_def_445,type,
sP58: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_446,type,
sP59: $i > $i > $i > $o ).
thf(func_def_447,type,
sP60: $i > $i > $o ).
thf(func_def_448,type,
sP61: $i > $i > $o ).
thf(func_def_449,type,
sP62: $i > $i > $o ).
thf(func_def_450,type,
sP63: $i > $i > $o ).
thf(func_def_451,type,
sP64: $i > $o ).
thf(func_def_452,type,
sP65: $i > $i > $i > $o ).
thf(func_def_453,type,
sP66: $i > $i > $i > $i > $i > $o ).
thf(func_def_455,type,
sP68: ( $i > $o ) > $i > $o ).
thf(func_def_457,type,
sP70: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_458,type,
sP71: $i > ( $i > $i > $o ) > $o ).
thf(func_def_460,type,
sP73: $i > $i > $o ).
thf(func_def_462,type,
sP75: ( $i > $o ) > $i > $o ).
thf(func_def_464,type,
sP77: $o > $i > $i > $o ).
thf(func_def_466,type,
sP79: $i > $i > ( $i > $i ) > $o ).
thf(func_def_468,type,
sP81: $i > $o ).
thf(func_def_469,type,
sP82: $i > ( $i > $o ) > $o ).
thf(func_def_470,type,
sP83: ( $i > $o ) > $i > $o ).
thf(func_def_473,type,
sP86: $i > $i > $i > $o ).
thf(func_def_475,type,
sP88: $i > ( $i > $o ) > $o ).
thf(func_def_476,type,
sP89: ( $i > $o ) > $i > $o ).
thf(func_def_478,type,
sP91: $i > ( $i > $o ) > $o ).
thf(func_def_480,type,
sP93: ( $i > $o ) > ( $i > $o ) > $i > $i > $o ).
thf(func_def_481,type,
sP94: ( $i > $i ) > $i > $i > $o ).
thf(func_def_483,type,
sP96: $i > $i > $o > $i > $i > $i > $o ).
thf(func_def_485,type,
sP98: $i > $i > $i > $o ).
thf(func_def_486,type,
sP99: $i > $i > ( $i > $o ) > $o ).
thf(func_def_487,type,
sP100: $i > $i > $o ).
thf(func_def_489,type,
sP102: ( $i > $o ) > $o ).
thf(func_def_491,type,
sK104: $i > $o ).
thf(func_def_493,type,
sK106: $i > $i > $i > $i ).
thf(func_def_494,type,
sK107: $i > $i > $i > $i > $i ).
thf(func_def_495,type,
sK108: $i > $i > $i > $i > $i ).
thf(func_def_496,type,
sK109: $i > $i > $i > $i > $i ).
thf(func_def_503,type,
sK116: $i > $i > $i > $i ).
thf(func_def_504,type,
sK117: $i > $i > $i > $i > $i ).
thf(func_def_513,type,
sK126: $i > $o ).
thf(func_def_521,type,
sK134: $i > ( $i > $o ) > $i ).
thf(func_def_522,type,
sK135: ( $i > $o ) > $i ).
thf(func_def_523,type,
sK136: $i > $o ).
thf(func_def_524,type,
sK137: $i > $o ).
thf(func_def_530,type,
sK143: ( $i > $o ) > $i ).
thf(func_def_531,type,
sK144: ( $i > $o ) > $i ).
thf(func_def_532,type,
sK145: $i > $o ).
thf(func_def_538,type,
sK151: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_539,type,
sK152: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_540,type,
sK153: $i > ( $i > $o ) > $i ).
thf(func_def_544,type,
sK157: $i > $o ).
thf(func_def_550,type,
sK163: $i > $i > $i ).
thf(func_def_551,type,
sK164: $i > ( $i > $o ) > $i > $i ).
thf(func_def_552,type,
sK165: $i > ( $i > $o ) > $i > $i ).
thf(func_def_553,type,
sK166: ( $i > $o ) > $i > $i ).
thf(func_def_556,type,
sK169: $i > $o ).
thf(func_def_568,type,
sK181: $i > $i > $i ).
thf(func_def_576,type,
sK189: $i > $i > $o ).
thf(func_def_581,type,
sK194: $i > $o ).
thf(func_def_582,type,
sK195: ( $i > $o ) > $i > $i ).
thf(func_def_583,type,
sK196: ( $i > $o ) > $i > $i ).
thf(func_def_592,type,
sK205: $i > $o ).
thf(func_def_594,type,
sK207: $i > ( $i > $o ) > $i ).
thf(func_def_595,type,
sK208: $i > ( $i > $o ) > $i ).
thf(func_def_596,type,
sK209: ( $i > $o ) > $i ).
thf(func_def_597,type,
sK210: $i > ( $i > $o ) > $i ).
thf(func_def_598,type,
sK211: $i > $o ).
thf(func_def_603,type,
sK216: $i > $o ).
thf(func_def_604,type,
sK217: ( $i > $o ) > $i > $i ).
thf(func_def_605,type,
sK218: $i > ( $i > $o ) > $i ).
thf(func_def_612,type,
sK225: $i > $o ).
thf(func_def_634,type,
sK247: $i > $i ).
thf(func_def_635,type,
sK248: ( $i > $i ) > $i > $i > $i ).
thf(func_def_645,type,
sK258: $i > $o ).
thf(func_def_651,type,
sK264: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_652,type,
sK265: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_653,type,
sK266: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_654,type,
sK267: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_656,type,
sK269: $i > $i > $o ).
thf(func_def_659,type,
sK272: $i > $o ).
thf(func_def_678,type,
sK291: $i > $i ).
thf(func_def_679,type,
sK292: $i > $i ).
thf(func_def_680,type,
sK293: ( $i > $i ) > $i > $i ).
thf(func_def_681,type,
sK294: $i > ( $i > $i ) > $i > $i ).
thf(func_def_685,type,
sK298: $i > $i > $o ).
thf(func_def_698,type,
sK311: $i > $i > $o ).
thf(func_def_707,type,
sK320: $i > $o > $i > $i ).
thf(func_def_719,type,
sK332: $i > $o ).
thf(func_def_720,type,
sK333: $i > $o ).
thf(func_def_721,type,
sK334: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_722,type,
sK335: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_725,type,
sK338: $i > $i > $o ).
thf(func_def_744,type,
sK357: $i > $o ).
thf(func_def_748,type,
sK361: $i > $i > $i > $i > $i ).
thf(func_def_749,type,
sK362: $i > $i > $i > $i > $i ).
thf(func_def_754,type,
sK367: $i > ( $i > $o ) > $i > $i ).
thf(func_def_755,type,
sK368: $i > ( $i > $o ) > $i > $i ).
thf(func_def_756,type,
sK369: $i > ( $i > $o ) > $i ).
thf(func_def_759,type,
sK372: $i > $o ).
thf(func_def_770,type,
sK383: ( $i > $o ) > $i ).
thf(func_def_771,type,
sK384: $i > ( $i > $o ) > $i ).
thf(func_def_772,type,
sK385: $i > $o ).
thf(func_def_797,type,
sK410: $i > $i ).
thf(func_def_798,type,
sK411: $i > $i ).
thf(func_def_809,type,
sK422: $i > $i > $i > $i > $o ).
thf(func_def_822,type,
sK435: $i > $i > $i > $i ).
thf(func_def_823,type,
sK436: $i > $i > $i > $i ).
thf(func_def_824,type,
sK437: $i > $i > $i > $i ).
thf(func_def_825,type,
sK438: $i > $i > $i > $i ).
thf(func_def_829,type,
sK442: $i > $i > $i > $o ).
thf(func_def_830,type,
sK443: $i > $i > $i > $i ).
thf(func_def_847,type,
sK460: $i > $i ).
thf(func_def_848,type,
sK461: ( $i > $i ) > $i > $i > $i ).
thf(func_def_849,type,
sK462: $i > $i > ( $i > $i ) > $i ).
thf(func_def_850,type,
sK463: $i > $i > $i > $i > $i ).
thf(func_def_851,type,
sK464: $i > $i > $i > $i > $i ).
thf(func_def_852,type,
sK465: $i > $i > $i > $i > $i ).
thf(func_def_853,type,
sK466: $i > $i > $i > $i > $i ).
thf(func_def_886,type,
sK499: $i > $i > $i > $i ).
thf(func_def_887,type,
sK500: $i > $i > $i > $i > $i ).
thf(func_def_899,type,
sK512: $i > $i > $i > $i ).
thf(func_def_919,type,
sK532: $i > $i ).
thf(func_def_955,type,
sK568: $i > $i > $i ).
thf(func_def_956,type,
sK569: $i > $i ).
thf(func_def_959,type,
sK572: $i > $i > ( $i > $i ) > $i ).
thf(func_def_982,type,
sK595: $i > $i > $o ).
thf(func_def_986,type,
sK599: $i > $o ).
thf(func_def_996,type,
sK609: $i > $o ).
thf(func_def_1027,type,
sK640: $i > $o ).
thf(func_def_1028,type,
sK641: $i > $o ).
thf(func_def_1029,type,
sK642: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1030,type,
sK643: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1032,type,
sK645: $i > $i ).
thf(func_def_1035,type,
sK648: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1040,type,
sK653: $i > $i > $i > $o ).
thf(func_def_1047,type,
sK660: $i > $i > $i ).
thf(func_def_1077,type,
sK690: $i > $i ).
thf(func_def_1079,type,
sK692: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1098,type,
sK711: $i > $i > $i > $i > $i ).
thf(func_def_1103,type,
sK716: $i > $i > $i > $i ).
thf(func_def_1104,type,
sK717: $i > $i > $i > $i ).
thf(func_def_1131,type,
sK744: $i > $i > $i > $i ).
thf(func_def_1132,type,
sK745: $i > $i > $i > $i ).
thf(func_def_1136,type,
sK749: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1137,type,
sK750: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_1138,type,
sK751: ( $i > $o ) > $i > $i ).
thf(func_def_1142,type,
sK755: $i > $o ).
thf(func_def_1145,type,
sK758: $i > $i ).
thf(func_def_1146,type,
sK759: $i > $i ).
thf(func_def_1147,type,
sK760: $i > $i ).
thf(func_def_1148,type,
sK761: $i > $i ).
thf(func_def_1149,type,
sK762: $i > $i > $i ).
thf(func_def_1150,type,
sK763: $i > $i > $i ).
thf(func_def_1151,type,
sK764: $i > $i > $i ).
thf(func_def_1152,type,
sK765: $i > $i > $i ).
thf(func_def_1153,type,
sK766: $i > $i > $i ).
thf(func_def_1154,type,
sK767: $i > $i > $i ).
thf(func_def_1155,type,
sK768: $i > $i > $i ).
thf(func_def_1156,type,
sK769: $i > $i > $i > $i ).
thf(func_def_1157,type,
sK770: $i > $i > $i ).
thf(func_def_1158,type,
sK771: $i > $i > $i ).
thf(func_def_1159,type,
sK772: $i > $i > $i ).
thf(func_def_1160,type,
sK773: $i > $i > $i > $i ).
thf(func_def_1161,type,
sK774: $i > $i > $i > $i ).
thf(func_def_1162,type,
sK775: $i > $i ).
thf(func_def_1166,type,
sK779: $i > $i > $i > $i ).
thf(func_def_1167,type,
sK780: $i > $i > $i > $i ).
thf(func_def_1171,type,
sK784: $i > $i > $i > $i > $i > $i ).
thf(func_def_1188,type,
sK801: $i > $i > $i ).
thf(func_def_1192,type,
sK805: $i > $i ).
thf(func_def_1218,type,
sK831: $i > $o ).
thf(func_def_1219,type,
sK832: ( $i > $o ) > $i > $i ).
thf(func_def_1220,type,
sK833: ( $i > $o ) > $i > $i ).
thf(func_def_1252,type,
sK865: $i > $i > $o ).
thf(func_def_1257,type,
sK870: $i > $i ).
thf(func_def_1258,type,
sK871: $i > $i > $o ).
thf(func_def_1260,type,
sK873: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1261,type,
sK874: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1262,type,
sK875: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1263,type,
sK876: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1266,type,
sK879: $i > $i > $i ).
thf(func_def_1267,type,
sK880: $i > $i > $i ).
thf(func_def_1271,type,
sK884: $i > $o ).
thf(func_def_1274,type,
sK887: $i > $o ).
thf(func_def_1276,type,
sK889: $i > ( $i > $o ) > $i ).
thf(func_def_1277,type,
sK890: ( $i > $o ) > $i > $i ).
thf(func_def_1293,type,
sK906: $i > $i ).
thf(func_def_1295,type,
sK908: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1296,type,
sK909: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1297,type,
sK910: $i > $i > $i ).
thf(func_def_1298,type,
sK911: $i > $i ).
thf(func_def_1301,type,
sK914: ( $i > $o ) > $i > $i ).
thf(func_def_1302,type,
sK915: ( $i > $o ) > $i > $i ).
thf(func_def_1303,type,
sK916: $i > ( $i > $o ) > $i ).
thf(func_def_1304,type,
sK917: $i > $o ).
thf(func_def_1307,type,
sK920: $i > $o ).
thf(func_def_1308,type,
sK921: ( $i > $o ) > $i > $i ).
thf(func_def_1315,type,
sK928: $i > $i > $i ).
thf(func_def_1321,type,
sK934: $i > $i > $o ).
thf(func_def_1337,type,
sK950: $i > $i > $o ).
thf(func_def_1344,type,
sK957: ( $i > $o ) > $i ).
thf(func_def_1345,type,
sK958: $i > $o ).
thf(func_def_1346,type,
sK959: ( $i > $o ) > $i > $i ).
thf(func_def_1347,type,
sK960: ( $i > $o ) > $i > $i ).
thf(func_def_1348,type,
sK961: $i > ( $i > $o ) > $i ).
thf(func_def_1350,type,
sK963: $i > $o ).
thf(func_def_1353,type,
sK966: $i > $i ).
thf(func_def_1377,type,
sK990: $i > $o ).
thf(func_def_1383,type,
sK996: $i > $i > $i ).
thf(func_def_1384,type,
sK997: $i > $o ).
thf(func_def_1386,type,
sK999: $i > ( $i > $o ) > $i ).
thf(func_def_1387,type,
sK1000: $i > ( $i > $o ) > $i ).
thf(func_def_1388,type,
sK1001: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1389,type,
sK1002: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1392,type,
sK1005: $i > $o ).
thf(func_def_1393,type,
sK1006: $i > $o ).
thf(func_def_1405,type,
sK1018: $i > $i ).
thf(func_def_1419,type,
sK1032: $i > $i ).
thf(func_def_1420,type,
sK1033: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1444,type,
sK1057: $i > $i > $o > $i > $i > $i > $i ).
thf(func_def_1462,type,
sK1075: $i > $i ).
thf(func_def_1465,type,
sK1078: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1466,type,
sK1079: $i > $i > $i > $i ).
thf(func_def_1473,type,
sK1086: $i > $o ).
thf(func_def_1485,type,
sK1098: $i > $i > ( $i > $o ) > $i ).
thf(func_def_1486,type,
sK1099: $i > $o ).
thf(func_def_1507,type,
sK1120: $i > $i > $i ).
thf(func_def_1509,type,
sK1122: $i > $i ).
thf(func_def_1516,type,
sK1129: $i > $o ).
thf(func_def_1526,type,
sK1139: $i > ( $i > $o ) > $i ).
thf(func_def_1527,type,
sK1140: ( $i > $o ) > $i ).
thf(func_def_1528,type,
sK1141: $i > $o ).
thf(func_def_1541,type,
ph1154:
!>[X0: $tType] : X0 ).
thf(f9456,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f9445,f4345]) ).
thf(f4345,plain,
( ( in @ sK619 @ ( funcSet @ sK618 @ sK620 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f2458]) ).
thf(f2458,plain,
( ( iffalseProp1 = $true )
& ( setunionI = $true )
& ( cartprodmempair = $true )
& ( setukpairinjR1 = $true )
& ( ubforcartprodlem3 = $true )
& ( upairsubunion = $true )
& ( dpsetconstrI = $true )
& ( demorgan1 = $true )
& ( eqimpsubset2 = $true )
& ( binunionLsub = $true )
& ( eta2 = $true )
& ( upairset2E = $true )
& ( cartprodpairsurjEq = $true )
& ( wellorderingAx = $true )
& ( setOfPairsIsBReln = $true )
& ( emptyset__Cong = $true )
& ( omega__Cong = $true )
& ( kpairsurjEq = $true )
& ( ifSingleton = $true )
& ( ex1I = $true )
& ( funcextLem = $true )
& ( powersetTE1 = $true )
& ( woz2W = $true )
& ( setadjoinIL = $true )
& ( breln1invI = $true )
& ( setunionsingleton2 = $true )
& ( setext = $true )
& ( setunionsingleton1 = $true )
& ( binunionEcases = $true )
& ( powersetE = $true )
& ( bs114d = $true )
& ( exuEu = $true )
& ( setminusELneg = $true )
& ( setukpairinjL2 = $true )
& ( contrasubsetT2 = $true )
& ( replAx = $true )
& ( upairset2IR = $true )
& ( injFuncInInjFuncSet = $true )
& ( binintersectSubset5 = $true )
& ( lam2lamEq = $true )
& ( setunion__Cong = $true )
& ( notequalI2 = $true )
& ( subbreln1 = $true )
& ( setadjoinAx = $true )
& ( demorgan2b = $true )
& ( emptyE1 = $true )
& ( ifp = $true )
& ( notdallE = $true )
& ( woz2A = $true )
& ( subsetemptysetimpeq = $true )
& ( notinemptyset = $true )
& ( eta1 = $true )
& ( setminusLsub = $true )
& ( complementTI1 = $true )
& ( breln1unionprop = $true )
& ( setadjoin__Cong = $true )
& ( binintersectRsub = $true )
& ( ap2apEq2 = $true )
& ( symdiffIneg2 = $true )
& ( woz1_3 = $true )
& ( setukpairinjR2 = $true )
& ( uniqinunit = $true )
& ( setoftrueEq = $true )
& ( dsetconstrI = $true )
& ( exuI3 = $true )
& ( demorgan1a = $true )
& ( contrasubsetT3 = $true )
& ( setukpairinjL = $true )
& ( woz13rule0 = $true )
& ( breln1unionIR = $true )
& ( setminusER = $true )
& ( eqbreln1 = $true )
& ( woz13rule1 = $true )
& ( woz13rule4 = $true )
& ( symdiffI2 = $true )
& ( nonemptyI1 = $true )
& ( breln1compE = $true )
& ( in__Cong = $true )
& ( breln1compI = $true )
& ( disjointsetsI1 = $true )
& ( demorgan2 = $true )
& ( image1Equiv = $true )
& ( brelnall1 = $true )
& ( breln1invprop = $true )
& ( binunionTIRcontra = $true )
& ( subsetI2 = $true )
& ( dpsetconstrSub = $true )
& ( symdiffI1 = $true )
& ( eqinunit = $true )
& ( binintersectI = $true )
& ( subsetE = $true )
& ( ubforcartprodlem2 = $true )
& ( foundationAx = $true )
& ( cartprodfstin = $true )
& ( cartprodsndpairEq = $true )
& ( choice2fnsingleton = $true )
& ( cartprodsndin = $true )
& ( breln1unionE = $true )
& ( setukpairIR = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( woz2Ex = $true )
& ( secondinupair = $true )
& ( setadjoinIR = $true )
& ( quantDeMorgan2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( cartprodpairmemEL = $true )
& ( binunionIR = $true )
& ( upairsetIL = $true )
& ( woz2B = $true )
& ( singletonsuniq = $true )
& ( complementImpComplementIntersect = $true )
& ( emptysetsubset = $true )
& ( setukpairinjR = $true )
& ( brelnall2 = $true )
& ( theeq = $true )
& ( doubleComplementSub1 = $true )
& ( iftrue = $true )
& ( omegaIndAx = $true )
& ( subsetE2 = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setukpairIL = $true )
& ( binintersectTERcontra = $true )
& ( funcext = $true )
& ( breln1unionCommutes = $true )
& ( app = $true )
& ( binintersectSubset3 = $true )
& ( funcGraphProp3 = $true )
& ( cartprodpairin = $true )
& ( exuE3e = $true )
& ( demorgan1b = $true )
& ( breln1unionIL = $true )
& ( contrasubsetT1 = $true )
& ( ap2apEq1 = $true )
& ( doubleComplementEq = $true )
& ( dsetconstr__Cong = $true )
& ( dpsetconstrER = $true )
& ( doubleComplementSub2 = $true )
& ( notinsingleton = $true )
& ( subsetRefl = $true )
& ( ap2p = $true )
& ( binunionE = $true )
& ( setunionsingleton = $true )
& ( binintersectSubset1 = $true )
& ( doubleComplementE1 = $true )
& ( complementTE1 = $true )
& ( powersetAx = $true )
& ( dpsetconstrEL2 = $true )
& ( ( in @ sK619 @ ( funcSet @ sK618 @ sK620 ) )
!= $true )
& ( ( in @ sK619 @ ( surjFuncSet @ sK618 @ sK620 ) )
= $true )
& ( sepSubset = $true )
& ( powersetsubset = $true )
& ( setadjoinOr = $true )
& ( complementTnotintersectT = $true )
& ( contrasubsetT = $true )
& ( complementT_lem = $true )
& ( woz13rule3 = $true )
& ( inCongP = $true )
& ( breln1all1 = $true )
& ( breln1unionEcases = $true )
& ( subsetTrans = $true )
& ( setminusERneg = $true )
& ( prop2setI = $true )
& ( inIntersectImpInUnion = $true )
& ( upairinpowunion = $true )
& ( iffalse = $true )
& ( noeltsimpempty = $true )
& ( subsetTI = $true )
& ( sepInPowerset = $true )
& ( emptysetAx = $true )
& ( setminusEL = $true )
& ( exuI2 = $true )
& ( setminusILneg = $true )
& ( setextT = $true )
& ( binunionTEcontra = $true )
& ( image1E = $true )
& ( woz1_2 = $true )
& ( funcinfuncset = $true )
& ( eqimpsubset1 = $true )
& ( setadjoinE = $true )
& ( binintersectLsub = $true )
& ( ex1E2 = $true )
& ( binintersectER = $true )
& ( woz13rule2 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( binintersectTELcontra = $true )
& ( binunionTE = $true )
& ( woz1_1 = $true )
& ( quantDeMorgan1 = $true )
& ( ex1E1 = $true )
& ( emptyinunitempty = $true )
& ( demorgan2b2 = $true )
& ( notequalI1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( powersetE1 = $true )
& ( kfstpairEq = $true )
& ( singletonsswitch = $true )
& ( nonemptyImpWitness = $true )
& ( singletoninpowunion = $true )
& ( breln1compprop = $true )
& ( eqbreln = $true )
& ( cartprodmempair1 = $true )
& ( doubleComplementI1 = $true )
& ( lam2p = $true )
& ( binintersectSubset4 = $true )
& ( ksndsingleton = $true )
& ( kpairiskpair = $true )
& ( setextsub = $true )
& ( setunionAx = $true )
& ( theprop = $true )
& ( contraSubsetComplement = $true )
& ( funcext2 = $true )
& ( breln1compEex = $true )
& ( beta2 = $true )
& ( quantDeMorgan4 = $true )
& ( breln1SetBreln1 = $true )
& ( ksndpairEq = $true )
& ( demorgan2a = $true )
& ( omegaSAx = $true )
& ( setminusIRneg = $true )
& ( setminusSubset2 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( funcGraphProp1 = $true )
& ( exu__Cong = $true )
& ( inIntersectImpInUnion2 = $true )
& ( exuE2 = $true )
& ( iffalseProp2 = $true )
& ( setminusT_lem = $true )
& ( setukpairinjL1 = $true )
& ( emptysetE = $true )
& ( lamProp = $true )
& ( powersetTI1 = $true )
& ( breln1invE = $true )
& ( prop2setE = $true )
& ( kpairp = $true )
& ( demorgan2a1 = $true )
& ( emptyInPowerset = $true )
& ( funcGraphProp2 = $true )
& ( setunionE2 = $true )
& ( injFuncSetFuncInj = $true )
& ( woz1_5 = $true )
& ( symdiffE = $true )
& ( setminusI = $true )
& ( infuncsetfunc = $true )
& ( binunionIL = $true )
& ( exuE3u = $true )
& ( cartprodpairmemER = $true )
& ( upairsetIR = $true )
& ( setunionE = $true )
& ( subbreln = $true )
& ( dpsetconstrERa = $true )
& ( setbeta = $true )
& ( demorgan2a2 = $true )
& ( cartprodmempaircEq = $true )
& ( complementUnionInPowersetComplement = $true )
& ( subset2powerset = $true )
& ( ex1I2 = $true )
& ( nonemptyI = $true )
& ( dpsetconstrEL1 = $true )
& ( dsetconstrEL = $true )
& ( binintersectEL = $true )
& ( nonemptyE1 = $true )
& ( binintersectT_lem = $true )
& ( injFuncSetFuncIn = $true )
& ( powerset__Cong = $true )
& ( breln1all2 = $true )
& ( subsetI1 = $true )
& ( singletonsubset = $true )
& ( ubforcartprodlem1 = $true )
& ( dsetconstrER = $true )
& ( lamp = $true )
& ( setadjoinSub = $true )
& ( notsubsetI = $true )
& ( setextAx = $true )
& ( emptysetimpfalse = $true )
& ( singletoninpowerset = $true )
& ( complementTcontraSubset = $true )
& ( inPowerset = $true )
& ( exuI1 = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( notdexE = $true )
& ( powersetT_lem = $true )
& ( powersetI1 = $true )
& ( descrp = $true )
& ( setukpairinjR11 = $true )
& ( setminusSubset1 = $true )
& ( omega0Ax = $true )
& ( funcImageSingleton = $true )
& ( beta1 = $true )
& ( subPowSU = $true )
& ( binunionRsub = $true )
& ( symdiffIneg1 = $true )
& ( prop2set2propI = $true )
& ( breln1unionI = $true )
& ( iftrueorfalse = $true )
& ( binunionT_lem = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( vacuousDall = $true )
& ( emptyI = $true )
& ( descr__Cong = $true )
& ( woz1_4 = $true )
& ( apProp = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setukpairinjR12 = $true )
& ( image1Ex1 = $true )
& ( kfstsingleton = $true )
& ( singletonprop = $true )
& ( upairequniteq = $true )
& ( binintersectSubset2 = $true )
& ( image1Ex = $true )
& ( binunionTILcontra = $true )
& ( exuE1 = $true )
& ( upairsetE = $true )
& ( setadjoinSub2 = $true )
& ( funcGraphProp4 = $true )
& ( image1I = $true )
& ( cartprodfstpairEq = $true )
& ( quantDeMorgan3 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK618,sK619,sK620])],[f1340,f2457]) ).
thf(f2457,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) )
& ( ( in @ X1 @ ( surjFuncSet @ X0 @ X2 ) )
= $true ) )
=> ( ( ( in @ sK619 @ ( funcSet @ sK618 @ sK620 ) )
!= $true )
& ( ( in @ sK619 @ ( surjFuncSet @ sK618 @ sK620 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1340,plain,
( ( iffalseProp1 = $true )
& ( setunionI = $true )
& ( cartprodmempair = $true )
& ( setukpairinjR1 = $true )
& ( ubforcartprodlem3 = $true )
& ( upairsubunion = $true )
& ( dpsetconstrI = $true )
& ( demorgan1 = $true )
& ( eqimpsubset2 = $true )
& ( binunionLsub = $true )
& ( eta2 = $true )
& ( upairset2E = $true )
& ( cartprodpairsurjEq = $true )
& ( wellorderingAx = $true )
& ( setOfPairsIsBReln = $true )
& ( emptyset__Cong = $true )
& ( omega__Cong = $true )
& ( kpairsurjEq = $true )
& ( ifSingleton = $true )
& ( ex1I = $true )
& ( funcextLem = $true )
& ( powersetTE1 = $true )
& ( woz2W = $true )
& ( setadjoinIL = $true )
& ( breln1invI = $true )
& ( setunionsingleton2 = $true )
& ( setext = $true )
& ( setunionsingleton1 = $true )
& ( binunionEcases = $true )
& ( powersetE = $true )
& ( bs114d = $true )
& ( exuEu = $true )
& ( setminusELneg = $true )
& ( setukpairinjL2 = $true )
& ( contrasubsetT2 = $true )
& ( replAx = $true )
& ( upairset2IR = $true )
& ( injFuncInInjFuncSet = $true )
& ( binintersectSubset5 = $true )
& ( lam2lamEq = $true )
& ( setunion__Cong = $true )
& ( notequalI2 = $true )
& ( subbreln1 = $true )
& ( setadjoinAx = $true )
& ( demorgan2b = $true )
& ( emptyE1 = $true )
& ( ifp = $true )
& ( notdallE = $true )
& ( woz2A = $true )
& ( subsetemptysetimpeq = $true )
& ( notinemptyset = $true )
& ( eta1 = $true )
& ( setminusLsub = $true )
& ( complementTI1 = $true )
& ( breln1unionprop = $true )
& ( setadjoin__Cong = $true )
& ( binintersectRsub = $true )
& ( ap2apEq2 = $true )
& ( symdiffIneg2 = $true )
& ( woz1_3 = $true )
& ( setukpairinjR2 = $true )
& ( uniqinunit = $true )
& ( setoftrueEq = $true )
& ( dsetconstrI = $true )
& ( exuI3 = $true )
& ( demorgan1a = $true )
& ( contrasubsetT3 = $true )
& ( setukpairinjL = $true )
& ( woz13rule0 = $true )
& ( breln1unionIR = $true )
& ( setminusER = $true )
& ( eqbreln1 = $true )
& ( woz13rule1 = $true )
& ( woz13rule4 = $true )
& ( symdiffI2 = $true )
& ( nonemptyI1 = $true )
& ( breln1compE = $true )
& ( in__Cong = $true )
& ( breln1compI = $true )
& ( disjointsetsI1 = $true )
& ( demorgan2 = $true )
& ( image1Equiv = $true )
& ( brelnall1 = $true )
& ( breln1invprop = $true )
& ( binunionTIRcontra = $true )
& ( subsetI2 = $true )
& ( dpsetconstrSub = $true )
& ( symdiffI1 = $true )
& ( eqinunit = $true )
& ( binintersectI = $true )
& ( subsetE = $true )
& ( ubforcartprodlem2 = $true )
& ( foundationAx = $true )
& ( cartprodfstin = $true )
& ( cartprodsndpairEq = $true )
& ( choice2fnsingleton = $true )
& ( cartprodsndin = $true )
& ( breln1unionE = $true )
& ( setukpairIR = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( woz2Ex = $true )
& ( secondinupair = $true )
& ( setadjoinIR = $true )
& ( quantDeMorgan2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( cartprodpairmemEL = $true )
& ( binunionIR = $true )
& ( upairsetIL = $true )
& ( woz2B = $true )
& ( singletonsuniq = $true )
& ( complementImpComplementIntersect = $true )
& ( emptysetsubset = $true )
& ( setukpairinjR = $true )
& ( brelnall2 = $true )
& ( theeq = $true )
& ( doubleComplementSub1 = $true )
& ( iftrue = $true )
& ( omegaIndAx = $true )
& ( subsetE2 = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setukpairIL = $true )
& ( binintersectTERcontra = $true )
& ( funcext = $true )
& ( breln1unionCommutes = $true )
& ( app = $true )
& ( binintersectSubset3 = $true )
& ( funcGraphProp3 = $true )
& ( cartprodpairin = $true )
& ( exuE3e = $true )
& ( demorgan1b = $true )
& ( breln1unionIL = $true )
& ( contrasubsetT1 = $true )
& ( ap2apEq1 = $true )
& ( doubleComplementEq = $true )
& ( dsetconstr__Cong = $true )
& ( dpsetconstrER = $true )
& ( doubleComplementSub2 = $true )
& ( notinsingleton = $true )
& ( subsetRefl = $true )
& ( ap2p = $true )
& ( binunionE = $true )
& ( setunionsingleton = $true )
& ( binintersectSubset1 = $true )
& ( doubleComplementE1 = $true )
& ( complementTE1 = $true )
& ( powersetAx = $true )
& ( dpsetconstrEL2 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) )
& ( ( in @ X1 @ ( surjFuncSet @ X0 @ X2 ) )
= $true ) )
& ( sepSubset = $true )
& ( powersetsubset = $true )
& ( setadjoinOr = $true )
& ( complementTnotintersectT = $true )
& ( contrasubsetT = $true )
& ( complementT_lem = $true )
& ( woz13rule3 = $true )
& ( inCongP = $true )
& ( breln1all1 = $true )
& ( breln1unionEcases = $true )
& ( subsetTrans = $true )
& ( setminusERneg = $true )
& ( prop2setI = $true )
& ( inIntersectImpInUnion = $true )
& ( upairinpowunion = $true )
& ( iffalse = $true )
& ( noeltsimpempty = $true )
& ( subsetTI = $true )
& ( sepInPowerset = $true )
& ( emptysetAx = $true )
& ( setminusEL = $true )
& ( exuI2 = $true )
& ( setminusILneg = $true )
& ( setextT = $true )
& ( binunionTEcontra = $true )
& ( image1E = $true )
& ( woz1_2 = $true )
& ( funcinfuncset = $true )
& ( eqimpsubset1 = $true )
& ( setadjoinE = $true )
& ( binintersectLsub = $true )
& ( ex1E2 = $true )
& ( binintersectER = $true )
& ( woz13rule2 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( binintersectTELcontra = $true )
& ( binunionTE = $true )
& ( woz1_1 = $true )
& ( quantDeMorgan1 = $true )
& ( ex1E1 = $true )
& ( emptyinunitempty = $true )
& ( demorgan2b2 = $true )
& ( notequalI1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( powersetE1 = $true )
& ( kfstpairEq = $true )
& ( singletonsswitch = $true )
& ( nonemptyImpWitness = $true )
& ( singletoninpowunion = $true )
& ( breln1compprop = $true )
& ( eqbreln = $true )
& ( cartprodmempair1 = $true )
& ( doubleComplementI1 = $true )
& ( lam2p = $true )
& ( binintersectSubset4 = $true )
& ( ksndsingleton = $true )
& ( kpairiskpair = $true )
& ( setextsub = $true )
& ( setunionAx = $true )
& ( theprop = $true )
& ( contraSubsetComplement = $true )
& ( funcext2 = $true )
& ( breln1compEex = $true )
& ( beta2 = $true )
& ( quantDeMorgan4 = $true )
& ( breln1SetBreln1 = $true )
& ( ksndpairEq = $true )
& ( demorgan2a = $true )
& ( omegaSAx = $true )
& ( setminusIRneg = $true )
& ( setminusSubset2 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( funcGraphProp1 = $true )
& ( exu__Cong = $true )
& ( inIntersectImpInUnion2 = $true )
& ( exuE2 = $true )
& ( iffalseProp2 = $true )
& ( setminusT_lem = $true )
& ( setukpairinjL1 = $true )
& ( emptysetE = $true )
& ( lamProp = $true )
& ( powersetTI1 = $true )
& ( breln1invE = $true )
& ( prop2setE = $true )
& ( kpairp = $true )
& ( demorgan2a1 = $true )
& ( emptyInPowerset = $true )
& ( funcGraphProp2 = $true )
& ( setunionE2 = $true )
& ( injFuncSetFuncInj = $true )
& ( woz1_5 = $true )
& ( symdiffE = $true )
& ( setminusI = $true )
& ( infuncsetfunc = $true )
& ( binunionIL = $true )
& ( exuE3u = $true )
& ( cartprodpairmemER = $true )
& ( upairsetIR = $true )
& ( setunionE = $true )
& ( subbreln = $true )
& ( dpsetconstrERa = $true )
& ( setbeta = $true )
& ( demorgan2a2 = $true )
& ( cartprodmempaircEq = $true )
& ( complementUnionInPowersetComplement = $true )
& ( subset2powerset = $true )
& ( ex1I2 = $true )
& ( nonemptyI = $true )
& ( dpsetconstrEL1 = $true )
& ( dsetconstrEL = $true )
& ( binintersectEL = $true )
& ( nonemptyE1 = $true )
& ( binintersectT_lem = $true )
& ( injFuncSetFuncIn = $true )
& ( powerset__Cong = $true )
& ( breln1all2 = $true )
& ( subsetI1 = $true )
& ( singletonsubset = $true )
& ( ubforcartprodlem1 = $true )
& ( dsetconstrER = $true )
& ( lamp = $true )
& ( setadjoinSub = $true )
& ( notsubsetI = $true )
& ( setextAx = $true )
& ( emptysetimpfalse = $true )
& ( singletoninpowerset = $true )
& ( complementTcontraSubset = $true )
& ( inPowerset = $true )
& ( exuI1 = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( notdexE = $true )
& ( powersetT_lem = $true )
& ( powersetI1 = $true )
& ( descrp = $true )
& ( setukpairinjR11 = $true )
& ( setminusSubset1 = $true )
& ( omega0Ax = $true )
& ( funcImageSingleton = $true )
& ( beta1 = $true )
& ( subPowSU = $true )
& ( binunionRsub = $true )
& ( symdiffIneg1 = $true )
& ( prop2set2propI = $true )
& ( breln1unionI = $true )
& ( iftrueorfalse = $true )
& ( binunionT_lem = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( vacuousDall = $true )
& ( emptyI = $true )
& ( descr__Cong = $true )
& ( woz1_4 = $true )
& ( apProp = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setukpairinjR12 = $true )
& ( image1Ex1 = $true )
& ( kfstsingleton = $true )
& ( singletonprop = $true )
& ( upairequniteq = $true )
& ( binintersectSubset2 = $true )
& ( image1Ex = $true )
& ( binunionTILcontra = $true )
& ( exuE1 = $true )
& ( upairsetE = $true )
& ( setadjoinSub2 = $true )
& ( funcGraphProp4 = $true )
& ( image1I = $true )
& ( cartprodfstpairEq = $true )
& ( quantDeMorgan3 = $true ) ),
inference(flattening,[status(thm)],[f1339]) ).
thf(f1339,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) )
& ( ( in @ X1 @ ( surjFuncSet @ X0 @ X2 ) )
= $true ) )
& ( injFuncSetFuncInj = $true )
& ( injFuncSetFuncIn = $true )
& ( injFuncInInjFuncSet = $true )
& ( image1I = $true )
& ( image1E = $true )
& ( image1Equiv = $true )
& ( image1Ex1 = $true )
& ( image1Ex = $true )
& ( woz2B = $true )
& ( woz2A = $true )
& ( woz2W = $true )
& ( woz2Ex = $true )
& ( breln1unionCommutes = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f699]) ).
thf(f699,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ( ( breln1unionE = $true )
=> ( ( breln1unionEcases = $true )
=> ( ( breln1unionCommutes = $true )
=> ( ( woz2Ex = $true )
=> ( ( woz2W = $true )
=> ( ( woz2A = $true )
=> ( ( woz2B = $true )
=> ( ( image1Ex = $true )
=> ( ( image1Ex1 = $true )
=> ( ( image1Equiv = $true )
=> ( ( image1E = $true )
=> ( ( image1I = $true )
=> ( ( injFuncInInjFuncSet = $true )
=> ( ( injFuncSetFuncIn = $true )
=> ( ( injFuncSetFuncInj = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( surjFuncSet @ X0 @ X2 ) )
= $true )
=> ( $true
= ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f698]) ).
thf(f698,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ ( surjFuncSet @ X0 @ X2 ) )
=> ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f323]) ).
thf(f323,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ! [X3: $i,X12: $i,X4: $i] :
( ( in @ X12 @ ( surjFuncSet @ X3 @ X4 ) )
=> ( in @ X12 @ ( funcSet @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f322]) ).
thf(f322,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ! [X3: $i,X12: $i,X4: $i] :
( ( in @ X12 @ ( surjFuncSet @ X3 @ X4 ) )
=> ( in @ X12 @ ( funcSet @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uXiymMPKZd/DTF2THF_27071.p',surjFuncSetFuncIn) ).
thf(f9445,plain,
( ( in @ sK619 @ ( funcSet @ sK618 @ sK620 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f9435]) ).
thf(f9435,plain,
( ( $true != $true )
| ( ( in @ sK619 @ ( funcSet @ sK618 @ sK620 ) )
= $true ) ),
inference(superposition,[status(thm)],[f6615,f6871]) ).
thf(f6871,plain,
( $true
= ( in @ sK619 @ ( dsetconstr @ ( funcSet @ sK618 @ sK620 ) @ ( surjective @ sK618 @ sK620 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f5901]) ).
thf(f5901,plain,
( ( in @ sK619
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( surjective @ Y0 @ Y1 @ Y2 ) )
@ sK618
@ sK620 ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f4344,f3397]) ).
thf(f3397,plain,
( surjFuncSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( surjective @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f739]) ).
thf(f739,plain,
( surjFuncSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( surjective @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f738]) ).
thf(f738,plain,
( surjFuncSet
= ( ^ [X0: $i,X1: $i] :
( dsetconstr @ ( funcSet @ X0 @ X1 )
@ ^ [X2: $i] : ( surjective @ X0 @ X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f321]) ).
thf(f321,axiom,
( surjFuncSet
= ( ^ [X3: $i,X4: $i] :
( dsetconstr @ ( funcSet @ X3 @ X4 )
@ ^ [X12: $i] : ( surjective @ X3 @ X4 @ X12 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uXiymMPKZd/DTF2THF_27071.p',surjFuncSet) ).
thf(f4344,plain,
( ( in @ sK619 @ ( surjFuncSet @ sK618 @ sK620 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f2458]) ).
thf(f6615,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( $true
!= ( in @ X1 @ ( dsetconstr @ X2 @ X0 ) ) )
| ( ( in @ X1 @ X2 )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f6614]) ).
thf(f6614,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( ( in @ X1 @ X2 )
= $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f6189]) ).
thf(f6189,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( $true != $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true )
| ( ( in @ X1 @ X2 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f4907,f4234]) ).
thf(f4234,plain,
dsetconstrEL = $true,
inference(cnf_transformation,[status(thm)],[f2458]) ).
thf(f4907,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( ( in @ X1 @ X2 )
= $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true )
| ( dsetconstrEL != $true ) ),
inference(cnf_transformation,[status(thm)],[f2931]) ).
thf(f2931,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) )
| ( dsetconstrEL != $true ) )
& ( ( dsetconstrEL = $true )
| ( ( ( in @ sK885 @ sK886 )
!= $true )
& ( ( in @ sK885
@ ( dsetconstr @ sK886
@ ^ [Y0: $i] : ( sK884 @ Y0 ) ) )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK884,sK885,sK886])],[f2929,f2930]) ).
thf(f2930,plain,
( ? [X3: $i > $o,X4: $i,X5: $i] :
( ( ( in @ X4 @ X5 )
!= $true )
& ( $true
= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) ) )
=> ( ( ( in @ sK885 @ sK886 )
!= $true )
& ( ( in @ sK885
@ ( dsetconstr @ sK886
@ ^ [Y0: $i] : ( sK884 @ Y0 ) ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2929,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) )
| ( dsetconstrEL != $true ) )
& ( ( dsetconstrEL = $true )
| ? [X3: $i > $o,X4: $i,X5: $i] :
( ( ( in @ X4 @ X5 )
!= $true )
& ( $true
= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2928]) ).
thf(f2928,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) )
| ( dsetconstrEL != $true ) )
& ( ( dsetconstrEL = $true )
| ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
& ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1074]) ).
thf(f1074,plain,
( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) )
<=> ( dsetconstrEL = $true ) ),
inference(ennf_transformation,[status(thm)],[f617]) ).
thf(f617,plain,
( ( dsetconstrEL = $true )
<=> ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
= $true )
=> ( ( in @ X1 @ X2 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f616]) ).
thf(f616,plain,
( dsetconstrEL
= ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X0 @ X3 ) ) )
=> ( in @ X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f15]) ).
thf(f15,axiom,
( dsetconstrEL
= ( ! [X0: $i > $o,X1: $i,X3: $i] :
( ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) )
=> ( in @ X1 @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uXiymMPKZd/DTF2THF_27071.p',dsetconstrEL) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU801^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.34 % Computer : n026.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Feb 24 12:34:40 EST 2026
% 0.14/0.34 % CPUTime :
% 0.14/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : SEU801^1 : TPTP v9.2.1. Released v3.7.0.
% 0.28/0.46 % Domain : Set Theory
% 0.28/0.46 % Problem : More about Functions - Surjective Functions
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English : (! A:i.! B:i.! f:i.in f (surjFuncSet A B) -> in f (funcSet A B))
% 0.28/0.46
% 0.28/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.28/0.46 % Source : [Bro08]
% 0.28/0.46 % Names : ZFC303g [Bro08]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.54 v8.1.0, 0.55 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v6.1.0, 0.86 v6.0.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.28/0.46 % Syntax : Number of formulae : 692 ( 321 unt; 370 typ; 321 def)
% 0.28/0.46 % Number of atoms : 1976 ( 462 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 319 ( 6 avg)
% 0.28/0.46 % Number of connectives : 4346 ( 73 ~; 14 |; 54 &;3167 @)
% 0.28/0.46 % ( 17 <=>;1021 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 325 ( 2 avg)
% 0.28/0.46 % Number of types : 2 ( 0 usr)
% 0.28/0.46 % Number of type conns : 175 ( 175 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 373 ( 370 usr; 321 con; 0-4 aty)
% 0.28/0.46 % Number of variables : 1138 ( 81 ^;1013 !; 44 ?;1138 :)
% 0.28/0.46 % SPC : TH0_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : http://mathgate.info/detsetitem.php?id=484
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 thf(in_type,type,
% 0.28/0.46 in: $i > $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(exu_type,type,
% 0.28/0.46 exu: ( $i > $o ) > $o ).
% 0.28/0.46
% 0.28/0.46 thf(exu,definition,
% 0.28/0.46 ( exu
% 0.28/0.46 = ( ^ [Xphi: $i > $o] :
% 0.28/0.46 ? [Xx: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 & ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setextAx_type,type,
% 0.28/0.46 setextAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setextAx,definition,
% 0.28/0.46 ( setextAx
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 <=> ( in @ Xx @ B ) )
% 0.28/0.46 => ( A = B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyset_type,type,
% 0.28/0.46 emptyset: $i ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetAx_type,type,
% 0.28/0.46 emptysetAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetAx,definition,
% 0.28/0.46 ( emptysetAx
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoin_type,type,
% 0.28/0.46 setadjoin: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinAx_type,type,
% 0.28/0.46 setadjoinAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinAx,definition,
% 0.28/0.46 ( setadjoinAx
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.46 <=> ( ( Xy = Xx )
% 0.28/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powerset_type,type,
% 0.28/0.46 powerset: $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(powersetAx_type,type,
% 0.28/0.46 powersetAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetAx,definition,
% 0.28/0.46 ( powersetAx
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.46 <=> ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setunion_type,type,
% 0.28/0.46 setunion: $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(setunionAx_type,type,
% 0.28/0.46 setunionAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setunionAx,definition,
% 0.28/0.46 ( setunionAx
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.46 <=> ? [B: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omega_type,type,
% 0.28/0.46 omega: $i ).
% 0.28/0.46
% 0.28/0.46 thf(omega0Ax_type,type,
% 0.28/0.46 omega0Ax: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omega0Ax,definition,
% 0.28/0.46 ( omega0Ax
% 0.28/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omegaSAx_type,type,
% 0.28/0.46 omegaSAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omegaSAx,definition,
% 0.28/0.46 ( omegaSAx
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ omega )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omegaIndAx_type,type,
% 0.28/0.46 omegaIndAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omegaIndAx,definition,
% 0.28/0.46 ( omegaIndAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( ( in @ emptyset @ A )
% 0.28/0.46 & ! [Xx: $i] :
% 0.28/0.46 ( ( ( in @ Xx @ omega )
% 0.28/0.46 & ( in @ Xx @ A ) )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ omega )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(replAx_type,type,
% 0.28/0.46 replAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(replAx,definition,
% 0.28/0.46 ( replAx
% 0.28/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( exu
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.28/0.46 => ? [B: $i] :
% 0.28/0.46 ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 <=> ? [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ A )
% 0.28/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(foundationAx_type,type,
% 0.28/0.46 foundationAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(foundationAx,definition,
% 0.28/0.46 ( foundationAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.46 => ? [B: $i] :
% 0.28/0.46 ( ( in @ B @ A )
% 0.28/0.46 & ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(wellorderingAx_type,type,
% 0.28/0.46 wellorderingAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(wellorderingAx,definition,
% 0.28/0.46 ( wellorderingAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ? [B: $i] :
% 0.28/0.46 ( ! [C: $i] :
% 0.28/0.46 ( ( in @ C @ B )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ A ) ) )
% 0.28/0.46 & ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( ( in @ Xx @ A )
% 0.28/0.46 & ( in @ Xy @ A ) )
% 0.28/0.46 => ( ! [C: $i] :
% 0.28/0.46 ( ( in @ C @ B )
% 0.28/0.46 => ( ( in @ Xx @ C )
% 0.28/0.46 <=> ( in @ Xy @ C ) ) )
% 0.28/0.46 => ( Xx = Xy ) ) )
% 0.28/0.46 & ! [C: $i,D: $i] :
% 0.28/0.46 ( ( ( in @ C @ B )
% 0.28/0.46 & ( in @ D @ B ) )
% 0.28/0.46 => ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ D ) )
% 0.28/0.46 | ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ D )
% 0.28/0.46 => ( in @ Xx @ C ) ) ) )
% 0.28/0.46 & ! [C: $i] :
% 0.28/0.46 ( ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ A ) )
% 0.28/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.28/0.46 => ? [D: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ D @ B )
% 0.28/0.46 & ( in @ Xx @ C )
% 0.28/0.46 & ~ ? [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ D )
% 0.28/0.46 & ( in @ Xy @ C ) )
% 0.28/0.46 & ! [E: $i] :
% 0.28/0.46 ( ( in @ E @ B )
% 0.28/0.46 => ( ! [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ E )
% 0.28/0.46 => ( in @ Xy @ D ) )
% 0.28/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(descr_type,type,
% 0.28/0.46 descr: ( $i > $o ) > $i ).
% 0.28/0.46
% 0.28/0.46 thf(descrp_type,type,
% 0.28/0.46 descrp: $o ).
% 0.28/0.46
% 0.28/0.46 thf(descrp,definition,
% 0.28/0.46 ( descrp
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ( Xphi
% 0.28/0.46 @ ( descr
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstr_type,type,
% 0.28/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrI_type,type,
% 0.28/0.46 dsetconstrI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrI,definition,
% 0.28/0.46 ( dsetconstrI
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( Xphi @ Xx )
% 0.28/0.46 => ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrEL_type,type,
% 0.28/0.46 dsetconstrEL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrEL,definition,
% 0.28/0.46 ( dsetconstrEL
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrER_type,type,
% 0.28/0.46 dsetconstrER: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrER,definition,
% 0.28/0.46 ( dsetconstrER
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE1_type,type,
% 0.28/0.46 exuE1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE1,definition,
% 0.28/0.46 ( exuE1
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 & ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(prop2set_type,type,
% 0.28/0.46 prop2set: $o > $i ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setE_type,type,
% 0.28/0.46 prop2setE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setE,definition,
% 0.28/0.46 ( prop2setE
% 0.28/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.28/0.46 => Xphi ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetE_type,type,
% 0.28/0.46 emptysetE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetE,definition,
% 0.28/0.46 ( emptysetE
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ emptyset )
% 0.28/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetimpfalse_type,type,
% 0.28/0.46 emptysetimpfalse: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetimpfalse,definition,
% 0.28/0.46 ( emptysetimpfalse
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ emptyset )
% 0.28/0.46 => $false ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notinemptyset_type,type,
% 0.28/0.46 notinemptyset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notinemptyset,definition,
% 0.28/0.46 ( notinemptyset
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE3e_type,type,
% 0.28/0.46 exuE3e: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE3e,definition,
% 0.28/0.46 ( exuE3e
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setext_type,type,
% 0.28/0.46 setext: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setext,definition,
% 0.28/0.46 ( setext
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ B ) )
% 0.28/0.46 => ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) )
% 0.28/0.46 => ( A = B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyI_type,type,
% 0.28/0.46 emptyI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyI,definition,
% 0.28/0.46 ( emptyI
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ A )
% 0.28/0.46 => ( A = emptyset ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(noeltsimpempty_type,type,
% 0.28/0.46 noeltsimpempty: $o ).
% 0.28/0.46
% 0.28/0.46 thf(noeltsimpempty,definition,
% 0.28/0.46 ( noeltsimpempty
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ A )
% 0.28/0.46 => ( A = emptyset ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setbeta_type,type,
% 0.28/0.46 setbeta: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setbeta,definition,
% 0.28/0.46 ( setbeta
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonempty_type,type,
% 0.28/0.46 nonempty: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonempty,definition,
% 0.28/0.46 ( nonempty
% 0.28/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyE1_type,type,
% 0.28/0.46 nonemptyE1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyE1,definition,
% 0.28/0.46 ( nonemptyE1
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( nonempty @ A )
% 0.28/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyI_type,type,
% 0.28/0.46 nonemptyI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyI,definition,
% 0.28/0.46 ( nonemptyI
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( Xphi @ Xx )
% 0.28/0.46 => ( nonempty
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyI1_type,type,
% 0.28/0.46 nonemptyI1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyI1,definition,
% 0.28/0.46 ( nonemptyI1
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinIL_type,type,
% 0.28/0.46 setadjoinIL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinIL,definition,
% 0.28/0.46 ( setadjoinIL
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyinunitempty_type,type,
% 0.28/0.46 emptyinunitempty: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyinunitempty,definition,
% 0.28/0.46 ( emptyinunitempty
% 0.28/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinIR_type,type,
% 0.28/0.46 setadjoinIR: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinIR,definition,
% 0.28/0.46 ( setadjoinIR
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ A )
% 0.28/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinE_type,type,
% 0.28/0.46 setadjoinE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinE,definition,
% 0.28/0.46 ( setadjoinE
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.46 => ! [Xphi: $o] :
% 0.28/0.46 ( ( ( Xy = Xx )
% 0.28/0.46 => Xphi )
% 0.28/0.46 => ( ( ( in @ Xy @ A )
% 0.28/0.46 => Xphi )
% 0.28/0.46 => Xphi ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinOr_type,type,
% 0.28/0.46 setadjoinOr: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinOr,definition,
% 0.28/0.46 ( setadjoinOr
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.46 => ( ( Xy = Xx )
% 0.28/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setoftrueEq_type,type,
% 0.28/0.46 setoftrueEq: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setoftrueEq,definition,
% 0.28/0.46 ( setoftrueEq
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xx: $i] : $true )
% 0.28/0.46 = A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powersetI_type,type,
% 0.28/0.46 powersetI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetI,definition,
% 0.28/0.46 ( powersetI
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) )
% 0.28/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyinPowerset_type,type,
% 0.28/0.46 emptyinPowerset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyinPowerset,definition,
% 0.28/0.46 ( emptyinPowerset
% 0.28/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyInPowerset_type,type,
% 0.28/0.46 emptyInPowerset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyInPowerset,definition,
% 0.28/0.46 ( emptyInPowerset
% 0.28/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powersetE_type,type,
% 0.28/0.46 powersetE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetE,definition,
% 0.28/0.46 ( powersetE
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.46 => ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setunionI_type,type,
% 0.28/0.46 setunionI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setunionI,definition,
% 0.28/0.46 ( setunionI
% 0.28/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( ( in @ B @ A )
% 0.28/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setunionE_type,type,
% 0.28/0.46 setunionE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setunionE,definition,
% 0.28/0.46 ( setunionE
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.46 => ! [Xphi: $o] :
% 0.28/0.46 ( ! [B: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( ( in @ B @ A )
% 0.28/0.46 => Xphi ) )
% 0.28/0.46 => Xphi ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subPowSU_type,type,
% 0.28/0.46 subPowSU: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subPowSU,definition,
% 0.28/0.46 ( subPowSU
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE2_type,type,
% 0.28/0.46 exuE2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE2,definition,
% 0.28/0.46 ( exuE2
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyImpWitness_type,type,
% 0.28/0.46 nonemptyImpWitness: $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyImpWitness,definition,
% 0.28/0.46 ( nonemptyImpWitness
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( nonempty @ A )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & $true ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(uniqinunit_type,type,
% 0.28/0.46 uniqinunit: $o ).
% 0.28/0.46
% 0.28/0.46 thf(uniqinunit,definition,
% 0.28/0.46 ( uniqinunit
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notinsingleton_type,type,
% 0.28/0.46 notinsingleton: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notinsingleton,definition,
% 0.28/0.46 ( notinsingleton
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx != Xy )
% 0.28/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(eqinunit_type,type,
% 0.28/0.46 eqinunit: $o ).
% 0.28/0.46
% 0.28/0.46 thf(eqinunit,definition,
% 0.28/0.46 ( eqinunit
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx = Xy )
% 0.28/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(singletonsswitch_type,type,
% 0.28/0.46 singletonsswitch: $o ).
% 0.28/0.46
% 0.28/0.46 thf(singletonsswitch,definition,
% 0.28/0.46 ( singletonsswitch
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetE_type,type,
% 0.28/0.46 upairsetE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetE,definition,
% 0.28/0.46 ( upairsetE
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.46 => ( ( Xz = Xx )
% 0.28/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetIL_type,type,
% 0.28/0.46 upairsetIL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetIL,definition,
% 0.28/0.46 ( upairsetIL
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetIR_type,type,
% 0.28/0.46 upairsetIR: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetIR,definition,
% 0.28/0.46 ( upairsetIR
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyE1_type,type,
% 0.28/0.46 emptyE1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyE1,definition,
% 0.28/0.46 ( emptyE1
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ( Xphi @ Xx ) )
% 0.28/0.46 => ( ( ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 = emptyset )
% 0.28/0.46 => $false ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(vacuousDall_type,type,
% 0.28/0.46 vacuousDall: $o ).
% 0.28/0.46
% 0.28/0.46 thf(vacuousDall,definition,
% 0.28/0.46 ( vacuousDall
% 0.28/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ emptyset )
% 0.28/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan1_type,type,
% 0.28/0.46 quantDeMorgan1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan1,definition,
% 0.28/0.46 ( quantDeMorgan1
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ~ ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan2_type,type,
% 0.28/0.46 quantDeMorgan2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan2,definition,
% 0.28/0.46 ( quantDeMorgan2
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ~ ( Xphi @ Xx ) )
% 0.28/0.46 => ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan3_type,type,
% 0.28/0.46 quantDeMorgan3: $o ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan3,definition,
% 0.28/0.46 ( quantDeMorgan3
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ( Xphi @ Xx ) )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan4_type,type,
% 0.28/0.46 quantDeMorgan4: $o ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan4,definition,
% 0.28/0.46 ( quantDeMorgan4
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ~ ( Xphi @ Xx ) )
% 0.28/0.46 => ~ ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setI_type,type,
% 0.28/0.46 prop2setI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setI,definition,
% 0.28/0.46 ( prop2setI
% 0.28/0.46 = ( ! [Xphi: $o] :
% 0.28/0.46 ( Xphi
% 0.28/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(set2prop_type,type,
% 0.28/0.46 set2prop: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2set2propI_type,type,
% 0.28/0.46 prop2set2propI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2set2propI,definition,
% 0.28/0.46 ( prop2set2propI
% 0.28/0.46 = ( ! [Xphi: $o] :
% 0.28/0.46 ( Xphi
% 0.28/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notdexE_type,type,
% 0.28/0.46 notdexE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notdexE,definition,
% 0.28/0.46 ( notdexE
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notdallE_type,type,
% 0.28/0.47 notdallE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notdallE,definition,
% 0.28/0.47 ( notdallE
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI1_type,type,
% 0.28/0.47 exuI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI1,definition,
% 0.28/0.47 ( exuI1
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 & ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI3_type,type,
% 0.28/0.47 exuI3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI3,definition,
% 0.28/0.47 ( exuI3
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.28/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI2_type,type,
% 0.28/0.47 exuI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI2,definition,
% 0.28/0.47 ( exuI2
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 <=> ( Xy = Xx ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inCongP_type,type,
% 0.28/0.47 inCongP: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inCongP,definition,
% 0.28/0.47 ( inCongP
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong_type,type,
% 0.28/0.47 in__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong,definition,
% 0.28/0.47 ( in__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3u_type,type,
% 0.28/0.47 exuE3u: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3u,definition,
% 0.28/0.47 ( exuE3u
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exu__Cong_type,type,
% 0.28/0.47 exu__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exu__Cong,definition,
% 0.28/0.47 ( exu__Cong
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 <=> ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset__Cong_type,type,
% 0.28/0.47 emptyset__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset__Cong,definition,
% 0.28/0.47 ( emptyset__Cong
% 0.28/0.47 = ( emptyset = emptyset ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin__Cong_type,type,
% 0.28/0.47 setadjoin__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin__Cong,definition,
% 0.28/0.47 ( setadjoin__Cong
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ! [Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( Xz = Xu )
% 0.28/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.28/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powerset__Cong_type,type,
% 0.28/0.47 powerset__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powerset__Cong,definition,
% 0.28/0.47 ( powerset__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( ( powerset @ A )
% 0.28/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunion__Cong_type,type,
% 0.28/0.47 setunion__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunion__Cong,definition,
% 0.28/0.47 ( setunion__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( ( setunion @ A )
% 0.28/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omega__Cong_type,type,
% 0.28/0.47 omega__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omega__Cong,definition,
% 0.28/0.47 ( omega__Cong
% 0.28/0.47 = ( omega = omega ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuEu_type,type,
% 0.28/0.47 exuEu: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuEu,definition,
% 0.28/0.47 ( exuEu
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(descr__Cong_type,type,
% 0.28/0.47 descr__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(descr__Cong,definition,
% 0.28/0.47 ( descr__Cong
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.28/0.47 => ( ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr__Cong_type,type,
% 0.28/0.47 dsetconstr__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr__Cong,definition,
% 0.28/0.47 ( dsetconstr__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.28/0.47 => ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset_type,type,
% 0.28/0.47 subset: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(disjoint_type,type,
% 0.28/0.47 disjoint: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setsmeet_type,type,
% 0.28/0.47 setsmeet: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI1_type,type,
% 0.28/0.47 subsetI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI1,definition,
% 0.28/0.47 ( subsetI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset2_type,type,
% 0.28/0.47 eqimpsubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset2,definition,
% 0.28/0.47 ( eqimpsubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset1_type,type,
% 0.28/0.47 eqimpsubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset1,definition,
% 0.28/0.47 ( eqimpsubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2_type,type,
% 0.28/0.47 subsetI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2,definition,
% 0.28/0.47 ( subsetI2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetsubset_type,type,
% 0.28/0.47 emptysetsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetsubset,definition,
% 0.28/0.47 ( emptysetsubset
% 0.28/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE_type,type,
% 0.28/0.47 subsetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE,definition,
% 0.28/0.47 ( subsetE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE2_type,type,
% 0.28/0.47 subsetE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE2,definition,
% 0.28/0.47 ( subsetE2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notsubsetI_type,type,
% 0.28/0.47 notsubsetI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notsubsetI,definition,
% 0.28/0.47 ( notsubsetI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI1_type,type,
% 0.28/0.47 notequalI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI1,definition,
% 0.28/0.47 ( notequalI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ~ ( subset @ A @ B )
% 0.28/0.47 => ( A != B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI2_type,type,
% 0.28/0.47 notequalI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI2,definition,
% 0.28/0.47 ( notequalI2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( A != B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetRefl_type,type,
% 0.28/0.47 subsetRefl: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetRefl,definition,
% 0.28/0.47 ( subsetRefl
% 0.28/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTrans_type,type,
% 0.28/0.47 subsetTrans: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTrans,definition,
% 0.28/0.47 ( subsetTrans
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( subset @ B @ C )
% 0.28/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub_type,type,
% 0.28/0.47 setadjoinSub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub,definition,
% 0.28/0.47 ( setadjoinSub
% 0.28/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub2_type,type,
% 0.28/0.47 setadjoinSub2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub2,definition,
% 0.28/0.47 ( setadjoinSub2
% 0.28/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset2powerset_type,type,
% 0.28/0.47 subset2powerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subset2powerset,definition,
% 0.28/0.47 ( subset2powerset
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setextsub_type,type,
% 0.28/0.47 setextsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setextsub,definition,
% 0.28/0.47 ( setextsub
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( subset @ B @ A )
% 0.28/0.47 => ( A = B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetemptysetimpeq_type,type,
% 0.28/0.47 subsetemptysetimpeq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetemptysetimpeq,definition,
% 0.28/0.47 ( subsetemptysetimpeq
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( subset @ A @ emptyset )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI1_type,type,
% 0.28/0.47 powersetI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI1,definition,
% 0.28/0.47 ( powersetI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ B @ A )
% 0.28/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE1_type,type,
% 0.28/0.47 powersetE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE1,definition,
% 0.28/0.47 ( powersetE1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inPowerset_type,type,
% 0.28/0.47 inPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inPowerset,definition,
% 0.28/0.47 ( inPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetsubset_type,type,
% 0.28/0.47 powersetsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetsubset,definition,
% 0.28/0.47 ( powersetsubset
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sepInPowerset_type,type,
% 0.28/0.47 sepInPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(sepInPowerset,definition,
% 0.28/0.47 ( sepInPowerset
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( in
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sepSubset_type,type,
% 0.28/0.47 sepSubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(sepSubset,definition,
% 0.28/0.47 ( sepSubset
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( subset
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunion_type,type,
% 0.28/0.47 binunion: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIL_type,type,
% 0.28/0.47 binunionIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIL,definition,
% 0.28/0.47 ( binunionIL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2IR_type,type,
% 0.28/0.47 upairset2IR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2IR,definition,
% 0.28/0.47 ( upairset2IR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIR_type,type,
% 0.28/0.47 binunionIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIR,definition,
% 0.28/0.47 ( binunionIR
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionEcases_type,type,
% 0.28/0.47 binunionEcases: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionEcases,definition,
% 0.28/0.47 ( binunionEcases
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.28/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.47 => ( ( ( in @ Xx @ A )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ Xx @ B )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionE_type,type,
% 0.28/0.47 binunionE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionE,definition,
% 0.28/0.47 ( binunionE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionLsub_type,type,
% 0.28/0.47 binunionLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionLsub,definition,
% 0.28/0.47 ( binunionLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionRsub_type,type,
% 0.28/0.47 binunionRsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionRsub,definition,
% 0.28/0.47 ( binunionRsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersect_type,type,
% 0.28/0.47 binintersect: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectI_type,type,
% 0.28/0.47 binintersectI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectI,definition,
% 0.28/0.47 ( binintersectI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset5_type,type,
% 0.28/0.47 binintersectSubset5: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset5,definition,
% 0.28/0.47 ( binintersectSubset5
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( subset @ C @ A )
% 0.28/0.47 => ( ( subset @ C @ B )
% 0.28/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectEL_type,type,
% 0.28/0.47 binintersectEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectEL,definition,
% 0.28/0.47 ( binintersectEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectLsub_type,type,
% 0.28/0.47 binintersectLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectLsub,definition,
% 0.28/0.47 ( binintersectLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset2_type,type,
% 0.28/0.47 binintersectSubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset2,definition,
% 0.28/0.47 ( binintersectSubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset3_type,type,
% 0.28/0.47 binintersectSubset3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset3,definition,
% 0.28/0.47 ( binintersectSubset3
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( binintersect @ A @ B )
% 0.28/0.47 = B )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectER_type,type,
% 0.28/0.47 binintersectER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectER,definition,
% 0.28/0.47 ( binintersectER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(disjointsetsI1_type,type,
% 0.28/0.47 disjointsetsI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(disjointsetsI1,definition,
% 0.28/0.47 ( disjointsetsI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( in @ Xx @ B ) )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectRsub_type,type,
% 0.28/0.47 binintersectRsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectRsub,definition,
% 0.28/0.47 ( binintersectRsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset4_type,type,
% 0.28/0.47 binintersectSubset4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset4,definition,
% 0.28/0.47 ( binintersectSubset4
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ B @ A )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset1_type,type,
% 0.28/0.47 binintersectSubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset1,definition,
% 0.28/0.47 ( binintersectSubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( binintersect @ A @ B )
% 0.28/0.47 = A )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(bs114d_type,type,
% 0.28/0.47 bs114d: $o ).
% 0.28/0.47
% 0.28/0.47 thf(bs114d,definition,
% 0.28/0.47 ( bs114d
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.28/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(regular_type,type,
% 0.28/0.47 regular: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminus_type,type,
% 0.28/0.47 setminus: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI_type,type,
% 0.28/0.47 setminusI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI,definition,
% 0.28/0.47 ( setminusI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusEL_type,type,
% 0.28/0.47 setminusEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusEL,definition,
% 0.28/0.47 ( setminusEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusER_type,type,
% 0.28/0.47 setminusER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusER,definition,
% 0.28/0.47 ( setminusER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset2_type,type,
% 0.28/0.47 setminusSubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset2,definition,
% 0.28/0.47 ( setminusSubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusERneg_type,type,
% 0.28/0.47 setminusERneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusERneg,definition,
% 0.28/0.47 ( setminusERneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusELneg_type,type,
% 0.28/0.47 setminusELneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusELneg,definition,
% 0.28/0.47 ( setminusELneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusILneg_type,type,
% 0.28/0.47 setminusILneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusILneg,definition,
% 0.28/0.47 ( setminusILneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusIRneg_type,type,
% 0.28/0.47 setminusIRneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusIRneg,definition,
% 0.28/0.47 ( setminusIRneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusLsub_type,type,
% 0.28/0.47 setminusLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusLsub,definition,
% 0.28/0.47 ( setminusLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1_type,type,
% 0.28/0.47 setminusSubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1,definition,
% 0.28/0.47 ( setminusSubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiff_type,type,
% 0.28/0.47 symdiff: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffE_type,type,
% 0.28/0.47 symdiffE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffE,definition,
% 0.28/0.47 ( symdiffE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI1_type,type,
% 0.28/0.47 symdiffI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI1,definition,
% 0.28/0.47 ( symdiffI1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI2_type,type,
% 0.28/0.47 symdiffI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI2,definition,
% 0.28/0.47 ( symdiffI2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg1_type,type,
% 0.28/0.47 symdiffIneg1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg1,definition,
% 0.28/0.47 ( symdiffIneg1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg2_type,type,
% 0.28/0.47 symdiffIneg2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg2,definition,
% 0.28/0.47 ( symdiffIneg2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iskpair_type,type,
% 0.28/0.47 iskpair: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(secondinupair_type,type,
% 0.28/0.47 secondinupair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(secondinupair,definition,
% 0.28/0.47 ( secondinupair
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIL_type,type,
% 0.28/0.47 setukpairIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIL,definition,
% 0.28/0.47 ( setukpairIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIR_type,type,
% 0.28/0.47 setukpairIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIR,definition,
% 0.28/0.47 ( setukpairIR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpairiskpair_type,type,
% 0.28/0.47 kpairiskpair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairiskpair,definition,
% 0.28/0.47 ( kpairiskpair
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpair_type,type,
% 0.28/0.47 kpair: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(kpairp_type,type,
% 0.28/0.47 kpairp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairp,definition,
% 0.28/0.47 ( kpairp
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprod_type,type,
% 0.28/0.47 cartprod: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsubset_type,type,
% 0.28/0.47 singletonsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsubset,definition,
% 0.28/0.47 ( singletonsubset
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowerset_type,type,
% 0.28/0.47 singletoninpowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowerset,definition,
% 0.28/0.47 ( singletoninpowerset
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowunion_type,type,
% 0.28/0.47 singletoninpowunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowunion,definition,
% 0.28/0.47 ( singletoninpowunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2E_type,type,
% 0.28/0.47 upairset2E: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2E,definition,
% 0.28/0.47 ( upairset2E
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xx )
% 0.28/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsubunion_type,type,
% 0.28/0.47 upairsubunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsubunion,definition,
% 0.28/0.47 ( upairsubunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairinpowunion_type,type,
% 0.28/0.47 upairinpowunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairinpowunion,definition,
% 0.28/0.47 ( upairinpowunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem1_type,type,
% 0.28/0.47 ubforcartprodlem1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem1,definition,
% 0.28/0.47 ( ubforcartprodlem1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem2_type,type,
% 0.28/0.47 ubforcartprodlem2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem2,definition,
% 0.28/0.47 ( ubforcartprodlem2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem3_type,type,
% 0.28/0.47 ubforcartprodlem3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem3,definition,
% 0.28/0.47 ( ubforcartprodlem3
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairin_type,type,
% 0.28/0.47 cartprodpairin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairin,definition,
% 0.28/0.47 ( cartprodpairin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair1_type,type,
% 0.28/0.47 cartprodmempair1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair1,definition,
% 0.28/0.47 ( cartprodmempair1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 & ( Xu
% 0.28/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair_type,type,
% 0.28/0.47 cartprodmempair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair,definition,
% 0.28/0.47 ( cartprodmempair
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE2_type,type,
% 0.28/0.47 setunionE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE2,definition,
% 0.28/0.47 ( setunionE2
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 => ? [X: $i] :
% 0.28/0.47 ( ( in @ X @ A )
% 0.28/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton1_type,type,
% 0.28/0.47 setunionsingleton1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton1,definition,
% 0.28/0.47 ( setunionsingleton1
% 0.28/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton2_type,type,
% 0.28/0.47 setunionsingleton2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton2,definition,
% 0.28/0.47 ( setunionsingleton2
% 0.28/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton_type,type,
% 0.28/0.47 setunionsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton,definition,
% 0.28/0.47 ( setunionsingleton
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.28/0.47 = Xx ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singleton_type,type,
% 0.28/0.47 singleton: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonprop_type,type,
% 0.28/0.47 singletonprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonprop,definition,
% 0.28/0.47 ( singletonprop
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) )
% 0.28/0.47 => ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1_type,type,
% 0.28/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E1_type,type,
% 0.28/0.47 ex1E1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E1,definition,
% 0.28/0.47 ( ex1E1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I_type,type,
% 0.28/0.47 ex1I: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I,definition,
% 0.28/0.47 ( ex1I
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xy = Xx ) ) )
% 0.28/0.47 => ( ex1 @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I2_type,type,
% 0.28/0.47 ex1I2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I2,definition,
% 0.28/0.47 ( ex1I2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) )
% 0.28/0.47 => ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsuniq_type,type,
% 0.28/0.47 singletonsuniq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsuniq,definition,
% 0.28/0.47 ( singletonsuniq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.28/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(atmost1p_type,type,
% 0.28/0.47 atmost1p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(atleast2p_type,type,
% 0.28/0.47 atleast2p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(atmost2p_type,type,
% 0.28/0.47 atmost2p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetp_type,type,
% 0.28/0.47 upairsetp: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL1_type,type,
% 0.28/0.47 setukpairinjL1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL1,definition,
% 0.28/0.47 ( setukpairinjL1
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kfstsingleton_type,type,
% 0.28/0.47 kfstsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kfstsingleton,definition,
% 0.28/0.47 ( kfstsingleton
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.28/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(theprop_type,type,
% 0.28/0.47 theprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(theprop,definition,
% 0.28/0.47 ( theprop
% 0.28/0.47 = ( ! [X: $i] :
% 0.28/0.47 ( ( singleton @ X )
% 0.28/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kfst_type,type,
% 0.28/0.47 kfst: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(kfstpairEq_type,type,
% 0.28/0.47 kfstpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kfstpairEq,definition,
% 0.28/0.47 ( kfstpairEq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xx ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstin_type,type,
% 0.28/0.47 cartprodfstin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstin,definition,
% 0.28/0.47 ( cartprodfstin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL2_type,type,
% 0.28/0.47 setukpairinjL2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL2,definition,
% 0.28/0.47 ( setukpairinjL2
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL_type,type,
% 0.28/0.47 setukpairinjL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL,definition,
% 0.28/0.47 ( setukpairinjL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xz @ Xu ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR11_type,type,
% 0.28/0.47 setukpairinjR11: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR11,definition,
% 0.28/0.47 ( setukpairinjR11
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR12_type,type,
% 0.28/0.47 setukpairinjR12: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR12,definition,
% 0.28/0.47 ( setukpairinjR12
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR1_type,type,
% 0.28/0.47 setukpairinjR1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR1,definition,
% 0.28/0.47 ( setukpairinjR1
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xu )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairequniteq_type,type,
% 0.28/0.47 upairequniteq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairequniteq,definition,
% 0.28/0.47 ( upairequniteq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR2_type,type,
% 0.28/0.47 setukpairinjR2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR2,definition,
% 0.28/0.47 ( setukpairinjR2
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR_type,type,
% 0.28/0.47 setukpairinjR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR,definition,
% 0.28/0.47 ( setukpairinjR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xz @ Xu ) )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ksndsingleton_type,type,
% 0.28/0.47 ksndsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ksndsingleton,definition,
% 0.28/0.47 ( ksndsingleton
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.28/0.47 @ ^ [Xx: $i] :
% 0.28/0.47 ( Xu
% 0.28/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ksnd_type,type,
% 0.28/0.47 ksnd: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(ksndpairEq_type,type,
% 0.28/0.47 ksndpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ksndpairEq,definition,
% 0.28/0.47 ( ksndpairEq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xy ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpairsurjEq_type,type,
% 0.28/0.47 kpairsurjEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairsurjEq,definition,
% 0.28/0.47 ( kpairsurjEq
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.28/0.47 = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndin_type,type,
% 0.28/0.47 cartprodsndin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndin,definition,
% 0.28/0.47 ( cartprodsndin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemEL_type,type,
% 0.28/0.47 cartprodpairmemEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemEL,definition,
% 0.28/0.47 ( cartprodpairmemEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemER_type,type,
% 0.28/0.47 cartprodpairmemER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemER,definition,
% 0.28/0.47 ( cartprodpairmemER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempaircEq_type,type,
% 0.28/0.47 cartprodmempaircEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempaircEq,definition,
% 0.28/0.47 ( cartprodmempaircEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstpairEq_type,type,
% 0.28/0.47 cartprodfstpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstpairEq,definition,
% 0.28/0.47 ( cartprodfstpairEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndpairEq_type,type,
% 0.28/0.47 cartprodsndpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndpairEq,definition,
% 0.28/0.47 ( cartprodsndpairEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairsurjEq_type,type,
% 0.28/0.47 cartprodpairsurjEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairsurjEq,definition,
% 0.28/0.47 ( cartprodpairsurjEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.28/0.47 = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln_type,type,
% 0.28/0.47 breln: $i > $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstr_type,type,
% 0.28/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrI_type,type,
% 0.28/0.47 dpsetconstrI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrI,definition,
% 0.28/0.47 ( dpsetconstrI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrSub_type,type,
% 0.28/0.47 dpsetconstrSub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrSub,definition,
% 0.28/0.47 ( dpsetconstrSub
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.47 ( subset
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.28/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setOfPairsIsBReln_type,type,
% 0.28/0.47 setOfPairsIsBReln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setOfPairsIsBReln,definition,
% 0.28/0.47 ( setOfPairsIsBReln
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.47 ( breln @ A @ B
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrERa_type,type,
% 0.28/0.47 dpsetconstrERa: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrERa,definition,
% 0.28/0.47 ( dpsetconstrERa
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL1_type,type,
% 0.28/0.47 dpsetconstrEL1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL1,definition,
% 0.28/0.47 ( dpsetconstrEL1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL2_type,type,
% 0.28/0.47 dpsetconstrEL2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL2,definition,
% 0.28/0.47 ( dpsetconstrEL2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrER_type,type,
% 0.28/0.47 dpsetconstrER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrER,definition,
% 0.28/0.47 ( dpsetconstrER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(func_type,type,
% 0.28/0.47 func: $i > $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcSet_type,type,
% 0.28/0.47 funcSet: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(funcImageSingleton_type,type,
% 0.28/0.47 funcImageSingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcImageSingleton,definition,
% 0.28/0.47 ( funcImageSingleton
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(apProp_type,type,
% 0.28/0.47 apProp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(apProp,definition,
% 0.28/0.47 ( apProp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in
% 0.28/0.47 @ ( setunion
% 0.28/0.47 @ ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.28/0.47 @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap_type,type,
% 0.28/0.47 ap: $i > $i > $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(app_type,type,
% 0.28/0.47 app: $o ).
% 0.28/0.47
% 0.28/0.47 thf(app,definition,
% 0.28/0.47 ( app
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(infuncsetfunc_type,type,
% 0.28/0.47 infuncsetfunc: $o ).
% 0.28/0.47
% 0.28/0.47 thf(infuncsetfunc,definition,
% 0.28/0.47 ( infuncsetfunc
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2p_type,type,
% 0.28/0.47 ap2p: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2p,definition,
% 0.28/0.47 ( ap2p
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcinfuncset_type,type,
% 0.28/0.47 funcinfuncset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcinfuncset,definition,
% 0.28/0.47 ( funcinfuncset
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lamProp_type,type,
% 0.28/0.47 lamProp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lamProp,definition,
% 0.28/0.47 ( lamProp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( func @ A @ B
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lam_type,type,
% 0.28/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(lamp_type,type,
% 0.28/0.47 lamp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lamp,definition,
% 0.28/0.47 ( lamp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( func @ A @ B
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lam2p_type,type,
% 0.28/0.47 lam2p: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lam2p,definition,
% 0.28/0.47 ( lam2p
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( in
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.28/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall1_type,type,
% 0.28/0.47 brelnall1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall1,definition,
% 0.28/0.47 ( brelnall1
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ R )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall2_type,type,
% 0.28/0.47 brelnall2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall2,definition,
% 0.28/0.47 ( brelnall2
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ R )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E2_type,type,
% 0.28/0.47 ex1E2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E2,definition,
% 0.28/0.47 ( ex1E2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp1_type,type,
% 0.28/0.47 funcGraphProp1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp1,definition,
% 0.28/0.47 ( funcGraphProp1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp3_type,type,
% 0.28/0.47 funcGraphProp3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp3,definition,
% 0.28/0.47 ( funcGraphProp3
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp2_type,type,
% 0.28/0.47 funcGraphProp2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp2,definition,
% 0.28/0.47 ( funcGraphProp2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcextLem_type,type,
% 0.28/0.47 funcextLem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcextLem,definition,
% 0.28/0.47 ( funcextLem
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xg )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp4_type,type,
% 0.28/0.47 funcGraphProp4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp4,definition,
% 0.28/0.47 ( funcGraphProp4
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subbreln_type,type,
% 0.28/0.47 subbreln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subbreln,definition,
% 0.28/0.47 ( subbreln
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ S )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqbreln_type,type,
% 0.28/0.47 eqbreln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqbreln,definition,
% 0.28/0.47 ( eqbreln
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ S )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.28/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcext_type,type,
% 0.28/0.47 funcext: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcext,definition,
% 0.28/0.47 ( funcext
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xg )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcext2_type,type,
% 0.28/0.47 funcext2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcext2,definition,
% 0.28/0.47 ( funcext2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq1_type,type,
% 0.28/0.47 ap2apEq1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq1,definition,
% 0.28/0.47 ( ap2apEq1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq2_type,type,
% 0.28/0.47 ap2apEq2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq2,definition,
% 0.28/0.47 ( ap2apEq2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(beta1_type,type,
% 0.28/0.47 beta1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(beta1,definition,
% 0.28/0.47 ( beta1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.28/0.47 @ Xx )
% 0.28/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eta1_type,type,
% 0.28/0.47 eta1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eta1,definition,
% 0.28/0.47 ( eta1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ( ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.28/0.47 = Xf ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lam2lamEq_type,type,
% 0.28/0.47 lam2lamEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lam2lamEq,definition,
% 0.28/0.47 ( lam2lamEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.28/0.47 = ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(beta2_type,type,
% 0.28/0.47 beta2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(beta2,definition,
% 0.28/0.47 ( beta2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.28/0.47 @ Xx )
% 0.28/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eta2_type,type,
% 0.28/0.47 eta2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eta2,definition,
% 0.28/0.47 ( eta2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ( ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.28/0.47 = Xf ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iffalseProp1_type,type,
% 0.28/0.47 iffalseProp1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iffalseProp1,definition,
% 0.28/0.47 ( iffalseProp1
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ~ Xphi
% 0.28/0.47 => ( in @ Xy
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iffalseProp2_type,type,
% 0.28/0.47 iffalseProp2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iffalseProp2,definition,
% 0.28/0.47 ( iffalseProp2
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ~ Xphi
% 0.28/0.47 => ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) )
% 0.28/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueProp1_type,type,
% 0.28/0.47 iftrueProp1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueProp1,definition,
% 0.28/0.47 ( iftrueProp1
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( Xphi
% 0.28/0.47 => ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueProp2_type,type,
% 0.28/0.47 iftrueProp2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueProp2,definition,
% 0.28/0.47 ( iftrueProp2
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( Xphi
% 0.28/0.47 => ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) )
% 0.28/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ifSingleton_type,type,
% 0.28/0.47 ifSingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ifSingleton,definition,
% 0.28/0.47 ( ifSingleton
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(if_type,type,
% 0.28/0.47 if: $i > $o > $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(ifp_type,type,
% 0.28/0.47 ifp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ifp,definition,
% 0.28/0.47 ( ifp
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(theeq_type,type,
% 0.28/0.47 theeq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(theeq,definition,
% 0.28/0.47 ( theeq
% 0.28/0.47 = ( ! [X: $i] :
% 0.28/0.47 ( ( singleton @ X )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ X )
% 0.28/0.47 => ( ( setunion @ X )
% 0.28/0.47 = Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iftrue_type,type,
% 0.28/0.47 iftrue: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iftrue,definition,
% 0.28/0.47 ( iftrue
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( Xphi
% 0.28/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.28/0.47 = Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iffalse_type,type,
% 0.28/0.47 iffalse: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iffalse,definition,
% 0.28/0.47 ( iffalse
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ~ Xphi
% 0.28/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.28/0.47 = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueorfalse_type,type,
% 0.28/0.47 iftrueorfalse: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueorfalse,definition,
% 0.28/0.47 ( iftrueorfalse
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectT_lem_type,type,
% 0.28/0.47 binintersectT_lem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectT_lem,definition,
% 0.28/0.47 ( binintersectT_lem
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionT_lem_type,type,
% 0.28/0.47 binunionT_lem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionT_lem,definition,
% 0.28/0.47 ( binunionT_lem
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetT_lem_type,type,
% 0.28/0.47 powersetT_lem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetT_lem,definition,
% 0.28/0.47 ( powersetT_lem
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusT_lem_type,type,
% 0.28/0.47 setminusT_lem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusT_lem,definition,
% 0.28/0.47 ( setminusT_lem
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementT_lem_type,type,
% 0.28/0.47 complementT_lem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementT_lem,definition,
% 0.28/0.47 ( complementT_lem
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setextT_type,type,
% 0.28/0.47 setextT: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setextT,definition,
% 0.28/0.47 ( setextT
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ X )
% 0.28/0.47 => ( in @ Xx @ Y ) ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ Y )
% 0.28/0.47 => ( in @ Xx @ X ) ) )
% 0.28/0.47 => ( X = Y ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTI_type,type,
% 0.28/0.47 subsetTI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTI,definition,
% 0.28/0.47 ( subsetTI
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ X )
% 0.28/0.47 => ( in @ Xx @ Y ) ) )
% 0.28/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetTI1_type,type,
% 0.28/0.47 powersetTI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetTI1,definition,
% 0.28/0.47 ( powersetTI1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ X )
% 0.28/0.47 => ( in @ Xx @ Y ) ) )
% 0.28/0.47 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetTE1_type,type,
% 0.28/0.47 powersetTE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetTE1,definition,
% 0.28/0.47 ( powersetTE1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ X @ ( powerset @ Y ) )
% 0.28/0.47 => ( ( in @ Xx @ X )
% 0.28/0.47 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementTI1_type,type,
% 0.28/0.47 complementTI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementTI1,definition,
% 0.28/0.47 ( complementTI1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ X )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementTE1_type,type,
% 0.28/0.47 complementTE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementTE1,definition,
% 0.28/0.47 ( complementTE1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectTELcontra_type,type,
% 0.28/0.47 binintersectTELcontra: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectTELcontra,definition,
% 0.28/0.47 ( binintersectTELcontra
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ X )
% 0.28/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectTERcontra_type,type,
% 0.28/0.47 binintersectTERcontra: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectTERcontra,definition,
% 0.28/0.47 ( binintersectTERcontra
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ Y )
% 0.28/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(contrasubsetT_type,type,
% 0.28/0.47 contrasubsetT: $o ).
% 0.28/0.47
% 0.28/0.47 thf(contrasubsetT,definition,
% 0.28/0.47 ( contrasubsetT
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.47 => ( ( in @ Xx @ Y )
% 0.28/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(contrasubsetT1_type,type,
% 0.28/0.47 contrasubsetT1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(contrasubsetT1,definition,
% 0.28/0.47 ( contrasubsetT1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( subset @ X @ Y )
% 0.28/0.47 => ( ~ ( in @ Xx @ Y )
% 0.28/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(contrasubsetT2_type,type,
% 0.28/0.47 contrasubsetT2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(contrasubsetT2,definition,
% 0.28/0.47 ( contrasubsetT2
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ X @ Y )
% 0.28/0.47 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(contrasubsetT3_type,type,
% 0.28/0.47 contrasubsetT3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(contrasubsetT3,definition,
% 0.28/0.47 ( contrasubsetT3
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.28/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementI1_type,type,
% 0.28/0.47 doubleComplementI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementI1,definition,
% 0.28/0.47 ( doubleComplementI1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ X )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementE1_type,type,
% 0.28/0.47 doubleComplementE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementE1,definition,
% 0.28/0.47 ( doubleComplementE1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.28/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementSub1_type,type,
% 0.28/0.47 doubleComplementSub1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementSub1,definition,
% 0.28/0.47 ( doubleComplementSub1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementSub2_type,type,
% 0.28/0.47 doubleComplementSub2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementSub2,definition,
% 0.28/0.47 ( doubleComplementSub2
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementEq_type,type,
% 0.28/0.47 doubleComplementEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(doubleComplementEq,definition,
% 0.28/0.47 ( doubleComplementEq
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ( X
% 0.28/0.47 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementTnotintersectT_type,type,
% 0.28/0.47 complementTnotintersectT: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementTnotintersectT,definition,
% 0.28/0.47 ( complementTnotintersectT
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementImpComplementIntersect_type,type,
% 0.28/0.47 complementImpComplementIntersect: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementImpComplementIntersect,definition,
% 0.28/0.47 ( complementImpComplementIntersect
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementSubsetComplementIntersect_type,type,
% 0.28/0.47 complementSubsetComplementIntersect: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementSubsetComplementIntersect,definition,
% 0.28/0.47 ( complementSubsetComplementIntersect
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementInPowersetComplementIntersect_type,type,
% 0.28/0.47 complementInPowersetComplementIntersect: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementInPowersetComplementIntersect,definition,
% 0.28/0.47 ( complementInPowersetComplementIntersect
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(contraSubsetComplement_type,type,
% 0.28/0.47 contraSubsetComplement: $o ).
% 0.28/0.47
% 0.28/0.47 thf(contraSubsetComplement,definition,
% 0.28/0.47 ( contraSubsetComplement
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ Y )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementTcontraSubset_type,type,
% 0.28/0.47 complementTcontraSubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementTcontraSubset,definition,
% 0.28/0.47 ( complementTcontraSubset
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.47 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionTILcontra_type,type,
% 0.28/0.47 binunionTILcontra: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionTILcontra,definition,
% 0.28/0.47 ( binunionTILcontra
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionTIRcontra_type,type,
% 0.28/0.47 binunionTIRcontra: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionTIRcontra,definition,
% 0.28/0.47 ( binunionTIRcontra
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.47 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inIntersectImpInUnion_type,type,
% 0.28/0.47 inIntersectImpInUnion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inIntersectImpInUnion,definition,
% 0.28/0.47 ( inIntersectImpInUnion
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inIntersectImpInUnion2_type,type,
% 0.28/0.47 inIntersectImpInUnion2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inIntersectImpInUnion2,definition,
% 0.28/0.47 ( inIntersectImpInUnion2
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inIntersectImpInIntersectUnions_type,type,
% 0.28/0.47 inIntersectImpInIntersectUnions: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inIntersectImpInIntersectUnions,definition,
% 0.28/0.47 ( inIntersectImpInIntersectUnions
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.47 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(intersectInPowersetIntersectUnions_type,type,
% 0.28/0.47 intersectInPowersetIntersectUnions: $o ).
% 0.28/0.47
% 0.28/0.47 thf(intersectInPowersetIntersectUnions,definition,
% 0.28/0.47 ( intersectInPowersetIntersectUnions
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inComplementUnionImpNotIn1_type,type,
% 0.28/0.47 inComplementUnionImpNotIn1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inComplementUnionImpNotIn1,definition,
% 0.28/0.47 ( inComplementUnionImpNotIn1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inComplementUnionImpInComplement1_type,type,
% 0.28/0.47 inComplementUnionImpInComplement1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inComplementUnionImpInComplement1,definition,
% 0.28/0.47 ( inComplementUnionImpInComplement1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionTE_type,type,
% 0.28/0.47 binunionTE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionTE,definition,
% 0.28/0.47 ( binunionTE
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.47 => ( ( ( in @ Xx @ X )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ Xx @ Y )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionTEcontra_type,type,
% 0.28/0.47 binunionTEcontra: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionTEcontra,definition,
% 0.28/0.47 ( binunionTEcontra
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ X )
% 0.28/0.47 => ( ~ ( in @ Xx @ Y )
% 0.28/0.47 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2a1_type,type,
% 0.28/0.47 demorgan2a1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2a1,definition,
% 0.28/0.47 ( demorgan2a1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(complementUnionInPowersetComplement_type,type,
% 0.28/0.47 complementUnionInPowersetComplement: $o ).
% 0.28/0.47
% 0.28/0.47 thf(complementUnionInPowersetComplement,definition,
% 0.28/0.47 ( complementUnionInPowersetComplement
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2a2_type,type,
% 0.28/0.47 demorgan2a2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2a2,definition,
% 0.28/0.47 ( demorgan2a2
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan1a_type,type,
% 0.28/0.47 demorgan1a: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan1a,definition,
% 0.28/0.47 ( demorgan1a
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan1b_type,type,
% 0.28/0.47 demorgan1b: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan1b,definition,
% 0.28/0.47 ( demorgan1b
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan1_type,type,
% 0.28/0.47 demorgan1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan1,definition,
% 0.28/0.47 ( demorgan1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.28/0.47 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2a_type,type,
% 0.28/0.47 demorgan2a: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2a,definition,
% 0.28/0.47 ( demorgan2a
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.47 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2b2_type,type,
% 0.28/0.47 demorgan2b2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2b2,definition,
% 0.28/0.47 ( demorgan2b2
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.47 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2b_type,type,
% 0.28/0.47 demorgan2b: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2b,definition,
% 0.28/0.47 ( demorgan2b
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2_type,type,
% 0.28/0.47 demorgan2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(demorgan2,definition,
% 0.28/0.47 ( demorgan2
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.28/0.47 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule0_type,type,
% 0.28/0.47 woz13rule0: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule0,definition,
% 0.28/0.47 ( woz13rule0
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule1_type,type,
% 0.28/0.47 woz13rule1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule1,definition,
% 0.28/0.47 ( woz13rule1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ X @ Z )
% 0.28/0.47 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule2_type,type,
% 0.28/0.47 woz13rule2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule2,definition,
% 0.28/0.47 ( woz13rule2
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ Y @ Z )
% 0.28/0.47 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule3_type,type,
% 0.28/0.47 woz13rule3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule3,definition,
% 0.28/0.47 ( woz13rule3
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ X @ Y )
% 0.28/0.47 => ( ( subset @ X @ Z )
% 0.28/0.47 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule4_type,type,
% 0.28/0.47 woz13rule4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz13rule4,definition,
% 0.28/0.47 ( woz13rule4
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ! [W: $i] :
% 0.28/0.47 ( ( in @ W @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ X @ Z )
% 0.28/0.47 => ( ( subset @ Y @ W )
% 0.28/0.47 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_1_type,type,
% 0.28/0.47 woz1_1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_1,definition,
% 0.28/0.47 ( woz1_1
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_2_type,type,
% 0.28/0.47 woz1_2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_2,definition,
% 0.28/0.47 ( woz1_2
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ! [W: $i] :
% 0.28/0.47 ( ( in @ W @ ( powerset @ A ) )
% 0.28/0.47 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.28/0.47 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_3_type,type,
% 0.28/0.47 woz1_3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_3,definition,
% 0.28/0.47 ( woz1_3
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ! [Z: $i] :
% 0.28/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_4_type,type,
% 0.28/0.47 woz1_4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_4,definition,
% 0.28/0.47 ( woz1_4
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.47 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_5_type,type,
% 0.28/0.47 woz1_5: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz1_5,definition,
% 0.28/0.47 ( woz1_5
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1_type,type,
% 0.28/0.47 breln1: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1all2_type,type,
% 0.28/0.47 breln1all2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1all2,definition,
% 0.28/0.47 ( breln1all2
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ R )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1Set_type,type,
% 0.28/0.47 breln1Set: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(breln1SetBreln1_type,type,
% 0.28/0.47 breln1SetBreln1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1SetBreln1,definition,
% 0.28/0.47 ( breln1SetBreln1
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( in @ R @ ( breln1Set @ A ) )
% 0.28/0.47 => ( breln1 @ A @ R ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(transitive_type,type,
% 0.28/0.47 transitive: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(antisymmetric_type,type,
% 0.28/0.47 antisymmetric: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(reflexive_type,type,
% 0.28/0.47 reflexive: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(refltransitive_type,type,
% 0.28/0.47 refltransitive: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(refllinearorder_type,type,
% 0.28/0.47 refllinearorder: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(reflwellordering_type,type,
% 0.28/0.47 reflwellordering: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(choice2fnsingleton_type,type,
% 0.28/0.47 choice2fnsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(choice2fnsingleton,definition,
% 0.28/0.47 ( choice2fnsingleton
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 & ( Xphi @ Xx @ Xy ) ) )
% 0.28/0.47 => ! [R: $i] :
% 0.28/0.47 ( ( in @ R @ ( breln1Set @ B ) )
% 0.28/0.47 => ( ( reflwellordering @ B @ R )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx @ Xy )
% 0.28/0.47 & ! [Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ B )
% 0.28/0.47 => ( ( Xphi @ Xx @ Xz )
% 0.28/0.47 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setOfPairsIsBReln1_type,type,
% 0.28/0.47 setOfPairsIsBReln1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setOfPairsIsBReln1,definition,
% 0.28/0.47 ( setOfPairsIsBReln1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.28/0.47 ( breln1 @ A
% 0.28/0.47 @ ( dpsetconstr @ A @ A
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1all1_type,type,
% 0.28/0.47 breln1all1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1all1,definition,
% 0.28/0.47 ( breln1all1
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ R )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subbreln1_type,type,
% 0.28/0.47 subbreln1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subbreln1,definition,
% 0.28/0.47 ( subbreln1
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqbreln1_type,type,
% 0.28/0.47 eqbreln1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqbreln1,definition,
% 0.28/0.47 ( eqbreln1
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.28/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1invset_type,type,
% 0.28/0.47 breln1invset: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(breln1invprop_type,type,
% 0.28/0.47 breln1invprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1invprop,definition,
% 0.28/0.47 ( breln1invprop
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1invI_type,type,
% 0.28/0.47 breln1invI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1invI,definition,
% 0.28/0.47 ( breln1invI
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1invE_type,type,
% 0.28/0.47 breln1invE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1invE,definition,
% 0.28/0.47 ( breln1invE
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compset_type,type,
% 0.28/0.47 breln1compset: $i > $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compprop_type,type,
% 0.28/0.47 breln1compprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compprop,definition,
% 0.28/0.47 ( breln1compprop
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compI_type,type,
% 0.28/0.47 breln1compI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compI,definition,
% 0.28/0.47 ( breln1compI
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ! [Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.28/0.47 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compE_type,type,
% 0.28/0.47 breln1compE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compE,definition,
% 0.28/0.47 ( breln1compE
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.28/0.47 => ? [Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ A )
% 0.28/0.47 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.28/0.47 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compEex_type,type,
% 0.28/0.47 breln1compEex: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1compEex,definition,
% 0.28/0.47 ( breln1compEex
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ! [Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.28/0.47 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.28/0.47 => Xphi ) ) )
% 0.28/0.47 => Xphi ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionprop_type,type,
% 0.28/0.47 breln1unionprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionprop,definition,
% 0.28/0.47 ( breln1unionprop
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionIL_type,type,
% 0.28/0.47 breln1unionIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionIL,definition,
% 0.28/0.47 ( breln1unionIL
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionIR_type,type,
% 0.28/0.47 breln1unionIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionIR,definition,
% 0.28/0.47 ( breln1unionIR
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionI_type,type,
% 0.28/0.47 breln1unionI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionI,definition,
% 0.28/0.47 ( breln1unionI
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionE_type,type,
% 0.28/0.47 breln1unionE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionE,definition,
% 0.28/0.47 ( breln1unionE
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionEcases_type,type,
% 0.28/0.47 breln1unionEcases: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionEcases,definition,
% 0.28/0.47 ( breln1unionEcases
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionCommutes_type,type,
% 0.28/0.47 breln1unionCommutes: $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln1unionCommutes,definition,
% 0.28/0.47 ( breln1unionCommutes
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ( ( binunion @ R @ S )
% 0.28/0.47 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz2Ex_type,type,
% 0.28/0.47 woz2Ex: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz2Ex,definition,
% 0.28/0.47 ( woz2Ex
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ( R
% 0.28/0.47 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz2W_type,type,
% 0.28/0.47 woz2W: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz2W,definition,
% 0.28/0.47 ( woz2W
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.28/0.47 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz2A_type,type,
% 0.28/0.47 woz2A: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz2A,definition,
% 0.28/0.47 ( woz2A
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [T: $i] :
% 0.28/0.47 ( ( breln1 @ A @ T )
% 0.28/0.47 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.28/0.47 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(woz2B_type,type,
% 0.28/0.47 woz2B: $o ).
% 0.28/0.47
% 0.28/0.47 thf(woz2B,definition,
% 0.28/0.47 ( woz2B
% 0.28/0.47 = ( ! [A: $i,R: $i] :
% 0.28/0.47 ( ( breln1 @ A @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln1 @ A @ S )
% 0.28/0.47 => ! [T: $i] :
% 0.28/0.47 ( ( breln1 @ A @ T )
% 0.28/0.47 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.28/0.47 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(image1Ex_type,type,
% 0.28/0.47 image1Ex: $o ).
% 0.28/0.47
% 0.28/0.47 thf(image1Ex,definition,
% 0.28/0.47 ( image1Ex
% 0.28/0.47 = ( ! [A: $i,Xf: $i > $i] :
% 0.28/0.47 ? [B: $i] :
% 0.28/0.47 ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 <=> ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 & ( Xx
% 0.28/0.47 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(image1Ex1_type,type,
% 0.28/0.47 image1Ex1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(image1Ex1,definition,
% 0.28/0.47 ( image1Ex1
% 0.28/0.47 = ( ! [A: $i,Xf: $i > $i] :
% 0.28/0.47 ( exu
% 0.28/0.47 @ ^ [B: $i] :
% 0.28/0.47 ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 <=> ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 & ( Xx
% 0.28/0.47 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(image1_type,type,
% 0.28/0.47 image1: $i > ( $i > $i ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(image1Equiv_type,type,
% 0.28/0.47 image1Equiv: $o ).
% 0.28/0.47
% 0.28/0.47 thf(image1Equiv,definition,
% 0.28/0.47 ( image1Equiv
% 0.28/0.47 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx
% 0.28/0.48 @ ( image1 @ A
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.28/0.48 <=> ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( Xx
% 0.28/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(image1E_type,type,
% 0.28/0.48 image1E: $o ).
% 0.28/0.48
% 0.28/0.48 thf(image1E,definition,
% 0.28/0.48 ( image1E
% 0.28/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx
% 0.28/0.48 @ ( image1 @ A
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.28/0.48 => ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( Xx
% 0.28/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(image1I_type,type,
% 0.28/0.48 image1I: $o ).
% 0.28/0.48
% 0.28/0.48 thf(image1I,definition,
% 0.28/0.48 ( image1I
% 0.28/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.28/0.48 ( ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( Xx
% 0.28/0.48 = ( Xf @ Xy ) ) )
% 0.28/0.48 => ( in @ Xx
% 0.28/0.48 @ ( image1 @ A
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(injective_type,type,
% 0.28/0.48 injective: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSet_type,type,
% 0.28/0.48 injFuncSet: $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncInInjFuncSet_type,type,
% 0.28/0.48 injFuncInInjFuncSet: $o ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncInInjFuncSet,definition,
% 0.28/0.48 ( injFuncInInjFuncSet
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ( ( injective @ A @ B @ Xf )
% 0.28/0.48 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSetFuncIn_type,type,
% 0.28/0.48 injFuncSetFuncIn: $o ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSetFuncIn,definition,
% 0.28/0.48 ( injFuncSetFuncIn
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.28/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSetFuncInj_type,type,
% 0.28/0.48 injFuncSetFuncInj: $o ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSetFuncInj,definition,
% 0.28/0.48 ( injFuncSetFuncInj
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.28/0.48 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(surjective_type,type,
% 0.28/0.48 surjective: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(surjective,definition,
% 0.28/0.48 ( surjective
% 0.28/0.48 = ( ^ [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ B )
% 0.28/0.48 => ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( ( ap @ A @ B @ Xf @ Xy )
% 0.28/0.48 = Xx ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(surjFuncSet_type,type,
% 0.28/0.48 surjFuncSet: $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(surjFuncSet,definition,
% 0.28/0.48 ( surjFuncSet
% 0.28/0.48 = ( ^ [A: $i,B: $i] :
% 0.28/0.48 ( dsetconstr @ ( funcSet @ A @ B )
% 0.28/0.48 @ ^ [Xf: $i] : ( surjective @ A @ B @ Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(surjFuncSetFuncIn,conjecture,
% 0.28/0.48 ( setextAx
% 0.28/0.48 => ( emptysetAx
% 0.28/0.48 => ( setadjoinAx
% 0.28/0.48 => ( powersetAx
% 0.28/0.48 => ( setunionAx
% 0.28/0.48 => ( omega0Ax
% 0.28/0.48 => ( omegaSAx
% 0.28/0.48 => ( omegaIndAx
% 0.28/0.48 => ( replAx
% 0.28/0.48 => ( foundationAx
% 0.28/0.48 => ( wellorderingAx
% 0.28/0.48 => ( descrp
% 0.28/0.48 => ( dsetconstrI
% 0.28/0.48 => ( dsetconstrEL
% 0.28/0.48 => ( dsetconstrER
% 0.28/0.48 => ( exuE1
% 0.28/0.48 => ( prop2setE
% 0.28/0.48 => ( emptysetE
% 0.28/0.48 => ( emptysetimpfalse
% 0.28/0.48 => ( notinemptyset
% 0.28/0.48 => ( exuE3e
% 0.28/0.48 => ( setext
% 0.28/0.48 => ( emptyI
% 0.28/0.48 => ( noeltsimpempty
% 0.28/0.48 => ( setbeta
% 0.28/0.48 => ( nonemptyE1
% 0.28/0.48 => ( nonemptyI
% 0.28/0.48 => ( nonemptyI1
% 0.28/0.48 => ( setadjoinIL
% 0.28/0.48 => ( emptyinunitempty
% 0.28/0.48 => ( setadjoinIR
% 0.28/0.48 => ( setadjoinE
% 0.28/0.48 => ( setadjoinOr
% 0.28/0.48 => ( setoftrueEq
% 0.28/0.48 => ( powersetI
% 0.28/0.48 => ( emptyinPowerset
% 0.28/0.48 => ( emptyInPowerset
% 0.28/0.48 => ( powersetE
% 0.28/0.48 => ( setunionI
% 0.28/0.48 => ( setunionE
% 0.28/0.48 => ( subPowSU
% 0.28/0.48 => ( exuE2
% 0.28/0.48 => ( nonemptyImpWitness
% 0.28/0.48 => ( uniqinunit
% 0.28/0.48 => ( notinsingleton
% 0.28/0.48 => ( eqinunit
% 0.28/0.48 => ( singletonsswitch
% 0.28/0.48 => ( upairsetE
% 0.28/0.48 => ( upairsetIL
% 0.28/0.48 => ( upairsetIR
% 0.28/0.48 => ( emptyE1
% 0.28/0.48 => ( vacuousDall
% 0.28/0.48 => ( quantDeMorgan1
% 0.28/0.48 => ( quantDeMorgan2
% 0.28/0.48 => ( quantDeMorgan3
% 0.28/0.48 => ( quantDeMorgan4
% 0.28/0.48 => ( prop2setI
% 0.28/0.48 => ( prop2set2propI
% 0.28/0.48 => ( notdexE
% 0.28/0.48 => ( notdallE
% 0.28/0.48 => ( exuI1
% 0.28/0.48 => ( exuI3
% 0.28/0.48 => ( exuI2
% 0.28/0.48 => ( inCongP
% 0.28/0.48 => ( in__Cong
% 0.28/0.48 => ( exuE3u
% 0.28/0.48 => ( exu__Cong
% 0.28/0.48 => ( emptyset__Cong
% 0.28/0.48 => ( setadjoin__Cong
% 0.28/0.48 => ( powerset__Cong
% 0.28/0.48 => ( setunion__Cong
% 0.28/0.48 => ( omega__Cong
% 0.28/0.48 => ( exuEu
% 0.28/0.48 => ( descr__Cong
% 0.28/0.48 => ( dsetconstr__Cong
% 0.28/0.48 => ( subsetI1
% 0.28/0.48 => ( eqimpsubset2
% 0.28/0.48 => ( eqimpsubset1
% 0.28/0.48 => ( subsetI2
% 0.28/0.48 => ( emptysetsubset
% 0.28/0.48 => ( subsetE
% 0.28/0.48 => ( subsetE2
% 0.28/0.48 => ( notsubsetI
% 0.28/0.48 => ( notequalI1
% 0.28/0.48 => ( notequalI2
% 0.28/0.48 => ( subsetRefl
% 0.28/0.48 => ( subsetTrans
% 0.28/0.48 => ( setadjoinSub
% 0.28/0.48 => ( setadjoinSub2
% 0.28/0.48 => ( subset2powerset
% 0.28/0.48 => ( setextsub
% 0.28/0.48 => ( subsetemptysetimpeq
% 0.28/0.48 => ( powersetI1
% 0.28/0.48 => ( powersetE1
% 0.28/0.48 => ( inPowerset
% 0.28/0.48 => ( powersetsubset
% 0.28/0.48 => ( sepInPowerset
% 0.28/0.48 => ( sepSubset
% 0.28/0.48 => ( binunionIL
% 0.28/0.48 => ( upairset2IR
% 0.28/0.48 => ( binunionIR
% 0.28/0.48 => ( binunionEcases
% 0.28/0.48 => ( binunionE
% 0.28/0.48 => ( binunionLsub
% 0.28/0.48 => ( binunionRsub
% 0.28/0.48 => ( binintersectI
% 0.28/0.48 => ( binintersectSubset5
% 0.28/0.48 => ( binintersectEL
% 0.28/0.48 => ( binintersectLsub
% 0.28/0.48 => ( binintersectSubset2
% 0.28/0.48 => ( binintersectSubset3
% 0.28/0.48 => ( binintersectER
% 0.28/0.48 => ( disjointsetsI1
% 0.28/0.48 => ( binintersectRsub
% 0.28/0.48 => ( binintersectSubset4
% 0.28/0.48 => ( binintersectSubset1
% 0.28/0.48 => ( bs114d
% 0.28/0.48 => ( setminusI
% 0.28/0.48 => ( setminusEL
% 0.28/0.48 => ( setminusER
% 0.28/0.48 => ( setminusSubset2
% 0.28/0.48 => ( setminusERneg
% 0.28/0.48 => ( setminusELneg
% 0.28/0.48 => ( setminusILneg
% 0.28/0.48 => ( setminusIRneg
% 0.28/0.48 => ( setminusLsub
% 0.28/0.48 => ( setminusSubset1
% 0.28/0.48 => ( symdiffE
% 0.28/0.48 => ( symdiffI1
% 0.28/0.48 => ( symdiffI2
% 0.28/0.48 => ( symdiffIneg1
% 0.28/0.48 => ( symdiffIneg2
% 0.28/0.48 => ( secondinupair
% 0.28/0.48 => ( setukpairIL
% 0.28/0.48 => ( setukpairIR
% 0.28/0.48 => ( kpairiskpair
% 0.28/0.48 => ( kpairp
% 0.28/0.48 => ( singletonsubset
% 0.28/0.48 => ( singletoninpowerset
% 0.28/0.48 => ( singletoninpowunion
% 0.28/0.48 => ( upairset2E
% 0.28/0.48 => ( upairsubunion
% 0.28/0.48 => ( upairinpowunion
% 0.28/0.48 => ( ubforcartprodlem1
% 0.28/0.48 => ( ubforcartprodlem2
% 0.28/0.48 => ( ubforcartprodlem3
% 0.28/0.48 => ( cartprodpairin
% 0.28/0.48 => ( cartprodmempair1
% 0.28/0.48 => ( cartprodmempair
% 0.28/0.48 => ( setunionE2
% 0.28/0.48 => ( setunionsingleton1
% 0.28/0.48 => ( setunionsingleton2
% 0.28/0.48 => ( setunionsingleton
% 0.28/0.48 => ( singletonprop
% 0.28/0.48 => ( ex1E1
% 0.28/0.48 => ( ex1I
% 0.28/0.48 => ( ex1I2
% 0.28/0.48 => ( singletonsuniq
% 0.28/0.48 => ( setukpairinjL1
% 0.28/0.48 => ( kfstsingleton
% 0.28/0.48 => ( theprop
% 0.28/0.48 => ( kfstpairEq
% 0.28/0.48 => ( cartprodfstin
% 0.28/0.48 => ( setukpairinjL2
% 0.28/0.48 => ( setukpairinjL
% 0.28/0.48 => ( setukpairinjR11
% 0.28/0.48 => ( setukpairinjR12
% 0.28/0.48 => ( setukpairinjR1
% 0.28/0.48 => ( upairequniteq
% 0.28/0.48 => ( setukpairinjR2
% 0.28/0.48 => ( setukpairinjR
% 0.28/0.48 => ( ksndsingleton
% 0.28/0.48 => ( ksndpairEq
% 0.28/0.48 => ( kpairsurjEq
% 0.28/0.48 => ( cartprodsndin
% 0.28/0.48 => ( cartprodpairmemEL
% 0.28/0.48 => ( cartprodpairmemER
% 0.28/0.48 => ( cartprodmempaircEq
% 0.28/0.48 => ( cartprodfstpairEq
% 0.28/0.48 => ( cartprodsndpairEq
% 0.28/0.48 => ( cartprodpairsurjEq
% 0.28/0.48 => ( dpsetconstrI
% 0.28/0.48 => ( dpsetconstrSub
% 0.28/0.48 => ( setOfPairsIsBReln
% 0.28/0.48 => ( dpsetconstrERa
% 0.28/0.48 => ( dpsetconstrEL1
% 0.28/0.48 => ( dpsetconstrEL2
% 0.28/0.48 => ( dpsetconstrER
% 0.28/0.48 => ( funcImageSingleton
% 0.28/0.48 => ( apProp
% 0.28/0.48 => ( app
% 0.28/0.48 => ( infuncsetfunc
% 0.28/0.48 => ( ap2p
% 0.28/0.48 => ( funcinfuncset
% 0.28/0.48 => ( lamProp
% 0.28/0.48 => ( lamp
% 0.28/0.48 => ( lam2p
% 0.28/0.48 => ( brelnall1
% 0.28/0.48 => ( brelnall2
% 0.28/0.48 => ( ex1E2
% 0.28/0.48 => ( funcGraphProp1
% 0.28/0.48 => ( funcGraphProp3
% 0.28/0.48 => ( funcGraphProp2
% 0.28/0.48 => ( funcextLem
% 0.28/0.48 => ( funcGraphProp4
% 0.28/0.48 => ( subbreln
% 0.28/0.48 => ( eqbreln
% 0.28/0.48 => ( funcext
% 0.28/0.48 => ( funcext2
% 0.28/0.48 => ( ap2apEq1
% 0.28/0.48 => ( ap2apEq2
% 0.28/0.48 => ( beta1
% 0.28/0.48 => ( eta1
% 0.28/0.48 => ( lam2lamEq
% 0.28/0.48 => ( beta2
% 0.28/0.48 => ( eta2
% 0.28/0.48 => ( iffalseProp1
% 0.28/0.48 => ( iffalseProp2
% 0.28/0.48 => ( iftrueProp1
% 0.28/0.48 => ( iftrueProp2
% 0.28/0.48 => ( ifSingleton
% 0.28/0.48 => ( ifp
% 0.28/0.48 => ( theeq
% 0.28/0.48 => ( iftrue
% 0.28/0.48 => ( iffalse
% 0.28/0.48 => ( iftrueorfalse
% 0.28/0.48 => ( binintersectT_lem
% 0.28/0.48 => ( binunionT_lem
% 0.28/0.48 => ( powersetT_lem
% 0.28/0.48 => ( setminusT_lem
% 0.28/0.48 => ( complementT_lem
% 0.28/0.48 => ( setextT
% 0.28/0.48 => ( subsetTI
% 0.28/0.48 => ( powersetTI1
% 0.28/0.48 => ( powersetTE1
% 0.28/0.48 => ( complementTI1
% 0.28/0.48 => ( complementTE1
% 0.28/0.48 => ( binintersectTELcontra
% 0.28/0.48 => ( binintersectTERcontra
% 0.28/0.48 => ( contrasubsetT
% 0.28/0.48 => ( contrasubsetT1
% 0.28/0.48 => ( contrasubsetT2
% 0.28/0.48 => ( contrasubsetT3
% 0.28/0.48 => ( doubleComplementI1
% 0.28/0.48 => ( doubleComplementE1
% 0.28/0.48 => ( doubleComplementSub1
% 0.28/0.48 => ( doubleComplementSub2
% 0.28/0.48 => ( doubleComplementEq
% 0.28/0.48 => ( complementTnotintersectT
% 0.28/0.48 => ( complementImpComplementIntersect
% 0.28/0.48 => ( complementSubsetComplementIntersect
% 0.28/0.48 => ( complementInPowersetComplementIntersect
% 0.28/0.48 => ( contraSubsetComplement
% 0.28/0.48 => ( complementTcontraSubset
% 0.28/0.48 => ( binunionTILcontra
% 0.28/0.48 => ( binunionTIRcontra
% 0.28/0.48 => ( inIntersectImpInUnion
% 0.28/0.48 => ( inIntersectImpInUnion2
% 0.28/0.48 => ( inIntersectImpInIntersectUnions
% 0.28/0.48 => ( intersectInPowersetIntersectUnions
% 0.28/0.48 => ( inComplementUnionImpNotIn1
% 0.28/0.48 => ( inComplementUnionImpInComplement1
% 0.28/0.48 => ( binunionTE
% 0.28/0.48 => ( binunionTEcontra
% 0.28/0.48 => ( demorgan2a1
% 0.28/0.48 => ( complementUnionInPowersetComplement
% 0.28/0.48 => ( demorgan2a2
% 0.28/0.48 => ( demorgan1a
% 0.28/0.48 => ( demorgan1b
% 0.28/0.48 => ( demorgan1
% 0.28/0.48 => ( demorgan2a
% 0.28/0.48 => ( demorgan2b2
% 0.28/0.48 => ( demorgan2b
% 0.28/0.48 => ( demorgan2
% 0.28/0.48 => ( woz13rule0
% 0.28/0.48 => ( woz13rule1
% 0.28/0.48 => ( woz13rule2
% 0.28/0.48 => ( woz13rule3
% 0.28/0.48 => ( woz13rule4
% 0.28/0.48 => ( woz1_1
% 0.28/0.48 => ( woz1_2
% 0.28/0.48 => ( woz1_3
% 0.28/0.48 => ( woz1_4
% 0.28/0.48 => ( woz1_5
% 0.28/0.48 => ( breln1all2
% 0.28/0.48 => ( breln1SetBreln1
% 0.28/0.48 => ( choice2fnsingleton
% 0.28/0.48 => ( setOfPairsIsBReln1
% 0.28/0.48 => ( breln1all1
% 0.28/0.48 => ( subbreln1
% 0.28/0.48 => ( eqbreln1
% 0.28/0.48 => ( breln1invprop
% 0.28/0.48 => ( breln1invI
% 0.28/0.48 => ( breln1invE
% 0.28/0.48 => ( breln1compprop
% 0.28/0.48 => ( breln1compI
% 0.28/0.48 => ( breln1compE
% 0.28/0.48 => ( breln1compEex
% 0.28/0.48 => ( breln1unionprop
% 0.28/0.48 => ( breln1unionIL
% 0.28/0.48 => ( breln1unionIR
% 0.28/0.48 => ( breln1unionI
% 0.28/0.48 => ( breln1unionE
% 0.28/0.48 => ( breln1unionEcases
% 0.28/0.48 => ( breln1unionCommutes
% 0.28/0.48 => ( woz2Ex
% 0.28/0.48 => ( woz2W
% 0.28/0.48 => ( woz2A
% 0.28/0.48 => ( woz2B
% 0.28/0.48 => ( image1Ex
% 0.28/0.48 => ( image1Ex1
% 0.28/0.48 => ( image1Equiv
% 0.28/0.48 => ( image1E
% 0.28/0.48 => ( image1I
% 0.28/0.48 => ( injFuncInInjFuncSet
% 0.28/0.48 => ( injFuncSetFuncIn
% 0.28/0.48 => ( injFuncSetFuncInj
% 0.28/0.48 => ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.28/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 %------------------------------------------------------------------------------
% 0.28/0.48 ------------------------
% 2.63/1.48 ---- Embedded in THF ---
% 2.63/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.63/1.48 %%% Generated on Tue Feb 24 12:34:41 EST 2026
% 2.63/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.63/1.48 %%% Logic specification used:
% 2.63/1.48 %%% thf(spec, logic, $$dhol).
% 2.63/1.48
% 2.63/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.uXiymMPKZd/DTF2DTF27071.p
% See solution above
% 2.63/1.48 ------------------------
% 2.95/1.52 ---- Cleaned THF ---
% 2.95/1.52 thf(in_type,type,
% 2.95/1.52 in: $i > $i > $o ).
% 2.95/1.52
% 2.95/1.52 thf(exu_type,type,
% 2.95/1.52 exu: ( $i > $o ) > $o ).
% 2.95/1.52
% 2.95/1.52 thf(exu,definition,
% 2.95/1.52 ( exu
% 2.95/1.52 = ( ^ [Xphi: $i > $o] :
% 2.95/1.52 ? [Xx: $i] :
% 2.95/1.52 ( ( Xphi @ Xx )
% 2.95/1.52 & ! [Xy: $i] :
% 2.95/1.52 ( ( Xphi @ Xy )
% 2.95/1.52 => ( Xx = Xy ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(setextAx_type,type,
% 2.95/1.52 setextAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(setextAx,definition,
% 2.95/1.52 ( setextAx
% 2.95/1.52 = ( ! [A: $i,B: $i] :
% 2.95/1.52 ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ A )
% 2.95/1.52 <=> ( in @ Xx @ B ) )
% 2.95/1.52 => ( A = B ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(emptyset_type,type,
% 2.95/1.52 emptyset: $i ).
% 2.95/1.52
% 2.95/1.52 thf(emptysetAx_type,type,
% 2.95/1.52 emptysetAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(emptysetAx,definition,
% 2.95/1.52 ( emptysetAx
% 2.95/1.52 = ( ! [Xx: $i] :
% 2.95/1.52 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(setadjoin_type,type,
% 2.95/1.52 setadjoin: $i > $i > $i ).
% 2.95/1.52
% 2.95/1.52 thf(setadjoinAx_type,type,
% 2.95/1.52 setadjoinAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(setadjoinAx,definition,
% 2.95/1.52 ( setadjoinAx
% 2.95/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.95/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.95/1.52 <=> ( ( Xy = Xx )
% 2.95/1.52 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(powerset_type,type,
% 2.95/1.52 powerset: $i > $i ).
% 2.95/1.52
% 2.95/1.52 thf(powersetAx_type,type,
% 2.95/1.52 powersetAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(powersetAx,definition,
% 2.95/1.52 ( powersetAx
% 2.95/1.52 = ( ! [A: $i,B: $i] :
% 2.95/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.95/1.52 <=> ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ B )
% 2.95/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(setunion_type,type,
% 2.95/1.52 setunion: $i > $i ).
% 2.95/1.52
% 2.95/1.52 thf(setunionAx_type,type,
% 2.95/1.52 setunionAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(setunionAx,definition,
% 2.95/1.52 ( setunionAx
% 2.95/1.52 = ( ! [A: $i,Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 2.95/1.52 <=> ? [B: $i] :
% 2.95/1.52 ( ( in @ Xx @ B )
% 2.95/1.52 & ( in @ B @ A ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(omega_type,type,
% 2.95/1.52 omega: $i ).
% 2.95/1.52
% 2.95/1.52 thf(omega0Ax_type,type,
% 2.95/1.52 omega0Ax: $o ).
% 2.95/1.52
% 2.95/1.52 thf(omega0Ax,definition,
% 2.95/1.52 ( omega0Ax
% 2.95/1.52 = ( in @ emptyset @ omega ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(omegaSAx_type,type,
% 2.95/1.52 omegaSAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(omegaSAx,definition,
% 2.95/1.52 ( omegaSAx
% 2.95/1.52 = ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ omega )
% 2.95/1.52 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(omegaIndAx_type,type,
% 2.95/1.52 omegaIndAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(omegaIndAx,definition,
% 2.95/1.52 ( omegaIndAx
% 2.95/1.52 = ( ! [A: $i] :
% 2.95/1.52 ( ( ( in @ emptyset @ A )
% 2.95/1.52 & ! [Xx: $i] :
% 2.95/1.52 ( ( ( in @ Xx @ omega )
% 2.95/1.52 & ( in @ Xx @ A ) )
% 2.95/1.52 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.95/1.52 => ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ omega )
% 2.95/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(replAx_type,type,
% 2.95/1.52 replAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(replAx,definition,
% 2.95/1.52 ( replAx
% 2.95/1.52 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.95/1.52 ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ A )
% 2.95/1.52 => ( exu
% 2.95/1.52 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.95/1.52 => ? [B: $i] :
% 2.95/1.52 ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ B )
% 2.95/1.52 <=> ? [Xy: $i] :
% 2.95/1.52 ( ( in @ Xy @ A )
% 2.95/1.52 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(foundationAx_type,type,
% 2.95/1.52 foundationAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(foundationAx,definition,
% 2.95/1.52 ( foundationAx
% 2.95/1.52 = ( ! [A: $i] :
% 2.95/1.52 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.95/1.52 => ? [B: $i] :
% 2.95/1.52 ( ( in @ B @ A )
% 2.95/1.52 & ~ ? [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ B )
% 2.95/1.52 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(wellorderingAx_type,type,
% 2.95/1.52 wellorderingAx: $o ).
% 2.95/1.52
% 2.95/1.52 thf(wellorderingAx,definition,
% 2.95/1.52 ( wellorderingAx
% 2.95/1.52 = ( ! [A: $i] :
% 2.95/1.52 ? [B: $i] :
% 2.95/1.52 ( ! [C: $i] :
% 2.95/1.52 ( ( in @ C @ B )
% 2.95/1.52 => ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ C )
% 2.95/1.52 => ( in @ Xx @ A ) ) )
% 2.95/1.52 & ! [Xx: $i,Xy: $i] :
% 2.95/1.52 ( ( ( in @ Xx @ A )
% 2.95/1.52 & ( in @ Xy @ A ) )
% 2.95/1.52 => ( ! [C: $i] :
% 2.95/1.52 ( ( in @ C @ B )
% 2.95/1.52 => ( ( in @ Xx @ C )
% 2.95/1.52 <=> ( in @ Xy @ C ) ) )
% 2.95/1.52 => ( Xx = Xy ) ) )
% 2.95/1.52 & ! [C: $i,D: $i] :
% 2.95/1.52 ( ( ( in @ C @ B )
% 2.95/1.52 & ( in @ D @ B ) )
% 2.95/1.52 => ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ C )
% 2.95/1.52 => ( in @ Xx @ D ) )
% 2.95/1.52 | ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ D )
% 2.95/1.52 => ( in @ Xx @ C ) ) ) )
% 2.95/1.52 & ! [C: $i] :
% 2.95/1.52 ( ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ C )
% 2.95/1.52 => ( in @ Xx @ A ) )
% 2.95/1.52 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.95/1.52 => ? [D: $i,Xx: $i] :
% 2.95/1.52 ( ( in @ D @ B )
% 2.95/1.52 & ( in @ Xx @ C )
% 2.95/1.52 & ~ ? [Xy: $i] :
% 2.95/1.52 ( ( in @ Xy @ D )
% 2.95/1.52 & ( in @ Xy @ C ) )
% 2.95/1.52 & ! [E: $i] :
% 2.95/1.52 ( ( in @ E @ B )
% 2.95/1.52 => ( ! [Xy: $i] :
% 2.95/1.52 ( ( in @ Xy @ E )
% 2.95/1.52 => ( in @ Xy @ D ) )
% 2.95/1.52 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(descr_type,type,
% 2.95/1.52 descr: ( $i > $o ) > $i ).
% 2.95/1.52
% 2.95/1.52 thf(descrp_type,type,
% 2.95/1.52 descrp: $o ).
% 2.95/1.52
% 2.95/1.52 thf(descrp,definition,
% 2.95/1.52 ( descrp
% 2.95/1.52 = ( ! [Xphi: $i > $o] :
% 2.95/1.52 ( ( exu
% 2.95/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.52 => ( Xphi
% 2.95/1.52 @ ( descr
% 2.95/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(dsetconstr_type,type,
% 2.95/1.52 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.95/1.52
% 2.95/1.52 thf(dsetconstrI_type,type,
% 2.95/1.52 dsetconstrI: $o ).
% 2.95/1.52
% 2.95/1.52 thf(dsetconstrI,definition,
% 2.95/1.52 ( dsetconstrI
% 2.95/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ A )
% 2.95/1.52 => ( ( Xphi @ Xx )
% 2.95/1.52 => ( in @ Xx
% 2.95/1.52 @ ( dsetconstr @ A
% 2.95/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(dsetconstrEL_type,type,
% 2.95/1.52 dsetconstrEL: $o ).
% 2.95/1.52
% 2.95/1.52 thf(dsetconstrEL,definition,
% 2.95/1.52 ( dsetconstrEL
% 2.95/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.95/1.52 ( ( in @ Xx
% 2.95/1.52 @ ( dsetconstr @ A
% 2.95/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.95/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(dsetconstrER_type,type,
% 2.95/1.52 dsetconstrER: $o ).
% 2.95/1.52
% 2.95/1.52 thf(dsetconstrER,definition,
% 2.95/1.52 ( dsetconstrER
% 2.95/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.95/1.52 ( ( in @ Xx
% 2.95/1.52 @ ( dsetconstr @ A
% 2.95/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.95/1.52 => ( Xphi @ Xx ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(exuE1_type,type,
% 2.95/1.52 exuE1: $o ).
% 2.95/1.52
% 2.95/1.52 thf(exuE1,definition,
% 2.95/1.52 ( exuE1
% 2.95/1.52 = ( ! [Xphi: $i > $o] :
% 2.95/1.52 ( ( exu
% 2.95/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.52 => ? [Xx: $i] :
% 2.95/1.52 ( ( Xphi @ Xx )
% 2.95/1.52 & ! [Xy: $i] :
% 2.95/1.52 ( ( Xphi @ Xy )
% 2.95/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(prop2set_type,type,
% 2.95/1.52 prop2set: $o > $i ).
% 2.95/1.52
% 2.95/1.52 thf(prop2setE_type,type,
% 2.95/1.52 prop2setE: $o ).
% 2.95/1.52
% 2.95/1.52 thf(prop2setE,definition,
% 2.95/1.52 ( prop2setE
% 2.95/1.52 = ( ! [Xphi: $o,Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.95/1.52 => Xphi ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(emptysetE_type,type,
% 2.95/1.52 emptysetE: $o ).
% 2.95/1.52
% 2.95/1.52 thf(emptysetE,definition,
% 2.95/1.52 ( emptysetE
% 2.95/1.52 = ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ emptyset )
% 2.95/1.52 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(emptysetimpfalse_type,type,
% 2.95/1.52 emptysetimpfalse: $o ).
% 2.95/1.52
% 2.95/1.52 thf(emptysetimpfalse,definition,
% 2.95/1.52 ( emptysetimpfalse
% 2.95/1.52 = ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ emptyset )
% 2.95/1.52 => $false ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(notinemptyset_type,type,
% 2.95/1.52 notinemptyset: $o ).
% 2.95/1.52
% 2.95/1.52 thf(notinemptyset,definition,
% 2.95/1.52 ( notinemptyset
% 2.95/1.52 = ( ! [Xx: $i] :
% 2.95/1.52 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(exuE3e_type,type,
% 2.95/1.52 exuE3e: $o ).
% 2.95/1.52
% 2.95/1.52 thf(exuE3e,definition,
% 2.95/1.52 ( exuE3e
% 2.95/1.52 = ( ! [Xphi: $i > $o] :
% 2.95/1.52 ( ( exu
% 2.95/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.52 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(setext_type,type,
% 2.95/1.52 setext: $o ).
% 2.95/1.52
% 2.95/1.52 thf(setext,definition,
% 2.95/1.52 ( setext
% 2.95/1.52 = ( ! [A: $i,B: $i] :
% 2.95/1.52 ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ A )
% 2.95/1.52 => ( in @ Xx @ B ) )
% 2.95/1.52 => ( ! [Xx: $i] :
% 2.95/1.52 ( ( in @ Xx @ B )
% 2.95/1.52 => ( in @ Xx @ A ) )
% 2.95/1.52 => ( A = B ) ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(emptyI_type,type,
% 2.95/1.52 emptyI: $o ).
% 2.95/1.52
% 2.95/1.52 thf(emptyI,definition,
% 2.95/1.52 ( emptyI
% 2.95/1.52 = ( ! [A: $i] :
% 2.95/1.52 ( ! [Xx: $i] :
% 2.95/1.52 ~ ( in @ Xx @ A )
% 2.95/1.52 => ( A = emptyset ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(noeltsimpempty_type,type,
% 2.95/1.52 noeltsimpempty: $o ).
% 2.95/1.52
% 2.95/1.52 thf(noeltsimpempty,definition,
% 2.95/1.52 ( noeltsimpempty
% 2.95/1.52 = ( ! [A: $i] :
% 2.95/1.52 ( ! [Xx: $i] :
% 2.95/1.52 ~ ( in @ Xx @ A )
% 2.95/1.52 => ( A = emptyset ) ) ) ) ).
% 2.95/1.52
% 2.95/1.52 thf(setbeta_type,type,
% 2.95/1.52 setbeta: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setbeta,definition,
% 2.95/1.53 ( setbeta
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx
% 2.95/1.53 @ ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.95/1.53 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(nonempty_type,type,
% 2.95/1.53 nonempty: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(nonempty,definition,
% 2.95/1.53 ( nonempty
% 2.95/1.53 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(nonemptyE1_type,type,
% 2.95/1.53 nonemptyE1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(nonemptyE1,definition,
% 2.95/1.53 ( nonemptyE1
% 2.95/1.53 = ( ! [A: $i] :
% 2.95/1.53 ( ( nonempty @ A )
% 2.95/1.53 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(nonemptyI_type,type,
% 2.95/1.53 nonemptyI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(nonemptyI,definition,
% 2.95/1.53 ( nonemptyI
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( Xphi @ Xx )
% 2.95/1.53 => ( nonempty
% 2.95/1.53 @ ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(nonemptyI1_type,type,
% 2.95/1.53 nonemptyI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(nonemptyI1,definition,
% 2.95/1.53 ( nonemptyI1
% 2.95/1.53 = ( ! [A: $i] :
% 2.95/1.53 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.95/1.53 => ( nonempty @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinIL_type,type,
% 2.95/1.53 setadjoinIL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinIL,definition,
% 2.95/1.53 ( setadjoinIL
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(emptyinunitempty_type,type,
% 2.95/1.53 emptyinunitempty: $o ).
% 2.95/1.53
% 2.95/1.53 thf(emptyinunitempty,definition,
% 2.95/1.53 ( emptyinunitempty
% 2.95/1.53 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinIR_type,type,
% 2.95/1.53 setadjoinIR: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinIR,definition,
% 2.95/1.53 ( setadjoinIR
% 2.95/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinE_type,type,
% 2.95/1.53 setadjoinE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinE,definition,
% 2.95/1.53 ( setadjoinE
% 2.95/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.95/1.53 => ! [Xphi: $o] :
% 2.95/1.53 ( ( ( Xy = Xx )
% 2.95/1.53 => Xphi )
% 2.95/1.53 => ( ( ( in @ Xy @ A )
% 2.95/1.53 => Xphi )
% 2.95/1.53 => Xphi ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinOr_type,type,
% 2.95/1.53 setadjoinOr: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinOr,definition,
% 2.95/1.53 ( setadjoinOr
% 2.95/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.95/1.53 => ( ( Xy = Xx )
% 2.95/1.53 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setoftrueEq_type,type,
% 2.95/1.53 setoftrueEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setoftrueEq,definition,
% 2.95/1.53 ( setoftrueEq
% 2.95/1.53 = ( ! [A: $i] :
% 2.95/1.53 ( ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xx: $i] : $true )
% 2.95/1.53 = A ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powersetI_type,type,
% 2.95/1.53 powersetI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powersetI,definition,
% 2.95/1.53 ( powersetI
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ B )
% 2.95/1.53 => ( in @ Xx @ A ) )
% 2.95/1.53 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(emptyinPowerset_type,type,
% 2.95/1.53 emptyinPowerset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(emptyinPowerset,definition,
% 2.95/1.53 ( emptyinPowerset
% 2.95/1.53 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(emptyInPowerset_type,type,
% 2.95/1.53 emptyInPowerset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(emptyInPowerset,definition,
% 2.95/1.53 ( emptyInPowerset
% 2.95/1.53 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powersetE_type,type,
% 2.95/1.53 powersetE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powersetE,definition,
% 2.95/1.53 ( powersetE
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.95/1.53 => ( ( in @ Xx @ B )
% 2.95/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setunionI_type,type,
% 2.95/1.53 setunionI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setunionI,definition,
% 2.95/1.53 ( setunionI
% 2.95/1.53 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.95/1.53 ( ( in @ Xx @ B )
% 2.95/1.53 => ( ( in @ B @ A )
% 2.95/1.53 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setunionE_type,type,
% 2.95/1.53 setunionE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setunionE,definition,
% 2.95/1.53 ( setunionE
% 2.95/1.53 = ( ! [A: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.95/1.53 => ! [Xphi: $o] :
% 2.95/1.53 ( ! [B: $i] :
% 2.95/1.53 ( ( in @ Xx @ B )
% 2.95/1.53 => ( ( in @ B @ A )
% 2.95/1.53 => Xphi ) )
% 2.95/1.53 => Xphi ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subPowSU_type,type,
% 2.95/1.53 subPowSU: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subPowSU,definition,
% 2.95/1.53 ( subPowSU
% 2.95/1.53 = ( ! [A: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(exuE2_type,type,
% 2.95/1.53 exuE2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(exuE2,definition,
% 2.95/1.53 ( exuE2
% 2.95/1.53 = ( ! [Xphi: $i > $o] :
% 2.95/1.53 ( ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 => ? [Xx: $i] :
% 2.95/1.53 ! [Xy: $i] :
% 2.95/1.53 ( ( Xphi @ Xy )
% 2.95/1.53 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(nonemptyImpWitness_type,type,
% 2.95/1.53 nonemptyImpWitness: $o ).
% 2.95/1.53
% 2.95/1.53 thf(nonemptyImpWitness,definition,
% 2.95/1.53 ( nonemptyImpWitness
% 2.95/1.53 = ( ! [A: $i] :
% 2.95/1.53 ( ( nonempty @ A )
% 2.95/1.53 => ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & $true ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(uniqinunit_type,type,
% 2.95/1.53 uniqinunit: $o ).
% 2.95/1.53
% 2.95/1.53 thf(uniqinunit,definition,
% 2.95/1.53 ( uniqinunit
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.95/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(notinsingleton_type,type,
% 2.95/1.53 notinsingleton: $o ).
% 2.95/1.53
% 2.95/1.53 thf(notinsingleton,definition,
% 2.95/1.53 ( notinsingleton
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx != Xy )
% 2.95/1.53 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(eqinunit_type,type,
% 2.95/1.53 eqinunit: $o ).
% 2.95/1.53
% 2.95/1.53 thf(eqinunit,definition,
% 2.95/1.53 ( eqinunit
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx = Xy )
% 2.95/1.53 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(singletonsswitch_type,type,
% 2.95/1.53 singletonsswitch: $o ).
% 2.95/1.53
% 2.95/1.53 thf(singletonsswitch,definition,
% 2.95/1.53 ( singletonsswitch
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.95/1.53 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(upairsetE_type,type,
% 2.95/1.53 upairsetE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairsetE,definition,
% 2.95/1.53 ( upairsetE
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.95/1.53 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.95/1.53 => ( ( Xz = Xx )
% 2.95/1.53 | ( Xz = Xy ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(upairsetIL_type,type,
% 2.95/1.53 upairsetIL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairsetIL,definition,
% 2.95/1.53 ( upairsetIL
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(upairsetIR_type,type,
% 2.95/1.53 upairsetIR: $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairsetIR,definition,
% 2.95/1.53 ( upairsetIR
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(emptyE1_type,type,
% 2.95/1.53 emptyE1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(emptyE1,definition,
% 2.95/1.53 ( emptyE1
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ( Xphi @ Xx ) )
% 2.95/1.53 => ( ( ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 = emptyset )
% 2.95/1.53 => $false ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(vacuousDall_type,type,
% 2.95/1.53 vacuousDall: $o ).
% 2.95/1.53
% 2.95/1.53 thf(vacuousDall,definition,
% 2.95/1.53 ( vacuousDall
% 2.95/1.53 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ emptyset )
% 2.95/1.53 => ( Xphi @ Xx ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(quantDeMorgan1_type,type,
% 2.95/1.53 quantDeMorgan1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(quantDeMorgan1,definition,
% 2.95/1.53 ( quantDeMorgan1
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ~ ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( Xphi @ Xx ) )
% 2.95/1.53 => ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(quantDeMorgan2_type,type,
% 2.95/1.53 quantDeMorgan2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(quantDeMorgan2,definition,
% 2.95/1.53 ( quantDeMorgan2
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ~ ( Xphi @ Xx ) )
% 2.95/1.53 => ~ ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(quantDeMorgan3_type,type,
% 2.95/1.53 quantDeMorgan3: $o ).
% 2.95/1.53
% 2.95/1.53 thf(quantDeMorgan3,definition,
% 2.95/1.53 ( quantDeMorgan3
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ~ ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ( Xphi @ Xx ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(quantDeMorgan4_type,type,
% 2.95/1.53 quantDeMorgan4: $o ).
% 2.95/1.53
% 2.95/1.53 thf(quantDeMorgan4,definition,
% 2.95/1.53 ( quantDeMorgan4
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ~ ( Xphi @ Xx ) )
% 2.95/1.53 => ~ ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(prop2setI_type,type,
% 2.95/1.53 prop2setI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(prop2setI,definition,
% 2.95/1.53 ( prop2setI
% 2.95/1.53 = ( ! [Xphi: $o] :
% 2.95/1.53 ( Xphi
% 2.95/1.53 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(set2prop_type,type,
% 2.95/1.53 set2prop: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(prop2set2propI_type,type,
% 2.95/1.53 prop2set2propI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(prop2set2propI,definition,
% 2.95/1.53 ( prop2set2propI
% 2.95/1.53 = ( ! [Xphi: $o] :
% 2.95/1.53 ( Xphi
% 2.95/1.53 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(notdexE_type,type,
% 2.95/1.53 notdexE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(notdexE,definition,
% 2.95/1.53 ( notdexE
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ~ ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ( Xphi @ Xx ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(notdallE_type,type,
% 2.95/1.53 notdallE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(notdallE,definition,
% 2.95/1.53 ( notdallE
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ~ ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( Xphi @ Xx ) )
% 2.95/1.53 => ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(exuI1_type,type,
% 2.95/1.53 exuI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(exuI1,definition,
% 2.95/1.53 ( exuI1
% 2.95/1.53 = ( ! [Xphi: $i > $o] :
% 2.95/1.53 ( ? [Xx: $i] :
% 2.95/1.53 ( ( Xphi @ Xx )
% 2.95/1.53 & ! [Xy: $i] :
% 2.95/1.53 ( ( Xphi @ Xy )
% 2.95/1.53 => ( Xx = Xy ) ) )
% 2.95/1.53 => ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(exuI3_type,type,
% 2.95/1.53 exuI3: $o ).
% 2.95/1.53
% 2.95/1.53 thf(exuI3,definition,
% 2.95/1.53 ( exuI3
% 2.95/1.53 = ( ! [Xphi: $i > $o] :
% 2.95/1.53 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.95/1.53 => ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xphi @ Xx )
% 2.95/1.53 => ( ( Xphi @ Xy )
% 2.95/1.53 => ( Xx = Xy ) ) )
% 2.95/1.53 => ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(exuI2_type,type,
% 2.95/1.53 exuI2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(exuI2,definition,
% 2.95/1.53 ( exuI2
% 2.95/1.53 = ( ! [Xphi: $i > $o] :
% 2.95/1.53 ( ? [Xx: $i] :
% 2.95/1.53 ! [Xy: $i] :
% 2.95/1.53 ( ( Xphi @ Xy )
% 2.95/1.53 <=> ( Xy = Xx ) )
% 2.95/1.53 => ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(inCongP_type,type,
% 2.95/1.53 inCongP: $o ).
% 2.95/1.53
% 2.95/1.53 thf(inCongP,definition,
% 2.95/1.53 ( inCongP
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( A = B )
% 2.95/1.53 => ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx = Xy )
% 2.95/1.53 => ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(in__Cong_type,type,
% 2.95/1.53 in__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(in__Cong,definition,
% 2.95/1.53 ( in__Cong
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( A = B )
% 2.95/1.53 => ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx = Xy )
% 2.95/1.53 => ( ( in @ Xx @ A )
% 2.95/1.53 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(exuE3u_type,type,
% 2.95/1.53 exuE3u: $o ).
% 2.95/1.53
% 2.95/1.53 thf(exuE3u,definition,
% 2.95/1.53 ( exuE3u
% 2.95/1.53 = ( ! [Xphi: $i > $o] :
% 2.95/1.53 ( ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 => ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xphi @ Xx )
% 2.95/1.53 => ( ( Xphi @ Xy )
% 2.95/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(exu__Cong_type,type,
% 2.95/1.53 exu__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(exu__Cong,definition,
% 2.95/1.53 ( exu__Cong
% 2.95/1.53 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx = Xy )
% 2.95/1.53 => ( ( Xphi @ Xx )
% 2.95/1.53 <=> ( Xpsi @ Xy ) ) )
% 2.95/1.53 => ( ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 <=> ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(emptyset__Cong_type,type,
% 2.95/1.53 emptyset__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(emptyset__Cong,definition,
% 2.95/1.53 ( emptyset__Cong
% 2.95/1.53 = ( emptyset = emptyset ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoin__Cong_type,type,
% 2.95/1.53 setadjoin__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoin__Cong,definition,
% 2.95/1.53 ( setadjoin__Cong
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx = Xy )
% 2.95/1.53 => ! [Xz: $i,Xu: $i] :
% 2.95/1.53 ( ( Xz = Xu )
% 2.95/1.53 => ( ( setadjoin @ Xx @ Xz )
% 2.95/1.53 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powerset__Cong_type,type,
% 2.95/1.53 powerset__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powerset__Cong,definition,
% 2.95/1.53 ( powerset__Cong
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( A = B )
% 2.95/1.53 => ( ( powerset @ A )
% 2.95/1.53 = ( powerset @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setunion__Cong_type,type,
% 2.95/1.53 setunion__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setunion__Cong,definition,
% 2.95/1.53 ( setunion__Cong
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( A = B )
% 2.95/1.53 => ( ( setunion @ A )
% 2.95/1.53 = ( setunion @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(omega__Cong_type,type,
% 2.95/1.53 omega__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(omega__Cong,definition,
% 2.95/1.53 ( omega__Cong
% 2.95/1.53 = ( omega = omega ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(exuEu_type,type,
% 2.95/1.53 exuEu: $o ).
% 2.95/1.53
% 2.95/1.53 thf(exuEu,definition,
% 2.95/1.53 ( exuEu
% 2.95/1.53 = ( ! [Xphi: $i > $o] :
% 2.95/1.53 ( ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 => ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xphi @ Xx )
% 2.95/1.53 => ( ( Xphi @ Xy )
% 2.95/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(descr__Cong_type,type,
% 2.95/1.53 descr__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(descr__Cong,definition,
% 2.95/1.53 ( descr__Cong
% 2.95/1.53 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx = Xy )
% 2.95/1.53 => ( ( Xphi @ Xx )
% 2.95/1.53 <=> ( Xpsi @ Xy ) ) )
% 2.95/1.53 => ( ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 => ( ( exu
% 2.95/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.95/1.53 => ( ( descr
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 = ( descr
% 2.95/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(dsetconstr__Cong_type,type,
% 2.95/1.53 dsetconstr__Cong: $o ).
% 2.95/1.53
% 2.95/1.53 thf(dsetconstr__Cong,definition,
% 2.95/1.53 ( dsetconstr__Cong
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( A = B )
% 2.95/1.53 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( Xx = Xy )
% 2.95/1.53 => ( ( Xphi @ Xx )
% 2.95/1.53 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.95/1.53 => ( ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 = ( dsetconstr @ B
% 2.95/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subset_type,type,
% 2.95/1.53 subset: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(disjoint_type,type,
% 2.95/1.53 disjoint: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(setsmeet_type,type,
% 2.95/1.53 setsmeet: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetI1_type,type,
% 2.95/1.53 subsetI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetI1,definition,
% 2.95/1.53 ( subsetI1
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ Xx @ B ) )
% 2.95/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(eqimpsubset2_type,type,
% 2.95/1.53 eqimpsubset2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(eqimpsubset2,definition,
% 2.95/1.53 ( eqimpsubset2
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( A = B )
% 2.95/1.53 => ( subset @ B @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(eqimpsubset1_type,type,
% 2.95/1.53 eqimpsubset1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(eqimpsubset1,definition,
% 2.95/1.53 ( eqimpsubset1
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( A = B )
% 2.95/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subsetI2_type,type,
% 2.95/1.53 subsetI2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetI2,definition,
% 2.95/1.53 ( subsetI2
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ Xx @ B ) )
% 2.95/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(emptysetsubset_type,type,
% 2.95/1.53 emptysetsubset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(emptysetsubset,definition,
% 2.95/1.53 ( emptysetsubset
% 2.95/1.53 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subsetE_type,type,
% 2.95/1.53 subsetE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetE,definition,
% 2.95/1.53 ( subsetE
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subsetE2_type,type,
% 2.95/1.53 subsetE2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetE2,definition,
% 2.95/1.53 ( subsetE2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( ~ ( in @ Xx @ B )
% 2.95/1.53 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(notsubsetI_type,type,
% 2.95/1.53 notsubsetI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(notsubsetI,definition,
% 2.95/1.53 ( notsubsetI
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ B )
% 2.95/1.53 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(notequalI1_type,type,
% 2.95/1.53 notequalI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(notequalI1,definition,
% 2.95/1.53 ( notequalI1
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ~ ( subset @ A @ B )
% 2.95/1.53 => ( A != B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(notequalI2_type,type,
% 2.95/1.53 notequalI2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(notequalI2,definition,
% 2.95/1.53 ( notequalI2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ B )
% 2.95/1.53 => ( A != B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subsetRefl_type,type,
% 2.95/1.53 subsetRefl: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetRefl,definition,
% 2.95/1.53 ( subsetRefl
% 2.95/1.53 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subsetTrans_type,type,
% 2.95/1.53 subsetTrans: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetTrans,definition,
% 2.95/1.53 ( subsetTrans
% 2.95/1.53 = ( ! [A: $i,B: $i,C: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( ( subset @ B @ C )
% 2.95/1.53 => ( subset @ A @ C ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinSub_type,type,
% 2.95/1.53 setadjoinSub: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinSub,definition,
% 2.95/1.53 ( setadjoinSub
% 2.95/1.53 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinSub2_type,type,
% 2.95/1.53 setadjoinSub2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setadjoinSub2,definition,
% 2.95/1.53 ( setadjoinSub2
% 2.95/1.53 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subset2powerset_type,type,
% 2.95/1.53 subset2powerset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subset2powerset,definition,
% 2.95/1.53 ( subset2powerset
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setextsub_type,type,
% 2.95/1.53 setextsub: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setextsub,definition,
% 2.95/1.53 ( setextsub
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( ( subset @ B @ A )
% 2.95/1.53 => ( A = B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subsetemptysetimpeq_type,type,
% 2.95/1.53 subsetemptysetimpeq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetemptysetimpeq,definition,
% 2.95/1.53 ( subsetemptysetimpeq
% 2.95/1.53 = ( ! [A: $i] :
% 2.95/1.53 ( ( subset @ A @ emptyset )
% 2.95/1.53 => ( A = emptyset ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powersetI1_type,type,
% 2.95/1.53 powersetI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powersetI1,definition,
% 2.95/1.53 ( powersetI1
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( subset @ B @ A )
% 2.95/1.53 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powersetE1_type,type,
% 2.95/1.53 powersetE1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powersetE1,definition,
% 2.95/1.53 ( powersetE1
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.95/1.53 => ( subset @ B @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(inPowerset_type,type,
% 2.95/1.53 inPowerset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(inPowerset,definition,
% 2.95/1.53 ( inPowerset
% 2.95/1.53 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powersetsubset_type,type,
% 2.95/1.53 powersetsubset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powersetsubset,definition,
% 2.95/1.53 ( powersetsubset
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(sepInPowerset_type,type,
% 2.95/1.53 sepInPowerset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(sepInPowerset,definition,
% 2.95/1.53 ( sepInPowerset
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( in
% 2.95/1.53 @ ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 @ ( powerset @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(sepSubset_type,type,
% 2.95/1.53 sepSubset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(sepSubset,definition,
% 2.95/1.53 ( sepSubset
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( subset
% 2.95/1.53 @ ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 @ A ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunion_type,type,
% 2.95/1.53 binunion: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(binunionIL_type,type,
% 2.95/1.53 binunionIL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionIL,definition,
% 2.95/1.53 ( binunionIL
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(upairset2IR_type,type,
% 2.95/1.53 upairset2IR: $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairset2IR,definition,
% 2.95/1.53 ( upairset2IR
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionIR_type,type,
% 2.95/1.53 binunionIR: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionIR,definition,
% 2.95/1.53 ( binunionIR
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ B )
% 2.95/1.53 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionEcases_type,type,
% 2.95/1.53 binunionEcases: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionEcases,definition,
% 2.95/1.53 ( binunionEcases
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.95/1.53 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.95/1.53 => ( ( ( in @ Xx @ A )
% 2.95/1.53 => Xphi )
% 2.95/1.53 => ( ( ( in @ Xx @ B )
% 2.95/1.53 => Xphi )
% 2.95/1.53 => Xphi ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionE_type,type,
% 2.95/1.53 binunionE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionE,definition,
% 2.95/1.53 ( binunionE
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.95/1.53 => ( ( in @ Xx @ A )
% 2.95/1.53 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionLsub_type,type,
% 2.95/1.53 binunionLsub: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionLsub,definition,
% 2.95/1.53 ( binunionLsub
% 2.95/1.53 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionRsub_type,type,
% 2.95/1.53 binunionRsub: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionRsub,definition,
% 2.95/1.53 ( binunionRsub
% 2.95/1.53 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersect_type,type,
% 2.95/1.53 binintersect: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectI_type,type,
% 2.95/1.53 binintersectI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectI,definition,
% 2.95/1.53 ( binintersectI
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ B )
% 2.95/1.53 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset5_type,type,
% 2.95/1.53 binintersectSubset5: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset5,definition,
% 2.95/1.53 ( binintersectSubset5
% 2.95/1.53 = ( ! [A: $i,B: $i,C: $i] :
% 2.95/1.53 ( ( subset @ C @ A )
% 2.95/1.53 => ( ( subset @ C @ B )
% 2.95/1.53 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectEL_type,type,
% 2.95/1.53 binintersectEL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectEL,definition,
% 2.95/1.53 ( binintersectEL
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.95/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectLsub_type,type,
% 2.95/1.53 binintersectLsub: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectLsub,definition,
% 2.95/1.53 ( binintersectLsub
% 2.95/1.53 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset2_type,type,
% 2.95/1.53 binintersectSubset2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset2,definition,
% 2.95/1.53 ( binintersectSubset2
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( ( binintersect @ A @ B )
% 2.95/1.53 = A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset3_type,type,
% 2.95/1.53 binintersectSubset3: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset3,definition,
% 2.95/1.53 ( binintersectSubset3
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( ( binintersect @ A @ B )
% 2.95/1.53 = B )
% 2.95/1.53 => ( subset @ B @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectER_type,type,
% 2.95/1.53 binintersectER: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectER,definition,
% 2.95/1.53 ( binintersectER
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.95/1.53 => ( in @ Xx @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(disjointsetsI1_type,type,
% 2.95/1.53 disjointsetsI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(disjointsetsI1,definition,
% 2.95/1.53 ( disjointsetsI1
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ~ ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ( in @ Xx @ B ) )
% 2.95/1.53 => ( ( binintersect @ A @ B )
% 2.95/1.53 = emptyset ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectRsub_type,type,
% 2.95/1.53 binintersectRsub: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectRsub,definition,
% 2.95/1.53 ( binintersectRsub
% 2.95/1.53 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset4_type,type,
% 2.95/1.53 binintersectSubset4: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset4,definition,
% 2.95/1.53 ( binintersectSubset4
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( subset @ B @ A )
% 2.95/1.53 => ( ( binintersect @ A @ B )
% 2.95/1.53 = B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset1_type,type,
% 2.95/1.53 binintersectSubset1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectSubset1,definition,
% 2.95/1.53 ( binintersectSubset1
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( ( binintersect @ A @ B )
% 2.95/1.53 = A )
% 2.95/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(bs114d_type,type,
% 2.95/1.53 bs114d: $o ).
% 2.95/1.53
% 2.95/1.53 thf(bs114d,definition,
% 2.95/1.53 ( bs114d
% 2.95/1.53 = ( ! [A: $i,B: $i,C: $i] :
% 2.95/1.53 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.95/1.53 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(regular_type,type,
% 2.95/1.53 regular: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminus_type,type,
% 2.95/1.53 setminus: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(setminusI_type,type,
% 2.95/1.53 setminusI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusI,definition,
% 2.95/1.53 ( setminusI
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ B )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusEL_type,type,
% 2.95/1.53 setminusEL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusEL,definition,
% 2.95/1.53 ( setminusEL
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.95/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusER_type,type,
% 2.95/1.53 setminusER: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusER,definition,
% 2.95/1.53 ( setminusER
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.95/1.53 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusSubset2_type,type,
% 2.95/1.53 setminusSubset2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusSubset2,definition,
% 2.95/1.53 ( setminusSubset2
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( subset @ A @ B )
% 2.95/1.53 => ( ( setminus @ A @ B )
% 2.95/1.53 = emptyset ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusERneg_type,type,
% 2.95/1.53 setminusERneg: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusERneg,definition,
% 2.95/1.53 ( setminusERneg
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.95/1.53 => ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusELneg_type,type,
% 2.95/1.53 setminusELneg: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusELneg,definition,
% 2.95/1.53 ( setminusELneg
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.95/1.53 => ( ~ ( in @ Xx @ B )
% 2.95/1.53 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusILneg_type,type,
% 2.95/1.53 setminusILneg: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusILneg,definition,
% 2.95/1.53 ( setminusILneg
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ~ ( in @ Xx @ A )
% 2.95/1.53 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusIRneg_type,type,
% 2.95/1.53 setminusIRneg: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusIRneg,definition,
% 2.95/1.53 ( setminusIRneg
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ B )
% 2.95/1.53 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusLsub_type,type,
% 2.95/1.53 setminusLsub: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusLsub,definition,
% 2.95/1.53 ( setminusLsub
% 2.95/1.53 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusSubset1_type,type,
% 2.95/1.53 setminusSubset1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusSubset1,definition,
% 2.95/1.53 ( setminusSubset1
% 2.95/1.53 = ( ! [A: $i,B: $i] :
% 2.95/1.53 ( ( ( setminus @ A @ B )
% 2.95/1.53 = emptyset )
% 2.95/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(symdiff_type,type,
% 2.95/1.53 symdiff: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffE_type,type,
% 2.95/1.53 symdiffE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffE,definition,
% 2.95/1.53 ( symdiffE
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.95/1.53 => ! [Xphi: $o] :
% 2.95/1.53 ( ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ B )
% 2.95/1.53 => Xphi ) )
% 2.95/1.53 => ( ( ~ ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ B )
% 2.95/1.53 => Xphi ) )
% 2.95/1.53 => Xphi ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffI1_type,type,
% 2.95/1.53 symdiffI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffI1,definition,
% 2.95/1.53 ( symdiffI1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ B )
% 2.95/1.53 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffI2_type,type,
% 2.95/1.53 symdiffI2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffI2,definition,
% 2.95/1.53 ( symdiffI2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ~ ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ B )
% 2.95/1.53 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffIneg1_type,type,
% 2.95/1.53 symdiffIneg1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffIneg1,definition,
% 2.95/1.53 ( symdiffIneg1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ B )
% 2.95/1.53 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffIneg2_type,type,
% 2.95/1.53 symdiffIneg2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(symdiffIneg2,definition,
% 2.95/1.53 ( symdiffIneg2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ~ ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ B )
% 2.95/1.53 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(iskpair_type,type,
% 2.95/1.53 iskpair: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(secondinupair_type,type,
% 2.95/1.53 secondinupair: $o ).
% 2.95/1.53
% 2.95/1.53 thf(secondinupair,definition,
% 2.95/1.53 ( secondinupair
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairIL_type,type,
% 2.95/1.53 setukpairIL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairIL,definition,
% 2.95/1.53 ( setukpairIL
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairIR_type,type,
% 2.95/1.53 setukpairIR: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairIR,definition,
% 2.95/1.53 ( setukpairIR
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(kpairiskpair_type,type,
% 2.95/1.53 kpairiskpair: $o ).
% 2.95/1.53
% 2.95/1.53 thf(kpairiskpair,definition,
% 2.95/1.53 ( kpairiskpair
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(kpair_type,type,
% 2.95/1.53 kpair: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(kpairp_type,type,
% 2.95/1.53 kpairp: $o ).
% 2.95/1.53
% 2.95/1.53 thf(kpairp,definition,
% 2.95/1.53 ( kpairp
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprod_type,type,
% 2.95/1.53 cartprod: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(singletonsubset_type,type,
% 2.95/1.53 singletonsubset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(singletonsubset,definition,
% 2.95/1.53 ( singletonsubset
% 2.95/1.53 = ( ! [A: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(singletoninpowerset_type,type,
% 2.95/1.53 singletoninpowerset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(singletoninpowerset,definition,
% 2.95/1.53 ( singletoninpowerset
% 2.95/1.53 = ( ! [A: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(singletoninpowunion_type,type,
% 2.95/1.53 singletoninpowunion: $o ).
% 2.95/1.53
% 2.95/1.53 thf(singletoninpowunion,definition,
% 2.95/1.53 ( singletoninpowunion
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(upairset2E_type,type,
% 2.95/1.53 upairset2E: $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairset2E,definition,
% 2.95/1.53 ( upairset2E
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.95/1.53 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.95/1.53 => ( ( Xz = Xx )
% 2.95/1.53 | ( Xz = Xy ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(upairsubunion_type,type,
% 2.95/1.53 upairsubunion: $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairsubunion,definition,
% 2.95/1.53 ( upairsubunion
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(upairinpowunion_type,type,
% 2.95/1.53 upairinpowunion: $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairinpowunion,definition,
% 2.95/1.53 ( upairinpowunion
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ubforcartprodlem1_type,type,
% 2.95/1.53 ubforcartprodlem1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ubforcartprodlem1,definition,
% 2.95/1.53 ( ubforcartprodlem1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ubforcartprodlem2_type,type,
% 2.95/1.53 ubforcartprodlem2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ubforcartprodlem2,definition,
% 2.95/1.53 ( ubforcartprodlem2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ubforcartprodlem3_type,type,
% 2.95/1.53 ubforcartprodlem3: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ubforcartprodlem3,definition,
% 2.95/1.53 ( ubforcartprodlem3
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodpairin_type,type,
% 2.95/1.53 cartprodpairin: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodpairin,definition,
% 2.95/1.53 ( cartprodpairin
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodmempair1_type,type,
% 2.95/1.53 cartprodmempair1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodmempair1,definition,
% 2.95/1.53 ( cartprodmempair1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.95/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.95/1.53 => ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ? [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 & ( Xu
% 2.95/1.53 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodmempair_type,type,
% 2.95/1.53 cartprodmempair: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodmempair,definition,
% 2.95/1.53 ( cartprodmempair
% 2.95/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.95/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.95/1.53 => ( iskpair @ Xu ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setunionE2_type,type,
% 2.95/1.53 setunionE2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setunionE2,definition,
% 2.95/1.53 ( setunionE2
% 2.95/1.53 = ( ! [A: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.95/1.53 => ? [X: $i] :
% 2.95/1.53 ( ( in @ X @ A )
% 2.95/1.53 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setunionsingleton1_type,type,
% 2.95/1.53 setunionsingleton1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setunionsingleton1,definition,
% 2.95/1.53 ( setunionsingleton1
% 2.95/1.53 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setunionsingleton2_type,type,
% 2.95/1.53 setunionsingleton2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setunionsingleton2,definition,
% 2.95/1.53 ( setunionsingleton2
% 2.95/1.53 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setunionsingleton_type,type,
% 2.95/1.53 setunionsingleton: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setunionsingleton,definition,
% 2.95/1.53 ( setunionsingleton
% 2.95/1.53 = ( ! [Xx: $i] :
% 2.95/1.53 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.95/1.53 = Xx ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(singleton_type,type,
% 2.95/1.53 singleton: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(singletonprop_type,type,
% 2.95/1.53 singletonprop: $o ).
% 2.95/1.53
% 2.95/1.53 thf(singletonprop,definition,
% 2.95/1.53 ( singletonprop
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( Xphi @ Xx )
% 2.95/1.53 => ( ( Xphi @ Xy )
% 2.95/1.53 => ( Xx = Xy ) ) ) ) )
% 2.95/1.53 => ( ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ( Xphi @ Xx ) )
% 2.95/1.53 => ( singleton
% 2.95/1.53 @ ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ex1_type,type,
% 2.95/1.53 ex1: $i > ( $i > $o ) > $o ).
% 2.95/1.53
% 2.95/1.53 thf(ex1E1_type,type,
% 2.95/1.53 ex1E1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ex1E1,definition,
% 2.95/1.53 ( ex1E1
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ( ex1 @ A
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 => ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ex1I_type,type,
% 2.95/1.53 ex1I: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ex1I,definition,
% 2.95/1.53 ( ex1I
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( Xphi @ Xx )
% 2.95/1.53 => ( ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( Xphi @ Xy )
% 2.95/1.53 => ( Xy = Xx ) ) )
% 2.95/1.53 => ( ex1 @ A
% 2.95/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ex1I2_type,type,
% 2.95/1.53 ex1I2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ex1I2,definition,
% 2.95/1.53 ( ex1I2
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( Xphi @ Xx )
% 2.95/1.53 => ( ( Xphi @ Xy )
% 2.95/1.53 => ( Xx = Xy ) ) ) ) )
% 2.95/1.53 => ( ? [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 & ( Xphi @ Xx ) )
% 2.95/1.53 => ( ex1 @ A
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(singletonsuniq_type,type,
% 2.95/1.53 singletonsuniq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(singletonsuniq,definition,
% 2.95/1.53 ( singletonsuniq
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( ( setadjoin @ Xx @ emptyset )
% 2.95/1.53 = ( setadjoin @ Xy @ emptyset ) )
% 2.95/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(atmost1p_type,type,
% 2.95/1.53 atmost1p: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(atleast2p_type,type,
% 2.95/1.53 atleast2p: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(atmost2p_type,type,
% 2.95/1.53 atmost2p: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairsetp_type,type,
% 2.95/1.53 upairsetp: $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjL1_type,type,
% 2.95/1.53 setukpairinjL1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjL1,definition,
% 2.95/1.53 ( setukpairinjL1
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.95/1.53 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.95/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(kfstsingleton_type,type,
% 2.95/1.53 kfstsingleton: $o ).
% 2.95/1.53
% 2.95/1.53 thf(kfstsingleton,definition,
% 2.95/1.53 ( kfstsingleton
% 2.95/1.53 = ( ! [Xu: $i] :
% 2.95/1.53 ( ( iskpair @ Xu )
% 2.95/1.53 => ( singleton
% 2.95/1.53 @ ( dsetconstr @ ( setunion @ Xu )
% 2.95/1.53 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(theprop_type,type,
% 2.95/1.53 theprop: $o ).
% 2.95/1.53
% 2.95/1.53 thf(theprop,definition,
% 2.95/1.53 ( theprop
% 2.95/1.53 = ( ! [X: $i] :
% 2.95/1.53 ( ( singleton @ X )
% 2.95/1.53 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(kfst_type,type,
% 2.95/1.53 kfst: $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(kfstpairEq_type,type,
% 2.95/1.53 kfstpairEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(kfstpairEq,definition,
% 2.95/1.53 ( kfstpairEq
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.95/1.53 = Xx ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodfstin_type,type,
% 2.95/1.53 cartprodfstin: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodfstin,definition,
% 2.95/1.53 ( cartprodfstin
% 2.95/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.95/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.95/1.53 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjL2_type,type,
% 2.95/1.53 setukpairinjL2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjL2,definition,
% 2.95/1.53 ( setukpairinjL2
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.95/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.95/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.95/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjL_type,type,
% 2.95/1.53 setukpairinjL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjL,definition,
% 2.95/1.53 ( setukpairinjL
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.95/1.53 ( ( ( kpair @ Xx @ Xy )
% 2.95/1.53 = ( kpair @ Xz @ Xu ) )
% 2.95/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR11_type,type,
% 2.95/1.53 setukpairinjR11: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR11,definition,
% 2.95/1.53 ( setukpairinjR11
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx = Xy )
% 2.95/1.53 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.95/1.53 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR12_type,type,
% 2.95/1.53 setukpairinjR12: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR12,definition,
% 2.95/1.53 ( setukpairinjR12
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xx = Xy )
% 2.95/1.53 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.95/1.53 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR1_type,type,
% 2.95/1.53 setukpairinjR1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR1,definition,
% 2.95/1.53 ( setukpairinjR1
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.95/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.95/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.95/1.53 => ( ( Xz = Xu )
% 2.95/1.53 => ( Xy = Xu ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(upairequniteq_type,type,
% 2.95/1.53 upairequniteq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(upairequniteq,definition,
% 2.95/1.53 ( upairequniteq
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.95/1.53 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.95/1.53 = ( setadjoin @ Xz @ emptyset ) )
% 2.95/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR2_type,type,
% 2.95/1.53 setukpairinjR2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR2,definition,
% 2.95/1.53 ( setukpairinjR2
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.95/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.95/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.95/1.53 => ( Xy = Xu ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR_type,type,
% 2.95/1.53 setukpairinjR: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setukpairinjR,definition,
% 2.95/1.53 ( setukpairinjR
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.95/1.53 ( ( ( kpair @ Xx @ Xy )
% 2.95/1.53 = ( kpair @ Xz @ Xu ) )
% 2.95/1.53 => ( Xy = Xu ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ksndsingleton_type,type,
% 2.95/1.53 ksndsingleton: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ksndsingleton,definition,
% 2.95/1.53 ( ksndsingleton
% 2.95/1.53 = ( ! [Xu: $i] :
% 2.95/1.53 ( ( iskpair @ Xu )
% 2.95/1.53 => ( singleton
% 2.95/1.53 @ ( dsetconstr @ ( setunion @ Xu )
% 2.95/1.53 @ ^ [Xx: $i] :
% 2.95/1.53 ( Xu
% 2.95/1.53 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ksnd_type,type,
% 2.95/1.53 ksnd: $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(ksndpairEq_type,type,
% 2.95/1.53 ksndpairEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ksndpairEq,definition,
% 2.95/1.53 ( ksndpairEq
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.95/1.53 = Xy ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(kpairsurjEq_type,type,
% 2.95/1.53 kpairsurjEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(kpairsurjEq,definition,
% 2.95/1.53 ( kpairsurjEq
% 2.95/1.53 = ( ! [Xu: $i] :
% 2.95/1.53 ( ( iskpair @ Xu )
% 2.95/1.53 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.95/1.53 = Xu ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodsndin_type,type,
% 2.95/1.53 cartprodsndin: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodsndin,definition,
% 2.95/1.53 ( cartprodsndin
% 2.95/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.95/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.95/1.53 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodpairmemEL_type,type,
% 2.95/1.53 cartprodpairmemEL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodpairmemEL,definition,
% 2.95/1.53 ( cartprodpairmemEL
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.95/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodpairmemER_type,type,
% 2.95/1.53 cartprodpairmemER: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodpairmemER,definition,
% 2.95/1.53 ( cartprodpairmemER
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.95/1.53 => ( in @ Xy @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodmempaircEq_type,type,
% 2.95/1.53 cartprodmempaircEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodmempaircEq,definition,
% 2.95/1.53 ( cartprodmempaircEq
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( kpair @ Xx @ Xy )
% 2.95/1.53 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodfstpairEq_type,type,
% 2.95/1.53 cartprodfstpairEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodfstpairEq,definition,
% 2.95/1.53 ( cartprodfstpairEq
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.95/1.53 = Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodsndpairEq_type,type,
% 2.95/1.53 cartprodsndpairEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodsndpairEq,definition,
% 2.95/1.53 ( cartprodsndpairEq
% 2.95/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.95/1.53 = Xy ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodpairsurjEq_type,type,
% 2.95/1.53 cartprodpairsurjEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(cartprodpairsurjEq,definition,
% 2.95/1.53 ( cartprodpairsurjEq
% 2.95/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.95/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.95/1.53 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.95/1.53 = Xu ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln_type,type,
% 2.95/1.53 breln: $i > $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstr_type,type,
% 2.95/1.53 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrI_type,type,
% 2.95/1.53 dpsetconstrI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrI,definition,
% 2.95/1.53 ( dpsetconstrI
% 2.95/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( Xphi @ Xx @ Xy )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy )
% 2.95/1.53 @ ( dpsetconstr @ A @ B
% 2.95/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrSub_type,type,
% 2.95/1.53 dpsetconstrSub: $o ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrSub,definition,
% 2.95/1.53 ( dpsetconstrSub
% 2.95/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.95/1.53 ( subset
% 2.95/1.53 @ ( dpsetconstr @ A @ B
% 2.95/1.53 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.95/1.53 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setOfPairsIsBReln_type,type,
% 2.95/1.53 setOfPairsIsBReln: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setOfPairsIsBReln,definition,
% 2.95/1.53 ( setOfPairsIsBReln
% 2.95/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.95/1.53 ( breln @ A @ B
% 2.95/1.53 @ ( dpsetconstr @ A @ B
% 2.95/1.53 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrERa_type,type,
% 2.95/1.53 dpsetconstrERa: $o ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrERa,definition,
% 2.95/1.53 ( dpsetconstrERa
% 2.95/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.95/1.53 @ ( dpsetconstr @ A @ B
% 2.95/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.95/1.53 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrEL1_type,type,
% 2.95/1.53 dpsetconstrEL1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrEL1,definition,
% 2.95/1.53 ( dpsetconstrEL1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.95/1.53 @ ( dpsetconstr @ A @ B
% 2.95/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.95/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrEL2_type,type,
% 2.95/1.53 dpsetconstrEL2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrEL2,definition,
% 2.95/1.53 ( dpsetconstrEL2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.95/1.53 @ ( dpsetconstr @ A @ B
% 2.95/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.95/1.53 => ( in @ Xy @ B ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrER_type,type,
% 2.95/1.53 dpsetconstrER: $o ).
% 2.95/1.53
% 2.95/1.53 thf(dpsetconstrER,definition,
% 2.95/1.53 ( dpsetconstrER
% 2.95/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.95/1.53 @ ( dpsetconstr @ A @ B
% 2.95/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.95/1.53 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(func_type,type,
% 2.95/1.53 func: $i > $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcSet_type,type,
% 2.95/1.53 funcSet: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(funcImageSingleton_type,type,
% 2.95/1.53 funcImageSingleton: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcImageSingleton,definition,
% 2.95/1.53 ( funcImageSingleton
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( singleton
% 2.95/1.53 @ ( dsetconstr @ B
% 2.95/1.53 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(apProp_type,type,
% 2.95/1.53 apProp: $o ).
% 2.95/1.53
% 2.95/1.53 thf(apProp,definition,
% 2.95/1.53 ( apProp
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in
% 2.95/1.53 @ ( setunion
% 2.95/1.53 @ ( dsetconstr @ B
% 2.95/1.53 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.95/1.53 @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ap_type,type,
% 2.95/1.53 ap: $i > $i > $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(app_type,type,
% 2.95/1.53 app: $o ).
% 2.95/1.53
% 2.95/1.53 thf(app,definition,
% 2.95/1.53 ( app
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(infuncsetfunc_type,type,
% 2.95/1.53 infuncsetfunc: $o ).
% 2.95/1.53
% 2.95/1.53 thf(infuncsetfunc,definition,
% 2.95/1.53 ( infuncsetfunc
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ap2p_type,type,
% 2.95/1.53 ap2p: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ap2p,definition,
% 2.95/1.53 ( ap2p
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(funcinfuncset_type,type,
% 2.95/1.53 funcinfuncset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcinfuncset,definition,
% 2.95/1.53 ( funcinfuncset
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(lamProp_type,type,
% 2.95/1.53 lamProp: $o ).
% 2.95/1.53
% 2.95/1.53 thf(lamProp,definition,
% 2.95/1.53 ( lamProp
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.95/1.53 => ( func @ A @ B
% 2.95/1.53 @ ( dpsetconstr @ A @ B
% 2.95/1.53 @ ^ [Xx: $i,Xy: $i] :
% 2.95/1.53 ( ( Xf @ Xx )
% 2.95/1.53 = Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(lam_type,type,
% 2.95/1.53 lam: $i > $i > ( $i > $i ) > $i ).
% 2.95/1.53
% 2.95/1.53 thf(lamp_type,type,
% 2.95/1.53 lamp: $o ).
% 2.95/1.53
% 2.95/1.53 thf(lamp,definition,
% 2.95/1.53 ( lamp
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.95/1.53 => ( func @ A @ B
% 2.95/1.53 @ ( lam @ A @ B
% 2.95/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(lam2p_type,type,
% 2.95/1.53 lam2p: $o ).
% 2.95/1.53
% 2.95/1.53 thf(lam2p,definition,
% 2.95/1.53 ( lam2p
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.95/1.53 => ( in
% 2.95/1.53 @ ( lam @ A @ B
% 2.95/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.95/1.53 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(brelnall1_type,type,
% 2.95/1.53 brelnall1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(brelnall1,definition,
% 2.95/1.53 ( brelnall1
% 2.95/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.95/1.53 ( ( breln @ A @ B @ R )
% 2.95/1.53 => ! [Xphi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ R )
% 2.95/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(brelnall2_type,type,
% 2.95/1.53 brelnall2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(brelnall2,definition,
% 2.95/1.53 ( brelnall2
% 2.95/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.95/1.53 ( ( breln @ A @ B @ R )
% 2.95/1.53 => ! [Xphi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ R )
% 2.95/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ex1E2_type,type,
% 2.95/1.53 ex1E2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ex1E2,definition,
% 2.95/1.53 ( ex1E2
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.95/1.53 ( ( ex1 @ A
% 2.95/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( Xphi @ Xx )
% 2.95/1.53 => ( ( Xphi @ Xy )
% 2.95/1.53 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(funcGraphProp1_type,type,
% 2.95/1.53 funcGraphProp1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcGraphProp1,definition,
% 2.95/1.53 ( funcGraphProp1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(funcGraphProp3_type,type,
% 2.95/1.53 funcGraphProp3: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcGraphProp3,definition,
% 2.95/1.53 ( funcGraphProp3
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(funcGraphProp2_type,type,
% 2.95/1.53 funcGraphProp2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcGraphProp2,definition,
% 2.95/1.53 ( funcGraphProp2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.95/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.95/1.53 = Xy ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(funcextLem_type,type,
% 2.95/1.53 funcextLem: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcextLem,definition,
% 2.95/1.53 ( funcextLem
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ! [Xg: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xg )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.95/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(funcGraphProp4_type,type,
% 2.95/1.53 funcGraphProp4: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcGraphProp4,definition,
% 2.95/1.53 ( funcGraphProp4
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.95/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.95/1.53 = Xy ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subbreln_type,type,
% 2.95/1.53 subbreln: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subbreln,definition,
% 2.95/1.53 ( subbreln
% 2.95/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.95/1.53 ( ( breln @ A @ B @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln @ A @ B @ S )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.95/1.53 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(eqbreln_type,type,
% 2.95/1.53 eqbreln: $o ).
% 2.95/1.53
% 2.95/1.53 thf(eqbreln,definition,
% 2.95/1.53 ( eqbreln
% 2.95/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.95/1.53 ( ( breln @ A @ B @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln @ A @ B @ S )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.95/1.53 => ( R = S ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(funcext_type,type,
% 2.95/1.53 funcext: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcext,definition,
% 2.95/1.53 ( funcext
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ! [Xg: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xg )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.95/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.95/1.53 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(funcext2_type,type,
% 2.95/1.53 funcext2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(funcext2,definition,
% 2.95/1.53 ( funcext2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ! [Xg: $i] :
% 2.95/1.53 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.95/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.95/1.53 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ap2apEq1_type,type,
% 2.95/1.53 ap2apEq1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ap2apEq1,definition,
% 2.95/1.53 ( ap2apEq1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.95/1.53 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ap2apEq2_type,type,
% 2.95/1.53 ap2apEq2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ap2apEq2,definition,
% 2.95/1.53 ( ap2apEq2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.95/1.53 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(beta1_type,type,
% 2.95/1.53 beta1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(beta1,definition,
% 2.95/1.53 ( beta1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( ap @ A @ B
% 2.95/1.53 @ ( lam @ A @ B
% 2.95/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.95/1.53 @ Xx )
% 2.95/1.53 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(eta1_type,type,
% 2.95/1.53 eta1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(eta1,definition,
% 2.95/1.53 ( eta1
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( func @ A @ B @ Xf )
% 2.95/1.53 => ( ( lam @ A @ B
% 2.95/1.53 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.95/1.53 = Xf ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(lam2lamEq_type,type,
% 2.95/1.53 lam2lamEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(lam2lamEq,definition,
% 2.95/1.53 ( lam2lamEq
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.95/1.53 => ( ( lam @ A @ B
% 2.95/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.95/1.53 = ( lam @ A @ B
% 2.95/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(beta2_type,type,
% 2.95/1.53 beta2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(beta2,definition,
% 2.95/1.53 ( beta2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( ap @ A @ B
% 2.95/1.53 @ ( lam @ A @ B
% 2.95/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.95/1.53 @ Xx )
% 2.95/1.53 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(eta2_type,type,
% 2.95/1.53 eta2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(eta2,definition,
% 2.95/1.53 ( eta2
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ( ( lam @ A @ B
% 2.95/1.53 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.95/1.53 = Xf ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(iffalseProp1_type,type,
% 2.95/1.53 iffalseProp1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(iffalseProp1,definition,
% 2.95/1.53 ( iffalseProp1
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ~ Xphi
% 2.95/1.53 => ( in @ Xy
% 2.95/1.53 @ ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xz: $i] :
% 2.95/1.53 ( ( Xphi
% 2.95/1.53 & ( Xz = Xx ) )
% 2.95/1.53 | ( ~ Xphi
% 2.95/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(iffalseProp2_type,type,
% 2.95/1.53 iffalseProp2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(iffalseProp2,definition,
% 2.95/1.53 ( iffalseProp2
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ~ Xphi
% 2.95/1.53 => ( ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xz: $i] :
% 2.95/1.53 ( ( Xphi
% 2.95/1.53 & ( Xz = Xx ) )
% 2.95/1.53 | ( ~ Xphi
% 2.95/1.53 & ( Xz = Xy ) ) ) )
% 2.95/1.53 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(iftrueProp1_type,type,
% 2.95/1.53 iftrueProp1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(iftrueProp1,definition,
% 2.95/1.53 ( iftrueProp1
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( Xphi
% 2.95/1.53 => ( in @ Xx
% 2.95/1.53 @ ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xz: $i] :
% 2.95/1.53 ( ( Xphi
% 2.95/1.53 & ( Xz = Xx ) )
% 2.95/1.53 | ( ~ Xphi
% 2.95/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(iftrueProp2_type,type,
% 2.95/1.53 iftrueProp2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(iftrueProp2,definition,
% 2.95/1.53 ( iftrueProp2
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( Xphi
% 2.95/1.53 => ( ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xz: $i] :
% 2.95/1.53 ( ( Xphi
% 2.95/1.53 & ( Xz = Xx ) )
% 2.95/1.53 | ( ~ Xphi
% 2.95/1.53 & ( Xz = Xy ) ) ) )
% 2.95/1.53 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(ifSingleton_type,type,
% 2.95/1.53 ifSingleton: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ifSingleton,definition,
% 2.95/1.53 ( ifSingleton
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( singleton
% 2.95/1.53 @ ( dsetconstr @ A
% 2.95/1.53 @ ^ [Xz: $i] :
% 2.95/1.53 ( ( Xphi
% 2.95/1.53 & ( Xz = Xx ) )
% 2.95/1.53 | ( ~ Xphi
% 2.95/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(if_type,type,
% 2.95/1.53 if: $i > $o > $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(ifp_type,type,
% 2.95/1.53 ifp: $o ).
% 2.95/1.53
% 2.95/1.53 thf(ifp,definition,
% 2.95/1.53 ( ifp
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(theeq_type,type,
% 2.95/1.53 theeq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(theeq,definition,
% 2.95/1.53 ( theeq
% 2.95/1.53 = ( ! [X: $i] :
% 2.95/1.53 ( ( singleton @ X )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ X )
% 2.95/1.53 => ( ( setunion @ X )
% 2.95/1.53 = Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(iftrue_type,type,
% 2.95/1.53 iftrue: $o ).
% 2.95/1.53
% 2.95/1.53 thf(iftrue,definition,
% 2.95/1.53 ( iftrue
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( Xphi
% 2.95/1.53 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.95/1.53 = Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(iffalse_type,type,
% 2.95/1.53 iffalse: $o ).
% 2.95/1.53
% 2.95/1.53 thf(iffalse,definition,
% 2.95/1.53 ( iffalse
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ~ Xphi
% 2.95/1.53 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.95/1.53 = Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(iftrueorfalse_type,type,
% 2.95/1.53 iftrueorfalse: $o ).
% 2.95/1.53
% 2.95/1.53 thf(iftrueorfalse,definition,
% 2.95/1.53 ( iftrueorfalse
% 2.95/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectT_lem_type,type,
% 2.95/1.53 binintersectT_lem: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectT_lem,definition,
% 2.95/1.53 ( binintersectT_lem
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionT_lem_type,type,
% 2.95/1.53 binunionT_lem: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionT_lem,definition,
% 2.95/1.53 ( binunionT_lem
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powersetT_lem_type,type,
% 2.95/1.53 powersetT_lem: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powersetT_lem,definition,
% 2.95/1.53 ( powersetT_lem
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setminusT_lem_type,type,
% 2.95/1.53 setminusT_lem: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setminusT_lem,definition,
% 2.95/1.53 ( setminusT_lem
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementT_lem_type,type,
% 2.95/1.53 complementT_lem: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementT_lem,definition,
% 2.95/1.53 ( complementT_lem
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setextT_type,type,
% 2.95/1.53 setextT: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setextT,definition,
% 2.95/1.53 ( setextT
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ X )
% 2.95/1.53 => ( in @ Xx @ Y ) ) )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ Y )
% 2.95/1.53 => ( in @ Xx @ X ) ) )
% 2.95/1.53 => ( X = Y ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subsetTI_type,type,
% 2.95/1.53 subsetTI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subsetTI,definition,
% 2.95/1.53 ( subsetTI
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ X )
% 2.95/1.53 => ( in @ Xx @ Y ) ) )
% 2.95/1.53 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powersetTI1_type,type,
% 2.95/1.53 powersetTI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powersetTI1,definition,
% 2.95/1.53 ( powersetTI1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ X )
% 2.95/1.53 => ( in @ Xx @ Y ) ) )
% 2.95/1.53 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(powersetTE1_type,type,
% 2.95/1.53 powersetTE1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(powersetTE1,definition,
% 2.95/1.53 ( powersetTE1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ X @ ( powerset @ Y ) )
% 2.95/1.53 => ( ( in @ Xx @ X )
% 2.95/1.53 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementTI1_type,type,
% 2.95/1.53 complementTI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementTI1,definition,
% 2.95/1.53 ( complementTI1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ X )
% 2.95/1.53 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementTE1_type,type,
% 2.95/1.53 complementTE1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementTE1,definition,
% 2.95/1.53 ( complementTE1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.95/1.53 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectTELcontra_type,type,
% 2.95/1.53 binintersectTELcontra: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectTELcontra,definition,
% 2.95/1.53 ( binintersectTELcontra
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ X )
% 2.95/1.53 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectTERcontra_type,type,
% 2.95/1.53 binintersectTERcontra: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binintersectTERcontra,definition,
% 2.95/1.53 ( binintersectTERcontra
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ Y )
% 2.95/1.53 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(contrasubsetT_type,type,
% 2.95/1.53 contrasubsetT: $o ).
% 2.95/1.53
% 2.95/1.53 thf(contrasubsetT,definition,
% 2.95/1.53 ( contrasubsetT
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.95/1.53 => ( ( in @ Xx @ Y )
% 2.95/1.53 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(contrasubsetT1_type,type,
% 2.95/1.53 contrasubsetT1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(contrasubsetT1,definition,
% 2.95/1.53 ( contrasubsetT1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( subset @ X @ Y )
% 2.95/1.53 => ( ~ ( in @ Xx @ Y )
% 2.95/1.53 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(contrasubsetT2_type,type,
% 2.95/1.53 contrasubsetT2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(contrasubsetT2,definition,
% 2.95/1.53 ( contrasubsetT2
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ X @ Y )
% 2.95/1.53 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(contrasubsetT3_type,type,
% 2.95/1.53 contrasubsetT3: $o ).
% 2.95/1.53
% 2.95/1.53 thf(contrasubsetT3,definition,
% 2.95/1.53 ( contrasubsetT3
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.95/1.53 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementI1_type,type,
% 2.95/1.53 doubleComplementI1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementI1,definition,
% 2.95/1.53 ( doubleComplementI1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ X )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementE1_type,type,
% 2.95/1.53 doubleComplementE1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementE1,definition,
% 2.95/1.53 ( doubleComplementE1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.95/1.53 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementSub1_type,type,
% 2.95/1.53 doubleComplementSub1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementSub1,definition,
% 2.95/1.53 ( doubleComplementSub1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementSub2_type,type,
% 2.95/1.53 doubleComplementSub2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementSub2,definition,
% 2.95/1.53 ( doubleComplementSub2
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementEq_type,type,
% 2.95/1.53 doubleComplementEq: $o ).
% 2.95/1.53
% 2.95/1.53 thf(doubleComplementEq,definition,
% 2.95/1.53 ( doubleComplementEq
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ( X
% 2.95/1.53 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementTnotintersectT_type,type,
% 2.95/1.53 complementTnotintersectT: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementTnotintersectT,definition,
% 2.95/1.53 ( complementTnotintersectT
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.95/1.53 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementImpComplementIntersect_type,type,
% 2.95/1.53 complementImpComplementIntersect: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementImpComplementIntersect,definition,
% 2.95/1.53 ( complementImpComplementIntersect
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementSubsetComplementIntersect_type,type,
% 2.95/1.53 complementSubsetComplementIntersect: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementSubsetComplementIntersect,definition,
% 2.95/1.53 ( complementSubsetComplementIntersect
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementInPowersetComplementIntersect_type,type,
% 2.95/1.53 complementInPowersetComplementIntersect: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementInPowersetComplementIntersect,definition,
% 2.95/1.53 ( complementInPowersetComplementIntersect
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(contraSubsetComplement_type,type,
% 2.95/1.53 contraSubsetComplement: $o ).
% 2.95/1.53
% 2.95/1.53 thf(contraSubsetComplement,definition,
% 2.95/1.53 ( contraSubsetComplement
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ Y )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementTcontraSubset_type,type,
% 2.95/1.53 complementTcontraSubset: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementTcontraSubset,definition,
% 2.95/1.53 ( complementTcontraSubset
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.95/1.53 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionTILcontra_type,type,
% 2.95/1.53 binunionTILcontra: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionTILcontra,definition,
% 2.95/1.53 ( binunionTILcontra
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.95/1.53 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionTIRcontra_type,type,
% 2.95/1.53 binunionTIRcontra: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionTIRcontra,definition,
% 2.95/1.53 ( binunionTIRcontra
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.95/1.53 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(inIntersectImpInUnion_type,type,
% 2.95/1.53 inIntersectImpInUnion: $o ).
% 2.95/1.53
% 2.95/1.53 thf(inIntersectImpInUnion,definition,
% 2.95/1.53 ( inIntersectImpInUnion
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.95/1.53 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(inIntersectImpInUnion2_type,type,
% 2.95/1.53 inIntersectImpInUnion2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(inIntersectImpInUnion2,definition,
% 2.95/1.53 ( inIntersectImpInUnion2
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.95/1.53 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(inIntersectImpInIntersectUnions_type,type,
% 2.95/1.53 inIntersectImpInIntersectUnions: $o ).
% 2.95/1.53
% 2.95/1.53 thf(inIntersectImpInIntersectUnions,definition,
% 2.95/1.53 ( inIntersectImpInIntersectUnions
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.95/1.53 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(intersectInPowersetIntersectUnions_type,type,
% 2.95/1.53 intersectInPowersetIntersectUnions: $o ).
% 2.95/1.53
% 2.95/1.53 thf(intersectInPowersetIntersectUnions,definition,
% 2.95/1.53 ( intersectInPowersetIntersectUnions
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(inComplementUnionImpNotIn1_type,type,
% 2.95/1.53 inComplementUnionImpNotIn1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(inComplementUnionImpNotIn1,definition,
% 2.95/1.53 ( inComplementUnionImpNotIn1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.95/1.53 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(inComplementUnionImpInComplement1_type,type,
% 2.95/1.53 inComplementUnionImpInComplement1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(inComplementUnionImpInComplement1,definition,
% 2.95/1.53 ( inComplementUnionImpInComplement1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionTE_type,type,
% 2.95/1.53 binunionTE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionTE,definition,
% 2.95/1.53 ( binunionTE
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xphi: $o,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.95/1.53 => ( ( ( in @ Xx @ X )
% 2.95/1.53 => Xphi )
% 2.95/1.53 => ( ( ( in @ Xx @ Y )
% 2.95/1.53 => Xphi )
% 2.95/1.53 => Xphi ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(binunionTEcontra_type,type,
% 2.95/1.53 binunionTEcontra: $o ).
% 2.95/1.53
% 2.95/1.53 thf(binunionTEcontra,definition,
% 2.95/1.53 ( binunionTEcontra
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ~ ( in @ Xx @ X )
% 2.95/1.53 => ( ~ ( in @ Xx @ Y )
% 2.95/1.53 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2a1_type,type,
% 2.95/1.53 demorgan2a1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2a1,definition,
% 2.95/1.53 ( demorgan2a1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(complementUnionInPowersetComplement_type,type,
% 2.95/1.53 complementUnionInPowersetComplement: $o ).
% 2.95/1.53
% 2.95/1.53 thf(complementUnionInPowersetComplement,definition,
% 2.95/1.53 ( complementUnionInPowersetComplement
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2a2_type,type,
% 2.95/1.53 demorgan2a2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2a2,definition,
% 2.95/1.53 ( demorgan2a2
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan1a_type,type,
% 2.95/1.53 demorgan1a: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan1a,definition,
% 2.95/1.53 ( demorgan1a
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.95/1.53 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan1b_type,type,
% 2.95/1.53 demorgan1b: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan1b,definition,
% 2.95/1.53 ( demorgan1b
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan1_type,type,
% 2.95/1.53 demorgan1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan1,definition,
% 2.95/1.53 ( demorgan1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.95/1.53 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2a_type,type,
% 2.95/1.53 demorgan2a: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2a,definition,
% 2.95/1.53 ( demorgan2a
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.95/1.53 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2b2_type,type,
% 2.95/1.53 demorgan2b2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2b2,definition,
% 2.95/1.53 ( demorgan2b2
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.95/1.53 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2b_type,type,
% 2.95/1.53 demorgan2b: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2b,definition,
% 2.95/1.53 ( demorgan2b
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.95/1.53 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2_type,type,
% 2.95/1.53 demorgan2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(demorgan2,definition,
% 2.95/1.53 ( demorgan2
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.95/1.53 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule0_type,type,
% 2.95/1.53 woz13rule0: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule0,definition,
% 2.95/1.53 ( woz13rule0
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.95/1.53 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule1_type,type,
% 2.95/1.53 woz13rule1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule1,definition,
% 2.95/1.53 ( woz13rule1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ X @ Z )
% 2.95/1.53 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule2_type,type,
% 2.95/1.53 woz13rule2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule2,definition,
% 2.95/1.53 ( woz13rule2
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ Y @ Z )
% 2.95/1.53 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule3_type,type,
% 2.95/1.53 woz13rule3: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule3,definition,
% 2.95/1.53 ( woz13rule3
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ X @ Y )
% 2.95/1.53 => ( ( subset @ X @ Z )
% 2.95/1.53 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule4_type,type,
% 2.95/1.53 woz13rule4: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz13rule4,definition,
% 2.95/1.53 ( woz13rule4
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ! [W: $i] :
% 2.95/1.53 ( ( in @ W @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ X @ Z )
% 2.95/1.53 => ( ( subset @ Y @ W )
% 2.95/1.53 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_1_type,type,
% 2.95/1.53 woz1_1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_1,definition,
% 2.95/1.53 ( woz1_1
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_2_type,type,
% 2.95/1.53 woz1_2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_2,definition,
% 2.95/1.53 ( woz1_2
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ! [W: $i] :
% 2.95/1.53 ( ( in @ W @ ( powerset @ A ) )
% 2.95/1.53 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.95/1.53 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_3_type,type,
% 2.95/1.53 woz1_3: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_3,definition,
% 2.95/1.53 ( woz1_3
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ! [Z: $i] :
% 2.95/1.53 ( ( in @ Z @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_4_type,type,
% 2.95/1.53 woz1_4: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_4,definition,
% 2.95/1.53 ( woz1_4
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.95/1.53 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_5_type,type,
% 2.95/1.53 woz1_5: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz1_5,definition,
% 2.95/1.53 ( woz1_5
% 2.95/1.53 = ( ! [A: $i,X: $i] :
% 2.95/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.95/1.53 => ! [Y: $i] :
% 2.95/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.95/1.53 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1_type,type,
% 2.95/1.53 breln1: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1all2_type,type,
% 2.95/1.53 breln1all2: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1all2,definition,
% 2.95/1.53 ( breln1all2
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [Xphi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ R )
% 2.95/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1Set_type,type,
% 2.95/1.53 breln1Set: $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(breln1SetBreln1_type,type,
% 2.95/1.53 breln1SetBreln1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1SetBreln1,definition,
% 2.95/1.53 ( breln1SetBreln1
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( in @ R @ ( breln1Set @ A ) )
% 2.95/1.53 => ( breln1 @ A @ R ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(transitive_type,type,
% 2.95/1.53 transitive: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(antisymmetric_type,type,
% 2.95/1.53 antisymmetric: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(reflexive_type,type,
% 2.95/1.53 reflexive: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(refltransitive_type,type,
% 2.95/1.53 refltransitive: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(refllinearorder_type,type,
% 2.95/1.53 refllinearorder: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(reflwellordering_type,type,
% 2.95/1.53 reflwellordering: $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(choice2fnsingleton_type,type,
% 2.95/1.53 choice2fnsingleton: $o ).
% 2.95/1.53
% 2.95/1.53 thf(choice2fnsingleton,definition,
% 2.95/1.53 ( choice2fnsingleton
% 2.95/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ? [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ B )
% 2.95/1.53 & ( Xphi @ Xx @ Xy ) ) )
% 2.95/1.53 => ! [R: $i] :
% 2.95/1.53 ( ( in @ R @ ( breln1Set @ B ) )
% 2.95/1.53 => ( ( reflwellordering @ B @ R )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ( singleton
% 2.95/1.53 @ ( dsetconstr @ B
% 2.95/1.53 @ ^ [Xy: $i] :
% 2.95/1.53 ( ( Xphi @ Xx @ Xy )
% 2.95/1.53 & ! [Xz: $i] :
% 2.95/1.53 ( ( in @ Xz @ B )
% 2.95/1.53 => ( ( Xphi @ Xx @ Xz )
% 2.95/1.53 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(setOfPairsIsBReln1_type,type,
% 2.95/1.53 setOfPairsIsBReln1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(setOfPairsIsBReln1,definition,
% 2.95/1.53 ( setOfPairsIsBReln1
% 2.95/1.53 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 2.95/1.53 ( breln1 @ A
% 2.95/1.53 @ ( dpsetconstr @ A @ A
% 2.95/1.53 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1all1_type,type,
% 2.95/1.53 breln1all1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1all1,definition,
% 2.95/1.53 ( breln1all1
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [Xphi: $i > $o] :
% 2.95/1.53 ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ R )
% 2.95/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(subbreln1_type,type,
% 2.95/1.53 subbreln1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(subbreln1,definition,
% 2.95/1.53 ( subbreln1
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.95/1.53 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(eqbreln1_type,type,
% 2.95/1.53 eqbreln1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(eqbreln1,definition,
% 2.95/1.53 ( eqbreln1
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.95/1.53 => ( ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.95/1.53 => ( R = S ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1invset_type,type,
% 2.95/1.53 breln1invset: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(breln1invprop_type,type,
% 2.95/1.53 breln1invprop: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1invprop,definition,
% 2.95/1.53 ( breln1invprop
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1invI_type,type,
% 2.95/1.53 breln1invI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1invI,definition,
% 2.95/1.53 ( breln1invI
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1invE_type,type,
% 2.95/1.53 breln1invE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1invE,definition,
% 2.95/1.53 ( breln1invE
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compset_type,type,
% 2.95/1.53 breln1compset: $i > $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compprop_type,type,
% 2.95/1.53 breln1compprop: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compprop,definition,
% 2.95/1.53 ( breln1compprop
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compI_type,type,
% 2.95/1.53 breln1compI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compI,definition,
% 2.95/1.53 ( breln1compI
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ! [Xz: $i] :
% 2.95/1.53 ( ( in @ Xz @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.95/1.53 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compE_type,type,
% 2.95/1.53 breln1compE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compE,definition,
% 2.95/1.53 ( breln1compE
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.95/1.53 => ? [Xz: $i] :
% 2.95/1.53 ( ( in @ Xz @ A )
% 2.95/1.53 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.95/1.53 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compEex_type,type,
% 2.95/1.53 breln1compEex: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1compEex,definition,
% 2.95/1.53 ( breln1compEex
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.95/1.53 => ! [Xphi: $o] :
% 2.95/1.53 ( ! [Xz: $i] :
% 2.95/1.53 ( ( in @ Xz @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.95/1.53 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.95/1.53 => Xphi ) ) )
% 2.95/1.53 => Xphi ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionprop_type,type,
% 2.95/1.53 breln1unionprop: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionprop,definition,
% 2.95/1.53 ( breln1unionprop
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionIL_type,type,
% 2.95/1.53 breln1unionIL: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionIL,definition,
% 2.95/1.53 ( breln1unionIL
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionIR_type,type,
% 2.95/1.53 breln1unionIR: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionIR,definition,
% 2.95/1.53 ( breln1unionIR
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionI_type,type,
% 2.95/1.53 breln1unionI: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionI,definition,
% 2.95/1.53 ( breln1unionI
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 2.95/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionE_type,type,
% 2.95/1.53 breln1unionE: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionE,definition,
% 2.95/1.53 ( breln1unionE
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionEcases_type,type,
% 2.95/1.53 breln1unionEcases: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionEcases,definition,
% 2.95/1.53 ( breln1unionEcases
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ A )
% 2.95/1.53 => ! [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.95/1.53 => ! [Xphi: $o] :
% 2.95/1.53 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.95/1.53 => Xphi )
% 2.95/1.53 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.95/1.53 => Xphi )
% 2.95/1.53 => Xphi ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionCommutes_type,type,
% 2.95/1.53 breln1unionCommutes: $o ).
% 2.95/1.53
% 2.95/1.53 thf(breln1unionCommutes,definition,
% 2.95/1.53 ( breln1unionCommutes
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ( ( binunion @ R @ S )
% 2.95/1.53 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz2Ex_type,type,
% 2.95/1.53 woz2Ex: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz2Ex,definition,
% 2.95/1.53 ( woz2Ex
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ( R
% 2.95/1.53 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz2W_type,type,
% 2.95/1.53 woz2W: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz2W,definition,
% 2.95/1.53 ( woz2W
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 2.95/1.53 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz2A_type,type,
% 2.95/1.53 woz2A: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz2A,definition,
% 2.95/1.53 ( woz2A
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [T: $i] :
% 2.95/1.53 ( ( breln1 @ A @ T )
% 2.95/1.53 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 2.95/1.53 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(woz2B_type,type,
% 2.95/1.53 woz2B: $o ).
% 2.95/1.53
% 2.95/1.53 thf(woz2B,definition,
% 2.95/1.53 ( woz2B
% 2.95/1.53 = ( ! [A: $i,R: $i] :
% 2.95/1.53 ( ( breln1 @ A @ R )
% 2.95/1.53 => ! [S: $i] :
% 2.95/1.53 ( ( breln1 @ A @ S )
% 2.95/1.53 => ! [T: $i] :
% 2.95/1.53 ( ( breln1 @ A @ T )
% 2.95/1.53 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 2.95/1.53 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(image1Ex_type,type,
% 2.95/1.53 image1Ex: $o ).
% 2.95/1.53
% 2.95/1.53 thf(image1Ex,definition,
% 2.95/1.53 ( image1Ex
% 2.95/1.53 = ( ! [A: $i,Xf: $i > $i] :
% 2.95/1.53 ? [B: $i] :
% 2.95/1.53 ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ B )
% 2.95/1.53 <=> ? [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 & ( Xx
% 2.95/1.53 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(image1Ex1_type,type,
% 2.95/1.53 image1Ex1: $o ).
% 2.95/1.53
% 2.95/1.53 thf(image1Ex1,definition,
% 2.95/1.53 ( image1Ex1
% 2.95/1.53 = ( ! [A: $i,Xf: $i > $i] :
% 2.95/1.53 ( exu
% 2.95/1.53 @ ^ [B: $i] :
% 2.95/1.53 ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ B )
% 2.95/1.53 <=> ? [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 & ( Xx
% 2.95/1.53 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(image1_type,type,
% 2.95/1.53 image1: $i > ( $i > $i ) > $i ).
% 2.95/1.53
% 2.95/1.53 thf(image1Equiv_type,type,
% 2.95/1.53 image1Equiv: $o ).
% 2.95/1.53
% 2.95/1.53 thf(image1Equiv,definition,
% 2.95/1.53 ( image1Equiv
% 2.95/1.53 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx
% 2.95/1.53 @ ( image1 @ A
% 2.95/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 2.95/1.53 <=> ? [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 & ( Xx
% 2.95/1.53 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(image1E_type,type,
% 2.95/1.53 image1E: $o ).
% 2.95/1.53
% 2.95/1.53 thf(image1E,definition,
% 2.95/1.53 ( image1E
% 2.95/1.53 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.95/1.53 ( ( in @ Xx
% 2.95/1.53 @ ( image1 @ A
% 2.95/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 2.95/1.53 => ? [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 & ( Xx
% 2.95/1.53 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(image1I_type,type,
% 2.95/1.53 image1I: $o ).
% 2.95/1.53
% 2.95/1.53 thf(image1I,definition,
% 2.95/1.53 ( image1I
% 2.95/1.53 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.95/1.53 ( ? [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 & ( Xx
% 2.95/1.53 = ( Xf @ Xy ) ) )
% 2.95/1.53 => ( in @ Xx
% 2.95/1.53 @ ( image1 @ A
% 2.95/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(injective_type,type,
% 2.95/1.53 injective: $i > $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(injFuncSet_type,type,
% 2.95/1.53 injFuncSet: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(injFuncInInjFuncSet_type,type,
% 2.95/1.53 injFuncInInjFuncSet: $o ).
% 2.95/1.53
% 2.95/1.53 thf(injFuncInInjFuncSet,definition,
% 2.95/1.53 ( injFuncInInjFuncSet
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.95/1.53 => ( ( injective @ A @ B @ Xf )
% 2.95/1.53 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(injFuncSetFuncIn_type,type,
% 2.95/1.53 injFuncSetFuncIn: $o ).
% 2.95/1.53
% 2.95/1.53 thf(injFuncSetFuncIn,definition,
% 2.95/1.53 ( injFuncSetFuncIn
% 2.95/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 2.95/1.53 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(injFuncSetFuncInj_type,type,
% 2.95/1.53 injFuncSetFuncInj: $o ).
% 2.95/1.53
% 2.95/1.53 thf(injFuncSetFuncInj,definition,
% 2.95/1.53 ( injFuncSetFuncInj
% 2.95/1.53 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 2.95/1.53 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(surjective_type,type,
% 2.95/1.53 surjective: $i > $i > $i > $o ).
% 2.95/1.53
% 2.95/1.53 thf(surjective,definition,
% 2.95/1.53 ( surjective
% 2.95/1.53 = ( ^ [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ! [Xx: $i] :
% 2.95/1.53 ( ( in @ Xx @ B )
% 2.95/1.53 => ? [Xy: $i] :
% 2.95/1.53 ( ( in @ Xy @ A )
% 2.95/1.53 & ( ( ap @ A @ B @ Xf @ Xy )
% 2.95/1.53 = Xx ) ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(surjFuncSet_type,type,
% 2.95/1.53 surjFuncSet: $i > $i > $i ).
% 2.95/1.53
% 2.95/1.53 thf(surjFuncSet,definition,
% 2.95/1.53 ( surjFuncSet
% 2.95/1.53 = ( ^ [A: $i,B: $i] :
% 2.95/1.53 ( dsetconstr @ ( funcSet @ A @ B )
% 2.95/1.53 @ ^ [Xf: $i] : ( surjective @ A @ B @ Xf ) ) ) ) ).
% 2.95/1.53
% 2.95/1.53 thf(surjFuncSetFuncIn,conjecture,
% 2.95/1.53 ( setextAx
% 2.95/1.53 => ( emptysetAx
% 2.95/1.53 => ( setadjoinAx
% 2.95/1.53 => ( powersetAx
% 2.95/1.53 => ( setunionAx
% 2.95/1.53 => ( omega0Ax
% 2.95/1.53 => ( omegaSAx
% 2.95/1.53 => ( omegaIndAx
% 2.95/1.53 => ( replAx
% 2.95/1.53 => ( foundationAx
% 2.95/1.53 => ( wellorderingAx
% 2.95/1.53 => ( descrp
% 2.95/1.53 => ( dsetconstrI
% 2.95/1.53 => ( dsetconstrEL
% 2.95/1.53 => ( dsetconstrER
% 2.95/1.53 => ( exuE1
% 2.95/1.53 => ( prop2setE
% 2.95/1.53 => ( emptysetE
% 2.95/1.53 => ( emptysetimpfalse
% 2.95/1.53 => ( notinemptyset
% 2.95/1.53 => ( exuE3e
% 2.95/1.53 => ( setext
% 2.95/1.53 => ( emptyI
% 2.95/1.53 => ( noeltsimpempty
% 2.95/1.53 => ( setbeta
% 2.95/1.53 => ( nonemptyE1
% 2.95/1.53 => ( nonemptyI
% 2.95/1.53 => ( nonemptyI1
% 2.95/1.53 => ( setadjoinIL
% 2.95/1.53 => ( emptyinunitempty
% 2.95/1.53 => ( setadjoinIR
% 2.95/1.53 => ( setadjoinE
% 2.95/1.53 => ( setadjoinOr
% 2.95/1.53 => ( setoftrueEq
% 2.95/1.53 => ( powersetI
% 2.95/1.53 => ( emptyinPowerset
% 2.95/1.53 => ( emptyInPowerset
% 2.95/1.53 => ( powersetE
% 2.95/1.53 => ( setunionI
% 2.95/1.53 => ( setunionE
% 2.95/1.53 => ( subPowSU
% 2.95/1.53 => ( exuE2
% 2.95/1.53 => ( nonemptyImpWitness
% 2.95/1.53 => ( uniqinunit
% 2.95/1.53 => ( notinsingleton
% 2.95/1.53 => ( eqinunit
% 2.95/1.53 => ( singletonsswitch
% 2.95/1.53 => ( upairsetE
% 2.95/1.53 => ( upairsetIL
% 2.95/1.53 => ( upairsetIR
% 2.95/1.53 => ( emptyE1
% 2.95/1.53 => ( vacuousDall
% 2.95/1.53 => ( quantDeMorgan1
% 2.95/1.53 => ( quantDeMorgan2
% 2.95/1.53 => ( quantDeMorgan3
% 2.95/1.53 => ( quantDeMorgan4
% 2.95/1.53 => ( prop2setI
% 2.95/1.53 => ( prop2set2propI
% 2.95/1.53 => ( notdexE
% 2.95/1.53 => ( notdallE
% 2.95/1.53 => ( exuI1
% 2.95/1.53 => ( exuI3
% 2.95/1.53 => ( exuI2
% 2.95/1.53 => ( inCongP
% 2.95/1.53 => ( in__Cong
% 2.95/1.53 => ( exuE3u
% 2.95/1.53 => ( exu__Cong
% 2.95/1.53 => ( emptyset__Cong
% 2.95/1.53 => ( setadjoin__Cong
% 2.95/1.53 => ( powerset__Cong
% 2.95/1.53 => ( setunion__Cong
% 2.95/1.53 => ( omega__Cong
% 2.95/1.53 => ( exuEu
% 2.95/1.53 => ( descr__Cong
% 2.95/1.53 => ( dsetconstr__Cong
% 2.95/1.53 => ( subsetI1
% 2.95/1.53 => ( eqimpsubset2
% 2.95/1.53 => ( eqimpsubset1
% 2.95/1.53 => ( subsetI2
% 2.95/1.53 => ( emptysetsubset
% 2.95/1.53 => ( subsetE
% 2.95/1.53 => ( subsetE2
% 2.95/1.53 => ( notsubsetI
% 2.95/1.53 => ( notequalI1
% 2.95/1.53 => ( notequalI2
% 2.95/1.53 => ( subsetRefl
% 2.95/1.53 => ( subsetTrans
% 2.95/1.53 => ( setadjoinSub
% 2.95/1.53 => ( setadjoinSub2
% 2.95/1.53 => ( subset2powerset
% 2.95/1.53 => ( setextsub
% 2.95/1.53 => ( subsetemptysetimpeq
% 2.95/1.53 => ( powersetI1
% 2.95/1.53 => ( powersetE1
% 2.95/1.53 => ( inPowerset
% 2.95/1.53 => ( powersetsubset
% 2.95/1.53 => ( sepInPowerset
% 2.95/1.53 => ( sepSubset
% 2.95/1.53 => ( binunionIL
% 2.95/1.53 => ( upairset2IR
% 2.95/1.53 => ( binunionIR
% 2.95/1.53 => ( binunionEcases
% 2.95/1.53 => ( binunionE
% 2.95/1.53 => ( binunionLsub
% 2.95/1.53 => ( binunionRsub
% 2.95/1.53 => ( binintersectI
% 2.95/1.53 => ( binintersectSubset5
% 2.95/1.53 => ( binintersectEL
% 2.95/1.53 => ( binintersectLsub
% 2.95/1.53 => ( binintersectSubset2
% 2.95/1.53 => ( binintersectSubset3
% 2.95/1.53 => ( binintersectER
% 2.95/1.53 => ( disjointsetsI1
% 2.95/1.53 => ( binintersectRsub
% 2.95/1.53 => ( binintersectSubset4
% 2.95/1.53 => ( binintersectSubset1
% 2.95/1.53 => ( bs114d
% 2.95/1.53 => ( setminusI
% 2.95/1.53 => ( setminusEL
% 2.95/1.53 => ( setminusER
% 2.95/1.53 => ( setminusSubset2
% 2.95/1.53 => ( setminusERneg
% 2.95/1.53 => ( setminusELneg
% 2.95/1.53 => ( setminusILneg
% 2.95/1.53 => ( setminusIRneg
% 2.95/1.53 => ( setminusLsub
% 2.95/1.53 => ( setminusSubset1
% 2.95/1.53 => ( symdiffE
% 2.95/1.53 => ( symdiffI1
% 2.95/1.53 => ( symdiffI2
% 2.95/1.53 => ( symdiffIneg1
% 2.95/1.53 => ( symdiffIneg2
% 2.95/1.53 => ( secondinupair
% 2.95/1.53 => ( setukpairIL
% 2.95/1.53 => ( setukpairIR
% 2.95/1.53 => ( kpairiskpair
% 2.95/1.53 => ( kpairp
% 2.95/1.53 => ( singletonsubset
% 2.95/1.53 => ( singletoninpowerset
% 2.95/1.53 => ( singletoninpowunion
% 2.95/1.53 => ( upairset2E
% 2.95/1.53 => ( upairsubunion
% 2.95/1.53 => ( upairinpowunion
% 2.95/1.53 => ( ubforcartprodlem1
% 2.95/1.53 => ( ubforcartprodlem2
% 2.95/1.53 => ( ubforcartprodlem3
% 2.95/1.53 => ( cartprodpairin
% 2.95/1.53 => ( cartprodmempair1
% 2.95/1.53 => ( cartprodmempair
% 2.95/1.53 => ( setunionE2
% 2.95/1.53 => ( setunionsingleton1
% 2.95/1.53 => ( setunionsingleton2
% 2.95/1.53 => ( setunionsingleton
% 2.95/1.53 => ( singletonprop
% 2.95/1.53 => ( ex1E1
% 2.95/1.53 => ( ex1I
% 2.95/1.53 => ( ex1I2
% 2.95/1.53 => ( singletonsuniq
% 2.95/1.53 => ( setukpairinjL1
% 2.95/1.53 => ( kfstsingleton
% 2.95/1.53 => ( theprop
% 2.95/1.53 => ( kfstpairEq
% 2.95/1.53 => ( cartprodfstin
% 2.95/1.53 => ( setukpairinjL2
% 2.95/1.53 => ( setukpairinjL
% 2.95/1.53 => ( setukpairinjR11
% 2.95/1.53 => ( setukpairinjR12
% 2.95/1.53 => ( setukpairinjR1
% 2.95/1.53 => ( upairequniteq
% 2.95/1.53 => ( setukpairinjR2
% 2.95/1.53 => ( setukpairinjR
% 2.95/1.53 => ( ksndsingleton
% 2.95/1.53 => ( ksndpairEq
% 2.95/1.53 => ( kpairsurjEq
% 2.95/1.53 => ( cartprodsndin
% 2.95/1.53 => ( cartprodpairmemEL
% 2.95/1.53 => ( cartprodpairmemER
% 2.95/1.53 => ( cartprodmempaircEq
% 2.95/1.53 => ( cartprodfstpairEq
% 2.95/1.53 => ( cartprodsndpairEq
% 2.95/1.53 => ( cartprodpairsurjEq
% 2.95/1.53 => ( dpsetconstrI
% 2.95/1.53 => ( dpsetconstrSub
% 2.95/1.53 => ( setOfPairsIsBReln
% 2.95/1.53 => ( dpsetconstrERa
% 2.95/1.53 => ( dpsetconstrEL1
% 2.95/1.53 => ( dpsetconstrEL2
% 2.95/1.53 => ( dpsetconstrER
% 2.95/1.53 => ( funcImageSingleton
% 2.95/1.53 => ( apProp
% 2.95/1.53 => ( app
% 2.95/1.53 => ( infuncsetfunc
% 2.95/1.53 => ( ap2p
% 2.95/1.53 => ( funcinfuncset
% 2.95/1.53 => ( lamProp
% 2.95/1.53 => ( lamp
% 2.95/1.53 => ( lam2p
% 2.95/1.53 => ( brelnall1
% 2.95/1.53 => ( brelnall2
% 2.95/1.53 => ( ex1E2
% 2.95/1.53 => ( funcGraphProp1
% 2.95/1.53 => ( funcGraphProp3
% 2.95/1.53 => ( funcGraphProp2
% 2.95/1.53 => ( funcextLem
% 2.95/1.53 => ( funcGraphProp4
% 2.95/1.53 => ( subbreln
% 2.95/1.53 => ( eqbreln
% 2.95/1.53 => ( funcext
% 2.95/1.53 => ( funcext2
% 2.95/1.53 => ( ap2apEq1
% 2.95/1.53 => ( ap2apEq2
% 2.95/1.53 => ( beta1
% 2.95/1.53 => ( eta1
% 2.95/1.53 => ( lam2lamEq
% 2.95/1.53 => ( beta2
% 2.95/1.53 => ( eta2
% 2.95/1.53 => ( iffalseProp1
% 2.95/1.53 => ( iffalseProp2
% 2.95/1.53 => ( iftrueProp1
% 2.95/1.53 => ( iftrueProp2
% 2.95/1.53 => ( ifSingleton
% 2.95/1.53 => ( ifp
% 2.95/1.53 => ( theeq
% 2.95/1.53 => ( iftrue
% 2.95/1.53 => ( iffalse
% 2.95/1.53 => ( iftrueorfalse
% 2.95/1.53 => ( binintersectT_lem
% 2.95/1.53 => ( binunionT_lem
% 2.95/1.53 => ( powersetT_lem
% 2.95/1.53 => ( setminusT_lem
% 2.95/1.53 => ( complementT_lem
% 2.95/1.53 => ( setextT
% 2.95/1.53 => ( subsetTI
% 2.95/1.53 => ( powersetTI1
% 2.95/1.53 => ( powersetTE1
% 2.95/1.53 => ( complementTI1
% 2.95/1.53 => ( complementTE1
% 2.95/1.53 => ( binintersectTELcontra
% 2.95/1.53 => ( binintersectTERcontra
% 2.95/1.53 => ( contrasubsetT
% 2.95/1.53 => ( contrasubsetT1
% 2.95/1.53 => ( contrasubsetT2
% 2.95/1.53 => ( contrasubsetT3
% 2.95/1.53 => ( doubleComplementI1
% 2.95/1.53 => ( doubleComplementE1
% 2.95/1.53 => ( doubleComplementSub1
% 2.95/1.53 => ( doubleComplementSub2
% 2.95/1.53 => ( doubleComplementEq
% 2.95/1.53 => ( complementTnotintersectT
% 2.95/1.53 => ( complementImpComplementIntersect
% 2.95/1.53 => ( complementSubsetComplementIntersect
% 2.95/1.53 => ( complementInPowersetComplementIntersect
% 2.95/1.53 => ( contraSubsetComplement
% 2.95/1.53 => ( complementTcontraSubset
% 2.95/1.53 => ( binunionTILcontra
% 2.95/1.53 => ( binunionTIRcontra
% 2.95/1.53 => ( inIntersectImpInUnion
% 2.95/1.53 => ( inIntersectImpInUnion2
% 2.95/1.53 => ( inIntersectImpInIntersectUnions
% 2.95/1.53 => ( intersectInPowersetIntersectUnions
% 2.95/1.53 => ( inComplementUnionImpNotIn1
% 2.95/1.53 => ( inComplementUnionImpInComplement1
% 2.95/1.53 => ( binunionTE
% 2.95/1.53 => ( binunionTEcontra
% 2.95/1.53 => ( demorgan2a1
% 2.95/1.53 => ( complementUnionInPowersetComplement
% 2.95/1.53 => ( demorgan2a2
% 2.95/1.53 => ( demorgan1a
% 2.95/1.53 => ( demorgan1b
% 2.95/1.53 => ( demorgan1
% 2.95/1.53 => ( demorgan2a
% 2.95/1.53 => ( demorgan2b2
% 2.95/1.53 => ( demorgan2b
% 2.95/1.53 => ( demorgan2
% 2.95/1.53 => ( woz13rule0
% 2.95/1.53 => ( woz13rule1
% 2.95/1.53 => ( woz13rule2
% 2.95/1.53 => ( woz13rule3
% 2.95/1.53 => ( woz13rule4
% 2.95/1.53 => ( woz1_1
% 2.95/1.53 => ( woz1_2
% 2.95/1.53 => ( woz1_3
% 2.95/1.53 => ( woz1_4
% 2.95/1.53 => ( woz1_5
% 2.95/1.53 => ( breln1all2
% 2.95/1.53 => ( breln1SetBreln1
% 2.95/1.53 => ( choice2fnsingleton
% 2.95/1.53 => ( setOfPairsIsBReln1
% 2.95/1.53 => ( breln1all1
% 2.95/1.53 => ( subbreln1
% 2.95/1.53 => ( eqbreln1
% 2.95/1.53 => ( breln1invprop
% 2.95/1.53 => ( breln1invI
% 2.95/1.53 => ( breln1invE
% 2.95/1.53 => ( breln1compprop
% 2.95/1.53 => ( breln1compI
% 2.95/1.53 => ( breln1compE
% 2.95/1.53 => ( breln1compEex
% 2.95/1.53 => ( breln1unionprop
% 2.95/1.53 => ( breln1unionIL
% 2.95/1.53 => ( breln1unionIR
% 2.95/1.53 => ( breln1unionI
% 2.95/1.53 => ( breln1unionE
% 2.95/1.53 => ( breln1unionEcases
% 2.95/1.53 => ( breln1unionCommutes
% 2.95/1.53 => ( woz2Ex
% 2.95/1.53 => ( woz2W
% 2.95/1.53 => ( woz2A
% 2.95/1.53 => ( woz2B
% 2.95/1.53 => ( image1Ex
% 2.95/1.53 => ( image1Ex1
% 2.95/1.53 => ( image1Equiv
% 2.95/1.53 => ( image1E
% 2.95/1.53 => ( image1I
% 2.95/1.53 => ( injFuncInInjFuncSet
% 2.95/1.53 => ( injFuncSetFuncIn
% 2.95/1.53 => ( injFuncSetFuncInj
% 2.95/1.53 => ! [A: $i,B: $i,Xf: $i] :
% 2.95/1.53 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 2.95/1.53 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.95/1.53 ------------------------
% 2.95/1.56 % (27173)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_27071 for (2999ds/4Mi)
% 2.95/1.56 % (27176)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_27071 for (2999ds/2Mi)
% 2.95/1.56 % (27176)Instruction limit reached!
% 2.95/1.56 % (27176)------------------------------
% 2.95/1.56 % (27176)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.56 % (27176)Termination reason: Unknown
% 2.95/1.56 % (27176)Termination phase: shuffling
% 2.95/1.56
% 2.95/1.56 % (27176)Memory used [KB]: 1663
% 2.95/1.56 % (27176)Time elapsed: 0.002 s
% 2.95/1.56 % (27176)Instructions burned: 2 (million)
% 2.95/1.56 % (27176)------------------------------
% 2.95/1.56 % (27176)------------------------------
% 2.95/1.56 % (27175)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/2Mi)
% 2.95/1.56 % (27173)Instruction limit reached!
% 2.95/1.56 % (27173)------------------------------
% 2.95/1.56 % (27173)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.56 % (27173)Termination reason: Unknown
% 2.95/1.56 % (27173)Termination phase: shuffling
% 2.95/1.56
% 2.95/1.56 % (27173)Memory used [KB]: 1663
% 2.95/1.56 % (27173)Time elapsed: 0.003 s
% 2.95/1.56 % (27173)Instructions burned: 5 (million)
% 2.95/1.56 % (27173)------------------------------
% 2.95/1.56 % (27173)------------------------------
% 2.95/1.56 % (27175)Instruction limit reached!
% 2.95/1.56 % (27175)------------------------------
% 2.95/1.56 % (27175)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.56 % (27175)Termination reason: Unknown
% 2.95/1.56 % (27175)Termination phase: shuffling
% 2.95/1.56
% 2.95/1.56 % (27175)Memory used [KB]: 1663
% 2.95/1.56 % (27175)Time elapsed: 0.002 s
% 2.95/1.56 % (27175)Instructions burned: 3 (million)
% 2.95/1.56 % (27175)------------------------------
% 2.95/1.56 % (27175)------------------------------
% 2.95/1.57 % (27177)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_27071 for (2999ds/275Mi)
% 2.95/1.57 % (27178)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/18Mi)
% 2.95/1.57 % (27174)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_27071 for (2999ds/27Mi)
% 2.95/1.57 % (27172)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/183Mi)
% 2.95/1.57 % (27180)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_27071 for (2999ds/37Mi)
% 2.95/1.57 % (27181)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_27071 for (2999ds/15Mi)
% 2.95/1.58 % (27178)Instruction limit reached!
% 2.95/1.58 % (27178)------------------------------
% 2.95/1.58 % (27178)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.58 % (27178)Termination reason: Unknown
% 2.95/1.58 % (27178)Termination phase: shuffling
% 2.95/1.58
% 2.95/1.58 % (27178)Memory used [KB]: 1918
% 2.95/1.58 % (27178)Time elapsed: 0.011 s
% 2.95/1.58 % (27178)Instructions burned: 19 (million)
% 2.95/1.58 % (27178)------------------------------
% 2.95/1.58 % (27178)------------------------------
% 2.95/1.58 % (27181)Instruction limit reached!
% 2.95/1.58 % (27181)------------------------------
% 2.95/1.58 % (27181)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.58 % (27181)Termination reason: Unknown
% 2.95/1.58 % (27181)Termination phase: shuffling
% 2.95/1.58
% 2.95/1.58 % (27181)Memory used [KB]: 1918
% 2.95/1.58 % (27181)Time elapsed: 0.007 s
% 2.95/1.58 % (27181)Instructions burned: 18 (million)
% 2.95/1.58 % (27181)------------------------------
% 2.95/1.58 % (27181)------------------------------
% 2.95/1.58 % (27179)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/3Mi)
% 2.95/1.58 % (27179)Instruction limit reached!
% 2.95/1.58 % (27179)------------------------------
% 2.95/1.58 % (27179)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.58 % (27179)Termination reason: Unknown
% 2.95/1.58 % (27179)Termination phase: shuffling
% 2.95/1.58
% 2.95/1.58 % (27179)Memory used [KB]: 1663
% 2.95/1.58 % (27179)Time elapsed: 0.004 s
% 2.95/1.58 % (27179)Instructions burned: 5 (million)
% 2.95/1.58 % (27179)------------------------------
% 2.95/1.58 % (27179)------------------------------
% 2.95/1.58 % (27174)Instruction limit reached!
% 2.95/1.58 % (27174)------------------------------
% 2.95/1.58 % (27174)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.58 % (27174)Termination reason: Unknown
% 2.95/1.58 % (27174)Termination phase: shuffling
% 2.95/1.58
% 2.95/1.58 % (27174)Memory used [KB]: 2174
% 2.95/1.58 % (27174)Time elapsed: 0.015 s
% 2.95/1.58 % (27174)Instructions burned: 29 (million)
% 2.95/1.58 % (27174)------------------------------
% 2.95/1.58 % (27174)------------------------------
% 2.95/1.58 % (27180)Instruction limit reached!
% 2.95/1.58 % (27180)------------------------------
% 2.95/1.58 % (27180)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.58 % (27180)Termination reason: Unknown
% 2.95/1.58 % (27180)Termination phase: shuffling
% 2.95/1.58
% 2.95/1.58 % (27180)Memory used [KB]: 2302
% 2.95/1.58 % (27180)Time elapsed: 0.011 s
% 2.95/1.58 % (27180)Instructions burned: 38 (million)
% 2.95/1.58 % (27180)------------------------------
% 2.95/1.58 % (27180)------------------------------
% 2.95/1.59 % (27183)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_27071 for (2999ds/1041Mi)
% 2.95/1.59 % (27186)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_27071 for (2999ds/3Mi)
% 2.95/1.59 % (27186)Instruction limit reached!
% 2.95/1.59 % (27186)------------------------------
% 2.95/1.59 % (27186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.59 % (27186)Termination reason: Unknown
% 2.95/1.59 % (27186)Termination phase: shuffling
% 2.95/1.59
% 2.95/1.59 % (27186)Memory used [KB]: 1663
% 2.95/1.59 % (27186)Time elapsed: 0.002 s
% 2.95/1.59 % (27186)Instructions burned: 3 (million)
% 2.95/1.59 % (27186)------------------------------
% 2.95/1.59 % (27186)------------------------------
% 2.95/1.59 % (27182)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_27071 for (2999ds/3Mi)
% 2.95/1.59 % (27182)Instruction limit reached!
% 2.95/1.59 % (27182)------------------------------
% 2.95/1.59 % (27182)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.59 % (27182)Termination reason: Unknown
% 2.95/1.59 % (27182)Termination phase: shuffling
% 2.95/1.59
% 2.95/1.59 % (27182)Memory used [KB]: 1663
% 2.95/1.59 % (27182)Time elapsed: 0.003 s
% 2.95/1.59 % (27182)Instructions burned: 3 (million)
% 2.95/1.59 % (27182)------------------------------
% 2.95/1.59 % (27182)------------------------------
% 2.95/1.60 % (27184)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/7Mi)
% 2.99/1.60 % (27184)Instruction limit reached!
% 2.99/1.60 % (27184)------------------------------
% 2.99/1.60 % (27184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.60 % (27184)Termination reason: Unknown
% 2.99/1.60 % (27184)Termination phase: shuffling
% 2.99/1.60
% 2.99/1.60 % (27184)Memory used [KB]: 1663
% 2.99/1.60 % (27184)Time elapsed: 0.005 s
% 2.99/1.60 % (27184)Instructions burned: 8 (million)
% 2.99/1.60 % (27184)------------------------------
% 2.99/1.60 % (27184)------------------------------
% 2.99/1.60 % (27187)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_27071 for (2999ds/3Mi)
% 2.99/1.60 % (27185)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/16Mi)
% 2.99/1.60 % (27187)Instruction limit reached!
% 2.99/1.60 % (27187)------------------------------
% 2.99/1.60 % (27187)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.60 % (27187)Termination reason: Unknown
% 2.99/1.60 % (27187)Termination phase: shuffling
% 2.99/1.60
% 2.99/1.60 % (27187)Memory used [KB]: 1663
% 2.99/1.60 % (27187)Time elapsed: 0.002 s
% 2.99/1.60 % (27187)Instructions burned: 3 (million)
% 2.99/1.60 % (27187)------------------------------
% 2.99/1.60 % (27187)------------------------------
% 2.99/1.61 % (27185)Instruction limit reached!
% 2.99/1.61 % (27185)------------------------------
% 2.99/1.61 % (27185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.61 % (27185)Termination reason: Unknown
% 2.99/1.61 % (27185)Termination phase: shuffling
% 2.99/1.61
% 2.99/1.61 % (27185)Memory used [KB]: 2046
% 2.99/1.61 % (27185)Time elapsed: 0.009 s
% 2.99/1.61 % (27185)Instructions burned: 16 (million)
% 2.99/1.61 % (27185)------------------------------
% 2.99/1.61 % (27185)------------------------------
% 2.99/1.61 % (27188)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/7Mi)
% 2.99/1.61 % (27190)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/4Mi)
% 2.99/1.61 % (27190)Instruction limit reached!
% 2.99/1.61 % (27190)------------------------------
% 2.99/1.61 % (27190)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.61 % (27190)Termination reason: Unknown
% 2.99/1.61 % (27190)Termination phase: shuffling
% 2.99/1.61
% 2.99/1.61 % (27190)Memory used [KB]: 1663
% 2.99/1.61 % (27190)Time elapsed: 0.003 s
% 2.99/1.61 % (27190)Instructions burned: 7 (million)
% 2.99/1.61 % (27190)------------------------------
% 2.99/1.61 % (27190)------------------------------
% 2.99/1.61 % (27188)Instruction limit reached!
% 2.99/1.61 % (27188)------------------------------
% 2.99/1.61 % (27188)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.61 % (27188)Termination reason: Unknown
% 2.99/1.61 % (27188)Termination phase: shuffling
% 2.99/1.61
% 2.99/1.61 % (27188)Memory used [KB]: 1663
% 2.99/1.61 % (27188)Time elapsed: 0.005 s
% 2.99/1.61 % (27188)Instructions burned: 7 (million)
% 2.99/1.61 % (27188)------------------------------
% 2.99/1.61 % (27188)------------------------------
% 2.99/1.61 % (27189)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/3Mi)
% 2.99/1.61 % (27172)Instruction limit reached!
% 2.99/1.61 % (27172)------------------------------
% 2.99/1.61 % (27172)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.61 % (27172)Termination reason: Unknown
% 2.99/1.61 % (27172)Termination phase: Property scanning
% 2.99/1.61
% 2.99/1.61 % (27172)Memory used [KB]: 3070
% 2.99/1.61 % (27172)Time elapsed: 0.047 s
% 2.99/1.61 % (27172)Instructions burned: 184 (million)
% 2.99/1.61 % (27172)------------------------------
% 2.99/1.61 % (27172)------------------------------
% 2.99/1.62 % (27189)Instruction limit reached!
% 2.99/1.62 % (27189)------------------------------
% 2.99/1.62 % (27189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.62 % (27189)Termination reason: Unknown
% 2.99/1.62 % (27189)Termination phase: shuffling
% 2.99/1.62
% 2.99/1.62 % (27189)Memory used [KB]: 1663
% 2.99/1.62 % (27189)Time elapsed: 0.003 s
% 2.99/1.62 % (27189)Instructions burned: 3 (million)
% 2.99/1.62 % (27189)------------------------------
% 2.99/1.62 % (27189)------------------------------
% 2.99/1.62 % (27192)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/710Mi)
% 2.99/1.62 % (27191)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/18Mi)
% 2.99/1.62 % (27193)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/6Mi)
% 2.99/1.63 % (27193)Instruction limit reached!
% 2.99/1.63 % (27193)------------------------------
% 2.99/1.63 % (27193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.63 % (27193)Termination reason: Unknown
% 2.99/1.63 % (27193)Termination phase: shuffling
% 2.99/1.63
% 2.99/1.63 % (27193)Memory used [KB]: 1663
% 2.99/1.63 % (27193)Time elapsed: 0.003 s
% 2.99/1.63 % (27193)Instructions burned: 7 (million)
% 2.99/1.63 % (27193)------------------------------
% 2.99/1.63 % (27193)------------------------------
% 2.99/1.63 % (27194)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_27071 for (2999ds/902Mi)
% 2.99/1.63 % (27195)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_27071 for (2999ds/21Mi)
% 2.99/1.63 % (27191)Instruction limit reached!
% 2.99/1.63 % (27191)------------------------------
% 2.99/1.63 % (27191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.63 % (27191)Termination reason: Unknown
% 2.99/1.63 % (27191)Termination phase: shuffling
% 2.99/1.63
% 2.99/1.63 % (27191)Memory used [KB]: 1918
% 2.99/1.63 % (27191)Time elapsed: 0.010 s
% 2.99/1.63 % (27191)Instructions burned: 18 (million)
% 2.99/1.63 % (27191)------------------------------
% 2.99/1.63 % (27191)------------------------------
% 2.99/1.64 % (27196)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/5Mi)
% 2.99/1.64 % (27196)Instruction limit reached!
% 2.99/1.64 % (27196)------------------------------
% 2.99/1.64 % (27196)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.64 % (27196)Termination reason: Unknown
% 2.99/1.64 % (27196)Termination phase: shuffling
% 2.99/1.64
% 2.99/1.64 % (27196)Memory used [KB]: 1663
% 2.99/1.64 % (27196)Time elapsed: 0.003 s
% 2.99/1.64 % (27196)Instructions burned: 7 (million)
% 2.99/1.64 % (27196)------------------------------
% 2.99/1.64 % (27196)------------------------------
% 2.99/1.64 % (27195)Instruction limit reached!
% 2.99/1.64 % (27195)------------------------------
% 2.99/1.64 % (27195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.64 % (27195)Termination reason: Unknown
% 2.99/1.64 % (27195)Termination phase: shuffling
% 2.99/1.64
% 2.99/1.64 % (27195)Memory used [KB]: 2046
% 2.99/1.64 % (27195)Time elapsed: 0.012 s
% 2.99/1.64 % (27195)Instructions burned: 22 (million)
% 2.99/1.64 % (27195)------------------------------
% 2.99/1.64 % (27195)------------------------------
% 2.99/1.64 % (27197)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_27071 for (2999ds/6Mi)
% 2.99/1.64 % (27197)Instruction limit reached!
% 2.99/1.64 % (27197)------------------------------
% 2.99/1.64 % (27197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.64 % (27197)Termination reason: Unknown
% 2.99/1.64 % (27197)Termination phase: shuffling
% 2.99/1.64
% 2.99/1.64 % (27197)Memory used [KB]: 1663
% 2.99/1.64 % (27197)Time elapsed: 0.003 s
% 2.99/1.64 % (27197)Instructions burned: 6 (million)
% 2.99/1.64 % (27197)------------------------------
% 2.99/1.64 % (27197)------------------------------
% 2.99/1.65 % (27198)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_27071 for (2999ds/377Mi)
% 2.99/1.65 % (27200)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_27071 for (2999ds/19Mi)
% 2.99/1.66 % (27199)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_27071 for (2999ds/779Mi)
% 2.99/1.66 % (27200)Instruction limit reached!
% 2.99/1.66 % (27200)------------------------------
% 2.99/1.66 % (27200)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.66 % (27200)Termination reason: Unknown
% 2.99/1.66 % (27200)Termination phase: shuffling
% 2.99/1.66
% 2.99/1.66 % (27200)Memory used [KB]: 2046
% 2.99/1.66 % (27200)Time elapsed: 0.007 s
% 2.99/1.66 % (27200)Instructions burned: 22 (million)
% 2.99/1.66 % (27200)------------------------------
% 2.99/1.66 % (27200)------------------------------
% 2.99/1.67 % (27201)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_27071 for (2998ds/879Mi)
% 2.99/1.69 % (27177)Instruction limit reached!
% 2.99/1.69 % (27177)------------------------------
% 2.99/1.69 % (27177)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.69 % (27177)Termination reason: Unknown
% 2.99/1.69 % (27177)Termination phase: Property scanning
% 2.99/1.69
% 2.99/1.69 % (27177)Memory used [KB]: 5884
% 2.99/1.69 % (27177)Time elapsed: 0.130 s
% 2.99/1.69 % (27177)Instructions burned: 275 (million)
% 2.99/1.69 % (27177)------------------------------
% 2.99/1.69 % (27177)------------------------------
% 2.99/1.71 % (27202)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_27071 for (2998ds/17Mi)
% 2.99/1.71 % (27202)Instruction limit reached!
% 2.99/1.71 % (27202)------------------------------
% 2.99/1.71 % (27202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.71 % (27202)Termination reason: Unknown
% 2.99/1.71 % (27202)Termination phase: shuffling
% 2.99/1.71
% 2.99/1.71 % (27202)Memory used [KB]: 1918
% 2.99/1.71 % (27202)Time elapsed: 0.006 s
% 2.99/1.71 % (27202)Instructions burned: 17 (million)
% 2.99/1.71 % (27202)------------------------------
% 2.99/1.71 % (27202)------------------------------
% 2.99/1.72 % (27203)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27071 for (2998ds/3Mi)
% 2.99/1.72 % (27203)Instruction limit reached!
% 2.99/1.72 % (27203)------------------------------
% 2.99/1.72 % (27203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.72 % (27203)Termination reason: Unknown
% 2.99/1.72 % (27203)Termination phase: shuffling
% 2.99/1.72
% 2.99/1.72 % (27203)Memory used [KB]: 1663
% 2.99/1.72 % (27203)Time elapsed: 0.003 s
% 2.99/1.72 % (27203)Instructions burned: 5 (million)
% 2.99/1.72 % (27203)------------------------------
% 2.99/1.72 % (27203)------------------------------
% 2.99/1.73 % (27204)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_27071 for (2998ds/30Mi)
% 2.99/1.74 % (27204)Instruction limit reached!
% 2.99/1.74 % (27204)------------------------------
% 2.99/1.74 % (27204)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.74 % (27204)Termination reason: Unknown
% 2.99/1.74 % (27204)Termination phase: shuffling
% 2.99/1.74
% 2.99/1.74 % (27204)Memory used [KB]: 2302
% 2.99/1.74 % (27204)Time elapsed: 0.010 s
% 2.99/1.74 % (27204)Instructions burned: 32 (million)
% 2.99/1.74 % (27204)------------------------------
% 2.99/1.74 % (27204)------------------------------
% 2.99/1.75 % (27205)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_27071 for (2998ds/127Mi)
% 2.99/1.78 % (27198)Instruction limit reached!
% 2.99/1.78 % (27198)------------------------------
% 2.99/1.78 % (27198)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.78 % (27198)Termination reason: Unknown
% 2.99/1.78 % (27198)Termination phase: Saturation
% 2.99/1.78
% 2.99/1.78 % (27198)Memory used [KB]: 9338
% 2.99/1.78 % (27198)Time elapsed: 0.129 s
% 2.99/1.78 % (27198)Instructions burned: 379 (million)
% 2.99/1.78 % (27198)------------------------------
% 2.99/1.78 % (27198)------------------------------
% 2.99/1.78 % (27205)Instruction limit reached!
% 2.99/1.78 % (27205)------------------------------
% 2.99/1.78 % (27205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.78 % (27205)Termination reason: Unknown
% 2.99/1.78 % (27205)Termination phase: Property scanning
% 2.99/1.78
% 2.99/1.78 % (27205)Memory used [KB]: 3070
% 2.99/1.78 % (27205)Time elapsed: 0.034 s
% 2.99/1.78 % (27205)Instructions burned: 128 (million)
% 2.99/1.78 % (27205)------------------------------
% 2.99/1.78 % (27205)------------------------------
% 2.99/1.79 % (27206)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_27071 for (2997ds/100Mi)
% 2.99/1.79 % (27207)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_27071 for (2997ds/3Mi)
% 2.99/1.79 % (27207)Instruction limit reached!
% 2.99/1.79 % (27207)------------------------------
% 2.99/1.79 % (27207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.79 % (27207)Termination reason: Unknown
% 2.99/1.79 % (27207)Termination phase: shuffling
% 2.99/1.79
% 2.99/1.79 % (27207)Memory used [KB]: 1663
% 2.99/1.79 % (27207)Time elapsed: 0.002 s
% 2.99/1.79 % (27207)Instructions burned: 3 (million)
% 2.99/1.79 % (27207)------------------------------
% 2.99/1.79 % (27207)------------------------------
% 2.99/1.80 % (27208)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_27071 for (2997ds/20Mi)
% 2.99/1.81 % (27208)Instruction limit reached!
% 2.99/1.81 % (27208)------------------------------
% 2.99/1.81 % (27208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.81 % (27208)Termination reason: Unknown
% 2.99/1.81 % (27208)Termination phase: shuffling
% 2.99/1.81
% 2.99/1.81 % (27208)Memory used [KB]: 2046
% 2.99/1.81 % (27208)Time elapsed: 0.008 s
% 2.99/1.81 % (27208)Instructions burned: 22 (million)
% 2.99/1.81 % (27208)------------------------------
% 2.99/1.81 % (27208)------------------------------
% 2.99/1.82 % (27209)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_27071 for (2997ds/86Mi)
% 2.99/1.84 % (27206)Instruction limit reached!
% 2.99/1.84 % (27206)------------------------------
% 2.99/1.84 % (27206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.84 % (27206)Termination reason: Unknown
% 2.99/1.84 % (27206)Termination phase: Preprocessing 3
% 2.99/1.84
% 2.99/1.84 % (27206)Memory used [KB]: 2814
% 2.99/1.84 % (27206)Time elapsed: 0.046 s
% 2.99/1.84 % (27206)Instructions burned: 100 (million)
% 2.99/1.84 % (27206)------------------------------
% 2.99/1.84 % (27206)------------------------------
% 2.99/1.84 % (27209)Instruction limit reached!
% 2.99/1.84 % (27209)------------------------------
% 2.99/1.84 % (27209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.84 % (27209)Termination reason: Unknown
% 2.99/1.84 % (27209)Termination phase: Property scanning
% 2.99/1.84
% 2.99/1.84 % (27209)Memory used [KB]: 2814
% 2.99/1.84 % (27209)Time elapsed: 0.024 s
% 2.99/1.84 % (27209)Instructions burned: 89 (million)
% 2.99/1.84 % (27209)------------------------------
% 2.99/1.84 % (27209)------------------------------
% 2.99/1.85 % (27183)Instruction limit reached!
% 2.99/1.85 % (27183)------------------------------
% 2.99/1.85 % (27183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.85 % (27183)Termination reason: Unknown
% 2.99/1.85 % (27183)Termination phase: Saturation
% 2.99/1.85
% 2.99/1.85 % (27183)Memory used [KB]: 18038
% 2.99/1.85 % (27183)Time elapsed: 0.261 s
% 2.99/1.85 % (27183)Instructions burned: 1042 (million)
% 2.99/1.85 % (27183)------------------------------
% 2.99/1.85 % (27183)------------------------------
% 2.99/1.85 % (27211)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_27071 for (2997ds/39Mi)
% 2.99/1.85 % (27210)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_27071 for (2997ds/107Mi)
% 2.99/1.86 % (27213)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_27071 for (2997ds/448Mi)
% 2.99/1.86 % (27211)Instruction limit reached!
% 2.99/1.86 % (27211)------------------------------
% 2.99/1.86 % (27211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.86 % (27211)Termination reason: Unknown
% 2.99/1.86 % (27211)Termination phase: shuffling
% 2.99/1.86
% 2.99/1.86 % (27211)Memory used [KB]: 2430
% 2.99/1.86 % (27211)Time elapsed: 0.012 s
% 2.99/1.86 % (27211)Instructions burned: 41 (million)
% 2.99/1.86 % (27211)------------------------------
% 2.99/1.86 % (27211)------------------------------
% 2.99/1.87 % (27214)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_27071 for (2996ds/46Mi)
% 2.99/1.89 % (27214)Instruction limit reached!
% 2.99/1.89 % (27214)------------------------------
% 2.99/1.89 % (27214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.89 % (27214)Termination reason: Unknown
% 2.99/1.89 % (27214)Termination phase: shuffling
% 2.99/1.89
% 2.99/1.89 % (27214)Memory used [KB]: 2558
% 2.99/1.89 % (27214)Time elapsed: 0.014 s
% 2.99/1.89 % (27214)Instructions burned: 47 (million)
% 2.99/1.89 % (27214)------------------------------
% 2.99/1.89 % (27214)------------------------------
% 2.99/1.90 % (27215)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_27071 for (2996ds/98Mi)
% 2.99/1.90 % (27210)Instruction limit reached!
% 2.99/1.90 % (27210)------------------------------
% 2.99/1.90 % (27210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.90 % (27210)Termination reason: Unknown
% 2.99/1.90 % (27210)Termination phase: SInE selection
% 2.99/1.90
% 2.99/1.90 % (27210)Memory used [KB]: 2814
% 2.99/1.90 % (27210)Time elapsed: 0.050 s
% 2.99/1.90 % (27210)Instructions burned: 108 (million)
% 2.99/1.90 % (27210)------------------------------
% 2.99/1.90 % (27210)------------------------------
% 2.99/1.92 % (27216)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_27071 for (2996ds/507Mi)
% 2.99/1.92 % (27215)Instruction limit reached!
% 2.99/1.92 % (27215)------------------------------
% 2.99/1.92 % (27215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.92 % (27215)Termination reason: Unknown
% 2.99/1.92 % (27215)Termination phase: Property scanning
% 2.99/1.92
% 2.99/1.92 % (27215)Memory used [KB]: 2814
% 2.99/1.92 % (27215)Time elapsed: 0.027 s
% 2.99/1.92 % (27215)Instructions burned: 101 (million)
% 2.99/1.92 % (27215)------------------------------
% 2.99/1.92 % (27215)------------------------------
% 2.99/1.92 % (27201)Instruction limit reached!
% 2.99/1.92 % (27201)------------------------------
% 2.99/1.92 % (27201)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.92 % (27201)Termination reason: Unknown
% 2.99/1.92 % (27201)Termination phase: Saturation
% 2.99/1.92
% 2.99/1.92 % (27201)Memory used [KB]: 10874
% 2.99/1.92 % (27201)Time elapsed: 0.256 s
% 2.99/1.92 % (27201)Instructions burned: 881 (million)
% 2.99/1.92 % (27201)------------------------------
% 2.99/1.92 % (27201)------------------------------
% 2.99/1.93 % (27217)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_27071 for (2996ds/149Mi)
% 2.99/1.93 % (27218)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_27071 for (2996ds/39Mi)
% 2.99/1.95 % (27218)Instruction limit reached!
% 2.99/1.95 % (27218)------------------------------
% 2.99/1.95 % (27218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.95 % (27218)Termination reason: Unknown
% 2.99/1.95 % (27218)Termination phase: shuffling
% 2.99/1.95
% 2.99/1.95 % (27218)Memory used [KB]: 2430
% 2.99/1.95 % (27218)Time elapsed: 0.012 s
% 2.99/1.95 % (27218)Instructions burned: 40 (million)
% 2.99/1.95 % (27218)------------------------------
% 2.99/1.95 % (27218)------------------------------
% 2.99/1.95 % (27219)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_27071 for (2996ds/6Mi)
% 2.99/1.96 % (27219)Instruction limit reached!
% 2.99/1.96 % (27219)------------------------------
% 2.99/1.96 % (27219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.96 % (27219)Termination reason: Unknown
% 2.99/1.96 % (27219)Termination phase: shuffling
% 2.99/1.96
% 2.99/1.96 % (27219)Memory used [KB]: 1663
% 2.99/1.96 % (27219)Time elapsed: 0.003 s
% 2.99/1.96 % (27219)Instructions burned: 7 (million)
% 2.99/1.96 % (27219)------------------------------
% 2.99/1.96 % (27219)------------------------------
% 2.99/1.97 % (27220)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_27071 for (2995ds/1087Mi)
% 2.99/1.97 % (27213)Instruction limit reached!
% 2.99/1.97 % (27213)------------------------------
% 2.99/1.97 % (27213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.97 % (27213)Termination reason: Unknown
% 2.99/1.97 % (27213)Termination phase: Saturation
% 2.99/1.97
% 2.99/1.97 % (27213)Memory used [KB]: 7803
% 2.99/1.97 % (27213)Time elapsed: 0.111 s
% 2.99/1.97 % (27213)Instructions burned: 450 (million)
% 2.99/1.97 % (27213)------------------------------
% 2.99/1.97 % (27213)------------------------------
% 2.99/1.97 % (27199)Instruction limit reached!
% 2.99/1.97 % (27199)------------------------------
% 2.99/1.97 % (27199)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.97 % (27199)Termination reason: Unknown
% 2.99/1.97 % (27199)Termination phase: Saturation
% 2.99/1.97
% 2.99/1.97 % (27199)Memory used [KB]: 7931
% 2.99/1.97 % (27199)Time elapsed: 0.309 s
% 2.99/1.97 % (27199)Instructions burned: 780 (million)
% 2.99/1.97 % (27199)------------------------------
% 2.99/1.97 % (27199)------------------------------
% 2.99/1.97 % (27217)Instruction limit reached!
% 2.99/1.97 % (27217)------------------------------
% 2.99/1.97 % (27217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.97 % (27217)Termination reason: Unknown
% 2.99/1.97 % (27217)Termination phase: Function definition elimination
% 2.99/1.97
% 2.99/1.97 % (27217)Memory used [KB]: 2942
% 2.99/1.97 % (27217)Time elapsed: 0.039 s
% 2.99/1.97 % (27217)Instructions burned: 151 (million)
% 2.99/1.97 % (27217)------------------------------
% 2.99/1.97 % (27217)------------------------------
% 2.99/1.97 % (27192)Instruction limit reached!
% 2.99/1.97 % (27192)------------------------------
% 2.99/1.97 % (27192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.97 % (27192)Termination reason: Unknown
% 2.99/1.97 % (27192)Termination phase: Saturation
% 2.99/1.97
% 2.99/1.97 % (27192)Memory used [KB]: 11769
% 2.99/1.97 % (27192)Time elapsed: 0.353 s
% 2.99/1.97 % (27192)Instructions burned: 710 (million)
% 2.99/1.97 % (27192)------------------------------
% 2.99/1.97 % (27192)------------------------------
% 2.99/1.98 % (27221)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_27071 for (2995ds/47Mi)
% 2.99/1.98 % (27223)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_27071 for (2995ds/432Mi)
% 2.99/1.99 % (27222)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_27071 for (2995ds/44Mi)
% 2.99/1.99 % (27224)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_27071 for (2995ds/62Mi)
% 2.99/2.01 % (27222)Instruction limit reached!
% 2.99/2.01 % (27222)------------------------------
% 2.99/2.01 % (27222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/2.01 % (27222)Termination reason: Unknown
% 2.99/2.01 % (27222)Termination phase: shuffling
% 2.99/2.01
% 2.99/2.01 % (27222)Memory used [KB]: 2430
% 2.99/2.01 % (27222)Time elapsed: 0.019 s
% 2.99/2.01 % (27222)Instructions burned: 45 (million)
% 2.99/2.01 % (27222)------------------------------
% 2.99/2.01 % (27222)------------------------------
% 6.09/2.02 % (27221)Instruction limit reached!
% 6.09/2.02 % (27221)------------------------------
% 6.09/2.02 % (27221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.02 % (27224)Instruction limit reached!
% 6.09/2.02 % (27224)------------------------------
% 6.09/2.02 % (27224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.02 % (27221)Termination reason: Unknown
% 6.09/2.02 % (27224)Termination reason: Unknown
% 6.09/2.02 % (27221)Termination phase: shuffling
% 6.09/2.02
% 6.09/2.02 % (27224)Termination phase: shuffling
% 6.09/2.02
% 6.09/2.02 % (27221)Memory used [KB]: 2430
% 6.09/2.02 % (27224)Memory used [KB]: 2686
% 6.09/2.02 % (27221)Time elapsed: 0.016 s
% 6.09/2.02 % (27221)Instructions burned: 48 (million)
% 6.09/2.02 % (27224)Time elapsed: 0.030 s
% 6.09/2.02 % (27221)------------------------------
% 6.09/2.02 % (27221)------------------------------
% 6.09/2.02 % (27224)Instructions burned: 63 (million)
% 6.09/2.02 % (27224)------------------------------
% 6.09/2.02 % (27224)------------------------------
% 6.09/2.03 % (27225)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_27071 for (2995ds/31Mi)
% 6.09/2.03 % (27226)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_27071 for (2995ds/96Mi)
% 6.09/2.04 % (27227)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_27071 for (2995ds/453Mi)
% 6.09/2.04 % (27225)Instruction limit reached!
% 6.09/2.04 % (27225)------------------------------
% 6.09/2.04 % (27225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.04 % (27225)Termination reason: Unknown
% 6.09/2.04 % (27225)Termination phase: shuffling
% 6.09/2.04
% 6.09/2.04 % (27225)Memory used [KB]: 2302
% 6.09/2.04 % (27225)Time elapsed: 0.017 s
% 6.09/2.04 % (27225)Instructions burned: 33 (million)
% 6.09/2.04 % (27225)------------------------------
% 6.09/2.04 % (27225)------------------------------
% 6.09/2.05 % (27226)Instruction limit reached!
% 6.09/2.05 % (27226)------------------------------
% 6.09/2.05 % (27226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.05 % (27226)Termination reason: Unknown
% 6.09/2.05 % (27226)Termination phase: Property scanning
% 6.09/2.05
% 6.09/2.05 % (27226)Memory used [KB]: 2814
% 6.09/2.05 % (27226)Time elapsed: 0.026 s
% 6.09/2.05 % (27226)Instructions burned: 96 (million)
% 6.09/2.05 % (27226)------------------------------
% 6.09/2.05 % (27226)------------------------------
% 6.09/2.06 % (27228)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_27071 for (2994ds/71Mi)
% 6.09/2.06 % (27229)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_27071 for (2994ds/83Mi)
% 6.09/2.07 % (27194)Instruction limit reached!
% 6.09/2.07 % (27194)------------------------------
% 6.09/2.07 % (27194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.07 % (27194)Termination reason: Unknown
% 6.09/2.07 % (27194)Termination phase: Saturation
% 6.09/2.07
% 6.09/2.07 % (27194)Memory used [KB]: 13176
% 6.09/2.07 % (27194)Time elapsed: 0.438 s
% 6.09/2.07 % (27194)Instructions burned: 902 (million)
% 6.09/2.07 % (27194)------------------------------
% 6.09/2.07 % (27194)------------------------------
% 6.09/2.08 % (27230)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_27071 for (2994ds/11Mi)
% 6.09/2.08 % (27223)Instruction limit reached!
% 6.09/2.08 % (27223)------------------------------
% 6.09/2.08 % (27223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.08 % (27223)Termination reason: Unknown
% 6.09/2.08 % (27223)Termination phase: Saturation
% 6.09/2.08
% 6.09/2.08 % (27223)Memory used [KB]: 3326
% 6.09/2.08 % (27223)Time elapsed: 0.107 s
% 6.09/2.08 % (27223)Instructions burned: 433 (million)
% 6.09/2.08 % (27223)------------------------------
% 6.09/2.08 % (27223)------------------------------
% 6.09/2.09 % (27229)Instruction limit reached!
% 6.09/2.09 % (27229)------------------------------
% 6.09/2.09 % (27229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.09 % (27229)Termination reason: Unknown
% 6.09/2.09 % (27229)Termination phase: Property scanning
% 6.09/2.09
% 6.09/2.09 % (27229)Memory used [KB]: 2814
% 6.09/2.09 % (27229)Time elapsed: 0.023 s
% 6.09/2.09 % (27229)Instructions burned: 83 (million)
% 6.09/2.09 % (27229)------------------------------
% 6.09/2.09 % (27229)------------------------------
% 6.09/2.09 % (27230)Instruction limit reached!
% 6.09/2.09 % (27230)------------------------------
% 6.09/2.09 % (27230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.09 % (27230)Termination reason: Unknown
% 6.09/2.09 % (27230)Termination phase: shuffling
% 6.09/2.09
% 6.09/2.09 % (27230)Memory used [KB]: 1791
% 6.09/2.09 % (27230)Time elapsed: 0.007 s
% 6.09/2.09 % (27230)Instructions burned: 11 (million)
% 6.09/2.09 % (27230)------------------------------
% 6.09/2.09 % (27230)------------------------------
% 6.09/2.09 % (27228)Instruction limit reached!
% 6.09/2.09 % (27228)------------------------------
% 6.09/2.09 % (27228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.09 % (27228)Termination reason: Unknown
% 6.09/2.09 % (27228)Termination phase: Property scanning
% 6.09/2.09
% 6.09/2.09 % (27228)Memory used [KB]: 2814
% 6.09/2.09 % (27228)Time elapsed: 0.034 s
% 6.09/2.09 % (27228)Instructions burned: 71 (million)
% 6.09/2.09 % (27228)------------------------------
% 6.09/2.09 % (27228)------------------------------
% 6.09/2.09 % (27231)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_27071 for (2994ds/41Mi)
% 6.09/2.09 % (27232)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27071 for (2994ds/6Mi)
% 6.09/2.10 % (27232)Instruction limit reached!
% 6.09/2.10 % (27232)------------------------------
% 6.09/2.10 % (27232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.10 % (27232)Termination reason: Unknown
% 6.09/2.10 % (27232)Termination phase: shuffling
% 6.09/2.10
% 6.09/2.10 % (27232)Memory used [KB]: 1663
% 6.09/2.10 % (27232)Time elapsed: 0.003 s
% 6.09/2.10 % (27232)Instructions burned: 7 (million)
% 6.09/2.10 % (27232)------------------------------
% 6.09/2.10 % (27232)------------------------------
% 6.09/2.11 % (27231)Instruction limit reached!
% 6.09/2.11 % (27231)------------------------------
% 6.09/2.11 % (27231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.09/2.11 % (27231)Termination reason: Unknown
% 6.09/2.11 % (27231)Termination phase: shuffling
% 6.09/2.11
% 6.09/2.11 % (27231)Memory used [KB]: 2430
% 6.09/2.11 % (27231)Time elapsed: 0.012 s
% 6.09/2.11 % (27231)Instructions burned: 41 (million)
% 6.09/2.11 % (27231)------------------------------
% 6.09/2.11 % (27231)------------------------------
% 6.09/2.11 % (27235)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_27071 for (2994ds/109Mi)
% 6.09/2.11 % (27233)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_27071 for (2994ds/150Mi)
% 6.09/2.11 % (27234)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_27071 for (2994ds/343Mi)
% 6.68/2.12 % (27236)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_27071 for (2994ds/253Mi)
% 6.68/2.14 % (27235)Instruction limit reached!
% 6.68/2.14 % (27235)------------------------------
% 6.68/2.14 % (27235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.14 % (27235)Termination reason: Unknown
% 6.68/2.14 % (27235)Termination phase: SInE selection
% 6.68/2.14
% 6.68/2.14 % (27235)Memory used [KB]: 2814
% 6.68/2.14 % (27235)Time elapsed: 0.030 s
% 6.68/2.14 % (27235)Instructions burned: 113 (million)
% 6.68/2.14 % (27235)------------------------------
% 6.68/2.14 % (27235)------------------------------
% 6.68/2.15 % (27237)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_27071 for (2994ds/140Mi)
% 6.68/2.16 % (27216)Instruction limit reached!
% 6.68/2.16 % (27216)------------------------------
% 6.68/2.16 % (27216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.16 % (27216)Termination reason: Unknown
% 6.68/2.16 % (27216)Termination phase: Saturation
% 6.68/2.16
% 6.68/2.16 % (27216)Memory used [KB]: 8315
% 6.68/2.16 % (27216)Time elapsed: 0.261 s
% 6.68/2.16 % (27216)Instructions burned: 507 (million)
% 6.68/2.16 % (27216)------------------------------
% 6.68/2.16 % (27216)------------------------------
% 6.68/2.18 % (27233)Instruction limit reached!
% 6.68/2.18 % (27233)------------------------------
% 6.68/2.18 % (27233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.18 % (27233)Termination reason: Unknown
% 6.68/2.18 % (27233)Termination phase: Preprocessing 3
% 6.68/2.18
% 6.68/2.18 % (27233)Memory used [KB]: 5245
% 6.68/2.18 % (27233)Time elapsed: 0.073 s
% 6.68/2.18 % (27233)Instructions burned: 150 (million)
% 6.68/2.18 % (27233)------------------------------
% 6.68/2.18 % (27233)------------------------------
% 6.68/2.18 % (27238)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_27071 for (2993ds/614Mi)
% 6.68/2.18 % (27236)Instruction limit reached!
% 6.68/2.18 % (27236)------------------------------
% 6.68/2.18 % (27236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.18 % (27236)Termination reason: Unknown
% 6.68/2.18 % (27236)Termination phase: Property scanning
% 6.68/2.18
% 6.68/2.18 % (27236)Memory used [KB]: 5884
% 6.68/2.18 % (27236)Time elapsed: 0.071 s
% 6.68/2.18 % (27236)Instructions burned: 254 (million)
% 6.68/2.18 % (27236)------------------------------
% 6.68/2.18 % (27236)------------------------------
% 6.68/2.19 % (27239)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_27071 for (2993ds/202Mi)
% 6.68/2.19 % (27237)Instruction limit reached!
% 6.68/2.19 % (27237)------------------------------
% 6.68/2.19 % (27237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.68/2.19 % (27237)Termination reason: Unknown
% 6.68/2.19 % (27237)Termination phase: Property scanning
% 6.68/2.19
% 6.68/2.19 % (27237)Memory used [KB]: 2942
% 6.68/2.19 % (27237)Time elapsed: 0.039 s
% 6.68/2.19 % (27237)Instructions burned: 142 (million)
% 6.68/2.19 % (27237)------------------------------
% 6.68/2.19 % (27237)------------------------------
% 6.68/2.19 % (27240)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_27071 for (2993ds/339Mi)
% 7.07/2.21 % (27241)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_27071 for (2993ds/283Mi)
% 7.07/2.22 % (27227)Instruction limit reached!
% 7.07/2.22 % (27227)------------------------------
% 7.07/2.22 % (27227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.07/2.22 % (27227)Termination reason: Unknown
% 7.07/2.22 % (27227)Termination phase: Saturation
% 7.07/2.22
% 7.07/2.22 % (27227)Memory used [KB]: 7803
% 7.07/2.22 % (27227)Time elapsed: 0.188 s
% 7.07/2.22 % (27227)Instructions burned: 455 (million)
% 7.07/2.22 % (27227)------------------------------
% 7.07/2.22 % (27227)------------------------------
% 7.07/2.23 % (27220)First to succeed.
% 7.07/2.24 % (27242)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_27071 for (2993ds/134Mi)
% 7.07/2.24 % (27220)Refutation found. Thanks to Tanya!
% 7.07/2.24 % SZS status Theorem for DTF2THF_27071
% 7.07/2.24 % SZS output start Proof for DTF2THF_27071
% 7.07/2.24 thf(func_def_0, type, in: $i > $i > $o).
% 7.07/2.24 thf(func_def_1, type, exu: ($i > $o) > $o).
% 7.07/2.24 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 7.07/2.24 thf(func_def_8, type, powerset: $i > $i).
% 7.07/2.24 thf(func_def_10, type, setunion: $i > $i).
% 7.07/2.24 thf(func_def_19, type, descr: ($i > $o) > $i).
% 7.07/2.24 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_26, type, prop2set: $o > $i).
% 7.07/2.24 thf(func_def_36, type, nonempty: $i > $o).
% 7.07/2.24 thf(func_def_69, type, set2prop: $i > $o).
% 7.07/2.24 thf(func_def_88, type, subset: $i > $i > $o).
% 7.07/2.24 thf(func_def_89, type, disjoint: $i > $i > $o).
% 7.07/2.24 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 7.07/2.24 thf(func_def_114, type, binunion: $i > $i > $i).
% 7.07/2.24 thf(func_def_122, type, binintersect: $i > $i > $i).
% 7.07/2.24 thf(func_def_135, type, regular: $i > $o).
% 7.07/2.24 thf(func_def_136, type, setminus: $i > $i > $i).
% 7.07/2.24 thf(func_def_147, type, symdiff: $i > $i > $i).
% 7.07/2.24 thf(func_def_153, type, iskpair: $i > $o).
% 7.07/2.24 thf(func_def_158, type, kpair: $i > $i > $i).
% 7.07/2.24 thf(func_def_160, type, cartprod: $i > $i > $i).
% 7.07/2.24 thf(func_def_177, type, singleton: $i > $o).
% 7.07/2.24 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_184, type, atmost1p: $i > $o).
% 7.07/2.24 thf(func_def_185, type, atleast2p: $i > $o).
% 7.07/2.24 thf(func_def_186, type, atmost2p: $i > $o).
% 7.07/2.24 thf(func_def_187, type, upairsetp: $i > $o).
% 7.07/2.24 thf(func_def_191, type, kfst: $i > $i).
% 7.07/2.24 thf(func_def_203, type, ksnd: $i > $i).
% 7.07/2.24 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 7.07/2.24 thf(func_def_222, type, func: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_223, type, funcSet: $i > $i > $i).
% 7.07/2.24 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 7.07/2.24 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 7.07/2.24 thf(func_def_323, type, breln1: $i > $i > $o).
% 7.07/2.24 thf(func_def_325, type, breln1Set: $i > $i).
% 7.07/2.24 thf(func_def_327, type, transitive: $i > $i > $o).
% 7.07/2.24 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 7.07/2.24 thf(func_def_329, type, reflexive: $i > $i > $o).
% 7.07/2.24 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 7.07/2.24 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 7.07/2.24 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 7.07/2.24 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 7.07/2.24 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 7.07/2.24 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 7.07/2.24 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 7.07/2.24 thf(func_def_387, type, sP0: $i > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_388, type, sP1: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_389, type, sP2: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_390, type, sP3: ($i > $o) > $o).
% 7.07/2.24 thf(func_def_391, type, sP4: ($i > $o) > $o).
% 7.07/2.24 thf(func_def_392, type, sP5: $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_393, type, sP6: $i > $i > ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_395, type, sP8: ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_396, type, sP9: $i > ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_397, type, sP10: $i > $i > $o).
% 7.07/2.24 thf(func_def_399, type, sP12: ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_401, type, sP14: $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_403, type, sP16: ($i > $o) > $o).
% 7.07/2.24 thf(func_def_404, type, sP17: $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_407, type, sP20: $i > $i > ($i > $i > $o) > $o).
% 7.07/2.24 thf(func_def_408, type, sP21: $i > ($i > $i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_409, type, sP22: $i > ($i > $i) > $i > $o).
% 7.07/2.24 thf(func_def_411, type, sP24: $i > $o > $i > $o).
% 7.07/2.24 thf(func_def_413, type, sP26: $i > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_414, type, sP27: $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_415, type, sP28: $i > ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_416, type, sP29: ($i > $o) > $o).
% 7.07/2.24 thf(func_def_417, type, sP30: $i > $o).
% 7.07/2.24 thf(func_def_418, type, sP31: $i > $o).
% 7.07/2.24 thf(func_def_419, type, sP32: $o > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_420, type, sP33: $i > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_421, type, sP34: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_422, type, sP35: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_423, type, sP36: $i > $i > $o > $o).
% 7.07/2.24 thf(func_def_424, type, sP37: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_425, type, sP38: $i > $i > ($i > $i) > $o).
% 7.07/2.24 thf(func_def_427, type, sP40: $i > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_428, type, sP41: $i > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_429, type, sP42: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_430, type, sP43: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_435, type, sP48: $o > $i > $i > $o).
% 7.07/2.24 thf(func_def_436, type, sP49: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_438, type, sP51: $i > $o > $i > $i > $o).
% 7.07/2.24 thf(func_def_439, type, sP52: $o > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_440, type, sP53: $i > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_441, type, sP54: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_442, type, sP55: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_443, type, sP56: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_444, type, sP57: ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_445, type, sP58: $i > $i > ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_446, type, sP59: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_447, type, sP60: $i > $i > $o).
% 7.07/2.24 thf(func_def_448, type, sP61: $i > $i > $o).
% 7.07/2.24 thf(func_def_449, type, sP62: $i > $i > $o).
% 7.07/2.24 thf(func_def_450, type, sP63: $i > $i > $o).
% 7.07/2.24 thf(func_def_451, type, sP64: $i > $o).
% 7.07/2.24 thf(func_def_452, type, sP65: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_453, type, sP66: $i > $i > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_455, type, sP68: ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_457, type, sP70: $i > ($i > $i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_458, type, sP71: $i > ($i > $i > $o) > $o).
% 7.07/2.24 thf(func_def_460, type, sP73: $i > $i > $o).
% 7.07/2.24 thf(func_def_462, type, sP75: ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_464, type, sP77: $o > $i > $i > $o).
% 7.07/2.24 thf(func_def_466, type, sP79: $i > $i > ($i > $i) > $o).
% 7.07/2.24 thf(func_def_468, type, sP81: $i > $o).
% 7.07/2.24 thf(func_def_469, type, sP82: $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_470, type, sP83: ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_473, type, sP86: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_475, type, sP88: $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_476, type, sP89: ($i > $o) > $i > $o).
% 7.07/2.24 thf(func_def_478, type, sP91: $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_480, type, sP93: ($i > $o) > ($i > $o) > $i > $i > $o).
% 7.07/2.24 thf(func_def_481, type, sP94: ($i > $i) > $i > $i > $o).
% 7.07/2.24 thf(func_def_483, type, sP96: $i > $i > $o > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_485, type, sP98: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_486, type, sP99: $i > $i > ($i > $o) > $o).
% 7.07/2.24 thf(func_def_487, type, sP100: $i > $i > $o).
% 7.07/2.24 thf(func_def_489, type, sP102: ($i > $o) > $o).
% 7.07/2.24 thf(func_def_491, type, sK104: $i > $o).
% 7.07/2.24 thf(func_def_493, type, sK106: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_494, type, sK107: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_495, type, sK108: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_496, type, sK109: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_503, type, sK116: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_504, type, sK117: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_513, type, sK126: $i > $o).
% 7.07/2.24 thf(func_def_521, type, sK134: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_522, type, sK135: ($i > $o) > $i).
% 7.07/2.24 thf(func_def_523, type, sK136: $i > $o).
% 7.07/2.24 thf(func_def_524, type, sK137: $i > $o).
% 7.07/2.24 thf(func_def_530, type, sK143: ($i > $o) > $i).
% 7.07/2.24 thf(func_def_531, type, sK144: ($i > $o) > $i).
% 7.07/2.24 thf(func_def_532, type, sK145: $i > $o).
% 7.07/2.24 thf(func_def_538, type, sK151: $i > $i > ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_539, type, sK152: $i > $i > ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_540, type, sK153: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_544, type, sK157: $i > $o).
% 7.07/2.24 thf(func_def_550, type, sK163: $i > $i > $i).
% 7.07/2.24 thf(func_def_551, type, sK164: $i > ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_552, type, sK165: $i > ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_553, type, sK166: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_556, type, sK169: $i > $o).
% 7.07/2.24 thf(func_def_568, type, sK181: $i > $i > $i).
% 7.07/2.24 thf(func_def_576, type, sK189: $i > $i > $o).
% 7.07/2.24 thf(func_def_581, type, sK194: $i > $o).
% 7.07/2.24 thf(func_def_582, type, sK195: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_583, type, sK196: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_592, type, sK205: $i > $o).
% 7.07/2.24 thf(func_def_594, type, sK207: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_595, type, sK208: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_596, type, sK209: ($i > $o) > $i).
% 7.07/2.24 thf(func_def_597, type, sK210: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_598, type, sK211: $i > $o).
% 7.07/2.24 thf(func_def_603, type, sK216: $i > $o).
% 7.07/2.24 thf(func_def_604, type, sK217: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_605, type, sK218: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_612, type, sK225: $i > $o).
% 7.07/2.24 thf(func_def_634, type, sK247: $i > $i).
% 7.07/2.24 thf(func_def_635, type, sK248: ($i > $i) > $i > $i > $i).
% 7.07/2.24 thf(func_def_645, type, sK258: $i > $o).
% 7.07/2.24 thf(func_def_651, type, sK264: $i > ($i > $i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_652, type, sK265: $i > ($i > $i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_653, type, sK266: $i > $i > ($i > $i > $o) > $i).
% 7.07/2.24 thf(func_def_654, type, sK267: $i > $i > ($i > $i > $o) > $i).
% 7.07/2.24 thf(func_def_656, type, sK269: $i > $i > $o).
% 7.07/2.24 thf(func_def_659, type, sK272: $i > $o).
% 7.07/2.24 thf(func_def_678, type, sK291: $i > $i).
% 7.07/2.24 thf(func_def_679, type, sK292: $i > $i).
% 7.07/2.24 thf(func_def_680, type, sK293: ($i > $i) > $i > $i).
% 7.07/2.24 thf(func_def_681, type, sK294: $i > ($i > $i) > $i > $i).
% 7.07/2.24 thf(func_def_685, type, sK298: $i > $i > $o).
% 7.07/2.24 thf(func_def_698, type, sK311: $i > $i > $o).
% 7.07/2.24 thf(func_def_707, type, sK320: $i > $o > $i > $i).
% 7.07/2.24 thf(func_def_719, type, sK332: $i > $o).
% 7.07/2.24 thf(func_def_720, type, sK333: $i > $o).
% 7.07/2.24 thf(func_def_721, type, sK334: ($i > $o) > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_722, type, sK335: ($i > $o) > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_725, type, sK338: $i > $i > $o).
% 7.07/2.24 thf(func_def_744, type, sK357: $i > $o).
% 7.07/2.24 thf(func_def_748, type, sK361: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_749, type, sK362: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_754, type, sK367: $i > ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_755, type, sK368: $i > ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_756, type, sK369: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_759, type, sK372: $i > $o).
% 7.07/2.24 thf(func_def_770, type, sK383: ($i > $o) > $i).
% 7.07/2.24 thf(func_def_771, type, sK384: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_772, type, sK385: $i > $o).
% 7.07/2.24 thf(func_def_797, type, sK410: $i > $i).
% 7.07/2.24 thf(func_def_798, type, sK411: $i > $i).
% 7.07/2.24 thf(func_def_809, type, sK422: $i > $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_822, type, sK435: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_823, type, sK436: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_824, type, sK437: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_825, type, sK438: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_829, type, sK442: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_830, type, sK443: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_847, type, sK460: $i > $i).
% 7.07/2.24 thf(func_def_848, type, sK461: ($i > $i) > $i > $i > $i).
% 7.07/2.24 thf(func_def_849, type, sK462: $i > $i > ($i > $i) > $i).
% 7.07/2.24 thf(func_def_850, type, sK463: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_851, type, sK464: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_852, type, sK465: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_853, type, sK466: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_886, type, sK499: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_887, type, sK500: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_899, type, sK512: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_919, type, sK532: $i > $i).
% 7.07/2.24 thf(func_def_955, type, sK568: $i > $i > $i).
% 7.07/2.24 thf(func_def_956, type, sK569: $i > $i).
% 7.07/2.24 thf(func_def_959, type, sK572: $i > $i > ($i > $i) > $i).
% 7.07/2.24 thf(func_def_982, type, sK595: $i > $i > $o).
% 7.07/2.24 thf(func_def_986, type, sK599: $i > $o).
% 7.07/2.24 thf(func_def_996, type, sK609: $i > $o).
% 7.07/2.24 thf(func_def_1027, type, sK640: $i > $o).
% 7.07/2.24 thf(func_def_1028, type, sK641: $i > $o).
% 7.07/2.24 thf(func_def_1029, type, sK642: ($i > $o) > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1030, type, sK643: ($i > $o) > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1032, type, sK645: $i > $i).
% 7.07/2.24 thf(func_def_1035, type, sK648: $i > $i > ($i > $i) > $i).
% 7.07/2.24 thf(func_def_1040, type, sK653: $i > $i > $i > $o).
% 7.07/2.24 thf(func_def_1047, type, sK660: $i > $i > $i).
% 7.07/2.24 thf(func_def_1077, type, sK690: $i > $i).
% 7.07/2.24 thf(func_def_1079, type, sK692: $i > ($i > $i) > $i > $i).
% 7.07/2.24 thf(func_def_1098, type, sK711: $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1103, type, sK716: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1104, type, sK717: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1131, type, sK744: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1132, type, sK745: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1136, type, sK749: $i > $i > ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1137, type, sK750: $i > $i > ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1138, type, sK751: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1142, type, sK755: $i > $o).
% 7.07/2.24 thf(func_def_1145, type, sK758: $i > $i).
% 7.07/2.24 thf(func_def_1146, type, sK759: $i > $i).
% 7.07/2.24 thf(func_def_1147, type, sK760: $i > $i).
% 7.07/2.24 thf(func_def_1148, type, sK761: $i > $i).
% 7.07/2.24 thf(func_def_1149, type, sK762: $i > $i > $i).
% 7.07/2.24 thf(func_def_1150, type, sK763: $i > $i > $i).
% 7.07/2.24 thf(func_def_1151, type, sK764: $i > $i > $i).
% 7.07/2.24 thf(func_def_1152, type, sK765: $i > $i > $i).
% 7.07/2.24 thf(func_def_1153, type, sK766: $i > $i > $i).
% 7.07/2.24 thf(func_def_1154, type, sK767: $i > $i > $i).
% 7.07/2.24 thf(func_def_1155, type, sK768: $i > $i > $i).
% 7.07/2.24 thf(func_def_1156, type, sK769: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1157, type, sK770: $i > $i > $i).
% 7.07/2.24 thf(func_def_1158, type, sK771: $i > $i > $i).
% 7.07/2.24 thf(func_def_1159, type, sK772: $i > $i > $i).
% 7.07/2.24 thf(func_def_1160, type, sK773: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1161, type, sK774: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1162, type, sK775: $i > $i).
% 7.07/2.24 thf(func_def_1166, type, sK779: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1167, type, sK780: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1171, type, sK784: $i > $i > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1188, type, sK801: $i > $i > $i).
% 7.07/2.24 thf(func_def_1192, type, sK805: $i > $i).
% 7.07/2.24 thf(func_def_1218, type, sK831: $i > $o).
% 7.07/2.24 thf(func_def_1219, type, sK832: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1220, type, sK833: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1252, type, sK865: $i > $i > $o).
% 7.07/2.24 thf(func_def_1257, type, sK870: $i > $i).
% 7.07/2.24 thf(func_def_1258, type, sK871: $i > $i > $o).
% 7.07/2.24 thf(func_def_1260, type, sK873: $i > ($i > $i > $o) > $i).
% 7.07/2.24 thf(func_def_1261, type, sK874: $i > $i > ($i > $i > $o) > $i).
% 7.07/2.24 thf(func_def_1262, type, sK875: $i > ($i > $i > $o) > $i).
% 7.07/2.24 thf(func_def_1263, type, sK876: $i > ($i > $i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1266, type, sK879: $i > $i > $i).
% 7.07/2.24 thf(func_def_1267, type, sK880: $i > $i > $i).
% 7.07/2.24 thf(func_def_1271, type, sK884: $i > $o).
% 7.07/2.24 thf(func_def_1274, type, sK887: $i > $o).
% 7.07/2.24 thf(func_def_1276, type, sK889: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1277, type, sK890: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1293, type, sK906: $i > $i).
% 7.07/2.24 thf(func_def_1295, type, sK908: $i > ($i > $i) > $i > $i).
% 7.07/2.24 thf(func_def_1296, type, sK909: $i > $i > ($i > $i) > $i).
% 7.07/2.24 thf(func_def_1297, type, sK910: $i > $i > $i).
% 7.07/2.24 thf(func_def_1298, type, sK911: $i > $i).
% 7.07/2.24 thf(func_def_1301, type, sK914: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1302, type, sK915: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1303, type, sK916: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1304, type, sK917: $i > $o).
% 7.07/2.24 thf(func_def_1307, type, sK920: $i > $o).
% 7.07/2.24 thf(func_def_1308, type, sK921: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1315, type, sK928: $i > $i > $i).
% 7.07/2.24 thf(func_def_1321, type, sK934: $i > $i > $o).
% 7.07/2.24 thf(func_def_1337, type, sK950: $i > $i > $o).
% 7.07/2.24 thf(func_def_1344, type, sK957: ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1345, type, sK958: $i > $o).
% 7.07/2.24 thf(func_def_1346, type, sK959: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1347, type, sK960: ($i > $o) > $i > $i).
% 7.07/2.24 thf(func_def_1348, type, sK961: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1350, type, sK963: $i > $o).
% 7.07/2.24 thf(func_def_1353, type, sK966: $i > $i).
% 7.07/2.24 thf(func_def_1377, type, sK990: $i > $o).
% 7.07/2.24 thf(func_def_1383, type, sK996: $i > $i > $i).
% 7.07/2.24 thf(func_def_1384, type, sK997: $i > $o).
% 7.07/2.24 thf(func_def_1386, type, sK999: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1387, type, sK1000: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1388, type, sK1001: ($i > $o) > ($i > $o) > $i > $i > $i).
% 7.07/2.24 thf(func_def_1389, type, sK1002: ($i > $o) > ($i > $o) > $i > $i > $i).
% 7.07/2.24 thf(func_def_1392, type, sK1005: $i > $o).
% 7.07/2.24 thf(func_def_1393, type, sK1006: $i > $o).
% 7.07/2.24 thf(func_def_1405, type, sK1018: $i > $i).
% 7.07/2.24 thf(func_def_1419, type, sK1032: $i > $i).
% 7.07/2.24 thf(func_def_1420, type, sK1033: ($i > $i) > $i > $i > $i).
% 7.07/2.24 thf(func_def_1444, type, sK1057: $i > $i > $o > $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1462, type, sK1075: $i > $i).
% 7.07/2.24 thf(func_def_1465, type, sK1078: $i > $i > ($i > $i) > $i).
% 7.07/2.24 thf(func_def_1466, type, sK1079: $i > $i > $i > $i).
% 7.07/2.24 thf(func_def_1473, type, sK1086: $i > $o).
% 7.07/2.24 thf(func_def_1485, type, sK1098: $i > $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1486, type, sK1099: $i > $o).
% 7.07/2.24 thf(func_def_1507, type, sK1120: $i > $i > $i).
% 7.07/2.24 thf(func_def_1509, type, sK1122: $i > $i).
% 7.07/2.24 thf(func_def_1516, type, sK1129: $i > $o).
% 7.07/2.24 thf(func_def_1526, type, sK1139: $i > ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1527, type, sK1140: ($i > $o) > $i).
% 7.07/2.24 thf(func_def_1528, type, sK1141: $i > $o).
% 7.07/2.24 thf(func_def_1541, type, ph1154: !>[X0: $tType]:(X0)).
% 7.07/2.24 thf(f9456,plain,(
% 7.07/2.24 $false),
% 7.07/2.24 inference(subsumption_resolution,[status(thm)],[f9445,f4345])).
% 7.07/2.24 thf(f4345,plain,(
% 7.07/2.24 ((in @ sK619 @ (funcSet @ sK618 @ sK620)) != $true)),
% 7.07/2.24 inference(cnf_transformation,[status(thm)],[f2458])).
% 7.07/2.24 thf(f2458,plain,(
% 7.07/2.24 (iffalseProp1 = $true) & (setunionI = $true) & (cartprodmempair = $true) & (setukpairinjR1 = $true) & (ubforcartprodlem3 = $true) & (upairsubunion = $true) & (dpsetconstrI = $true) & (demorgan1 = $true) & (eqimpsubset2 = $true) & (binunionLsub = $true) & (eta2 = $true) & (upairset2E = $true) & (cartprodpairsurjEq = $true) & (wellorderingAx = $true) & (setOfPairsIsBReln = $true) & (emptyset__Cong = $true) & (omega__Cong = $true) & (kpairsurjEq = $true) & (ifSingleton = $true) & (ex1I = $true) & (funcextLem = $true) & (powersetTE1 = $true) & (woz2W = $true) & (setadjoinIL = $true) & (breln1invI = $true) & (setunionsingleton2 = $true) & (setext = $true) & (setunionsingleton1 = $true) & (binunionEcases = $true) & (powersetE = $true) & (bs114d = $true) & (exuEu = $true) & (setminusELneg = $true) & (setukpairinjL2 = $true) & (contrasubsetT2 = $true) & (replAx = $true) & (upairset2IR = $true) & (injFuncInInjFuncSet = $true) & (binintersectSubset5 = $true) & (lam2lamEq = $true) & (setunion__Cong = $true) & (notequalI2 = $true) & (subbreln1 = $true) & (setadjoinAx = $true) & (demorgan2b = $true) & (emptyE1 = $true) & (ifp = $true) & (notdallE = $true) & (woz2A = $true) & (subsetemptysetimpeq = $true) & (notinemptyset = $true) & (eta1 = $true) & (setminusLsub = $true) & (complementTI1 = $true) & (breln1unionprop = $true) & (setadjoin__Cong = $true) & (binintersectRsub = $true) & (ap2apEq2 = $true) & (symdiffIneg2 = $true) & (woz1_3 = $true) & (setukpairinjR2 = $true) & (uniqinunit = $true) & (setoftrueEq = $true) & (dsetconstrI = $true) & (exuI3 = $true) & (demorgan1a = $true) & (contrasubsetT3 = $true) & (setukpairinjL = $true) & (woz13rule0 = $true) & (breln1unionIR = $true) & (setminusER = $true) & (eqbreln1 = $true) & (woz13rule1 = $true) & (woz13rule4 = $true) & (symdiffI2 = $true) & (nonemptyI1 = $true) & (breln1compE = $true) & (in__Cong = $true) & (breln1compI = $true) & (disjointsetsI1 = $true) & (demorgan2 = $true) & (image1Equiv = $true) & (brelnall1 = $true) & (breln1invprop = $true) & (binunionTIRcontra = $true) & (subsetI2 = $true) & (dpsetconstrSub = $true) & (symdiffI1 = $true) & (eqinunit = $true) & (binintersectI = $true) & (subsetE = $true) & (ubforcartprodlem2 = $true) & (foundationAx = $true) & (cartprodfstin = $true) & (cartprodsndpairEq = $true) & (choice2fnsingleton = $true) & (cartprodsndin = $true) & (breln1unionE = $true) & (setukpairIR = $true) & (inComplementUnionImpInComplement1 = $true) & (woz2Ex = $true) & (secondinupair = $true) & (setadjoinIR = $true) & (quantDeMorgan2 = $true) & (complementSubsetComplementIntersect = $true) & (cartprodpairmemEL = $true) & (binunionIR = $true) & (upairsetIL = $true) & (woz2B = $true) & (singletonsuniq = $true) & (complementImpComplementIntersect = $true) & (emptysetsubset = $true) & (setukpairinjR = $true) & (brelnall2 = $true) & (theeq = $true) & (doubleComplementSub1 = $true) & (iftrue = $true) & (omegaIndAx = $true) & (subsetE2 = $true) & (emptyinPowerset = $true) & (powersetI = $true) & (setukpairIL = $true) & (binintersectTERcontra = $true) & (funcext = $true) & (breln1unionCommutes = $true) & (app = $true) & (binintersectSubset3 = $true) & (funcGraphProp3 = $true) & (cartprodpairin = $true) & (exuE3e = $true) & (demorgan1b = $true) & (breln1unionIL = $true) & (contrasubsetT1 = $true) & (ap2apEq1 = $true) & (doubleComplementEq = $true) & (dsetconstr__Cong = $true) & (dpsetconstrER = $true) & (doubleComplementSub2 = $true) & (notinsingleton = $true) & (subsetRefl = $true) & (ap2p = $true) & (binunionE = $true) & (setunionsingleton = $true) & (binintersectSubset1 = $true) & (doubleComplementE1 = $true) & (complementTE1 = $true) & (powersetAx = $true) & (dpsetconstrEL2 = $true) & (((in @ sK619 @ (funcSet @ sK618 @ sK620)) != $true) & ((in @ sK619 @ (surjFuncSet @ sK618 @ sK620)) = $true)) & (sepSubset = $true) & (powersetsubset = $true) & (setadjoinOr = $true) & (complementTnotintersectT = $true) & (contrasubsetT = $true) & (complementT_lem = $true) & (woz13rule3 = $true) & (inCongP = $true) & (breln1all1 = $true) & (breln1unionEcases = $true) & (subsetTrans = $true) & (setminusERneg = $true) & (prop2setI = $true) & (inIntersectImpInUnion = $true) & (upairinpowunion = $true) & (iffalse = $true) & (noeltsimpempty = $true) & (subsetTI = $true) & (sepInPowerset = $true) & (emptysetAx = $true) & (setminusEL = $true) & (exuI2 = $true) & (setminusILneg = $true) & (setextT = $true) & (binunionTEcontra = $true) & (image1E = $true) & (woz1_2 = $true) & (funcinfuncset = $true) & (eqimpsubset1 = $true) & (setadjoinE = $true) & (binintersectLsub = $true) & (ex1E2 = $true) & (binintersectER = $true) & (woz13rule2 = $true) & (setOfPairsIsBReln1 = $true) & (binintersectTELcontra = $true) & (binunionTE = $true) & (woz1_1 = $true) & (quantDeMorgan1 = $true) & (ex1E1 = $true) & (emptyinunitempty = $true) & (demorgan2b2 = $true) & (notequalI1 = $true) & (intersectInPowersetIntersectUnions = $true) & (powersetE1 = $true) & (kfstpairEq = $true) & (singletonsswitch = $true) & (nonemptyImpWitness = $true) & (singletoninpowunion = $true) & (breln1compprop = $true) & (eqbreln = $true) & (cartprodmempair1 = $true) & (doubleComplementI1 = $true) & (lam2p = $true) & (binintersectSubset4 = $true) & (ksndsingleton = $true) & (kpairiskpair = $true) & (setextsub = $true) & (setunionAx = $true) & (theprop = $true) & (contraSubsetComplement = $true) & (funcext2 = $true) & (breln1compEex = $true) & (beta2 = $true) & (quantDeMorgan4 = $true) & (breln1SetBreln1 = $true) & (ksndpairEq = $true) & (demorgan2a = $true) & (omegaSAx = $true) & (setminusIRneg = $true) & (setminusSubset2 = $true) & (inIntersectImpInIntersectUnions = $true) & (funcGraphProp1 = $true) & (exu__Cong = $true) & (inIntersectImpInUnion2 = $true) & (exuE2 = $true) & (iffalseProp2 = $true) & (setminusT_lem = $true) & (setukpairinjL1 = $true) & (emptysetE = $true) & (lamProp = $true) & (powersetTI1 = $true) & (breln1invE = $true) & (prop2setE = $true) & (kpairp = $true) & (demorgan2a1 = $true) & (emptyInPowerset = $true) & (funcGraphProp2 = $true) & (setunionE2 = $true) & (injFuncSetFuncInj = $true) & (woz1_5 = $true) & (symdiffE = $true) & (setminusI = $true) & (infuncsetfunc = $true) & (binunionIL = $true) & (exuE3u = $true) & (cartprodpairmemER = $true) & (upairsetIR = $true) & (setunionE = $true) & (subbreln = $true) & (dpsetconstrERa = $true) & (setbeta = $true) & (demorgan2a2 = $true) & (cartprodmempaircEq = $true) & (complementUnionInPowersetComplement = $true) & (subset2powerset = $true) & (ex1I2 = $true) & (nonemptyI = $true) & (dpsetconstrEL1 = $true) & (dsetconstrEL = $true) & (binintersectEL = $true) & (nonemptyE1 = $true) & (binintersectT_lem = $true) & (injFuncSetFuncIn = $true) & (powerset__Cong = $true) & (breln1all2 = $true) & (subsetI1 = $true) & (singletonsubset = $true) & (ubforcartprodlem1 = $true) & (dsetconstrER = $true) & (lamp = $true) & (setadjoinSub = $true) & (notsubsetI = $true) & (setextAx = $true) & (emptysetimpfalse = $true) & (singletoninpowerset = $true) & (complementTcontraSubset = $true) & (inPowerset = $true) & (exuI1 = $true) & (iftrueProp2 = $true) & (iftrueProp1 = $true) & (notdexE = $true) & (powersetT_lem = $true) & (powersetI1 = $true) & (descrp = $true) & (setukpairinjR11 = $true) & (setminusSubset1 = $true) & (omega0Ax = $true) & (funcImageSingleton = $true) & (beta1 = $true) & (subPowSU = $true) & (binunionRsub = $true) & (symdiffIneg1 = $true) & (prop2set2propI = $true) & (breln1unionI = $true) & (iftrueorfalse = $true) & (binunionT_lem = $true) & (complementInPowersetComplementIntersect = $true) & (vacuousDall = $true) & (emptyI = $true) & (descr__Cong = $true) & (woz1_4 = $true) & (apProp = $true) & (inComplementUnionImpNotIn1 = $true) & (setukpairinjR12 = $true) & (image1Ex1 = $true) & (kfstsingleton = $true) & (singletonprop = $true) & (upairequniteq = $true) & (binintersectSubset2 = $true) & (image1Ex = $true) & (binunionTILcontra = $true) & (exuE1 = $true) & (upairsetE = $true) & (setadjoinSub2 = $true) & (funcGraphProp4 = $true) & (image1I = $true) & (cartprodfstpairEq = $true) & (quantDeMorgan3 = $true)),
% 7.07/2.24 inference(skolemisation,[status(esa),new_symbols(skolem,[sK618,sK619,sK620])],[f1340,f2457])).
% 7.07/2.24 thf(f2457,plain,(
% 7.07/2.24 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ (funcSet @ X0 @ X2))) & ((in @ X1 @ (surjFuncSet @ X0 @ X2)) = $true)) => (((in @ sK619 @ (funcSet @ sK618 @ sK620)) != $true) & ((in @ sK619 @ (surjFuncSet @ sK618 @ sK620)) = $true))),
% 7.07/2.24 introduced(definition,[],[choice_axiom])).
% 7.07/2.24 thf(f1340,plain,(
% 7.07/2.24 (iffalseProp1 = $true) & (setunionI = $true) & (cartprodmempair = $true) & (setukpairinjR1 = $true) & (ubforcartprodlem3 = $true) & (upairsubunion = $true) & (dpsetconstrI = $true) & (demorgan1 = $true) & (eqimpsubset2 = $true) & (binunionLsub = $true) & (eta2 = $true) & (upairset2E = $true) & (cartprodpairsurjEq = $true) & (wellorderingAx = $true) & (setOfPairsIsBReln = $true) & (emptyset__Cong = $true) & (omega__Cong = $true) & (kpairsurjEq = $true) & (ifSingleton = $true) & (ex1I = $true) & (funcextLem = $true) & (powersetTE1 = $true) & (woz2W = $true) & (setadjoinIL = $true) & (breln1invI = $true) & (setunionsingleton2 = $true) & (setext = $true) & (setunionsingleton1 = $true) & (binunionEcases = $true) & (powersetE = $true) & (bs114d = $true) & (exuEu = $true) & (setminusELneg = $true) & (setukpairinjL2 = $true) & (contrasubsetT2 = $true) & (replAx = $true) & (upairset2IR = $true) & (injFuncInInjFuncSet = $true) & (binintersectSubset5 = $true) & (lam2lamEq = $true) & (setunion__Cong = $true) & (notequalI2 = $true) & (subbreln1 = $true) & (setadjoinAx = $true) & (demorgan2b = $true) & (emptyE1 = $true) & (ifp = $true) & (notdallE = $true) & (woz2A = $true) & (subsetemptysetimpeq = $true) & (notinemptyset = $true) & (eta1 = $true) & (setminusLsub = $true) & (complementTI1 = $true) & (breln1unionprop = $true) & (setadjoin__Cong = $true) & (binintersectRsub = $true) & (ap2apEq2 = $true) & (symdiffIneg2 = $true) & (woz1_3 = $true) & (setukpairinjR2 = $true) & (uniqinunit = $true) & (setoftrueEq = $true) & (dsetconstrI = $true) & (exuI3 = $true) & (demorgan1a = $true) & (contrasubsetT3 = $true) & (setukpairinjL = $true) & (woz13rule0 = $true) & (breln1unionIR = $true) & (setminusER = $true) & (eqbreln1 = $true) & (woz13rule1 = $true) & (woz13rule4 = $true) & (symdiffI2 = $true) & (nonemptyI1 = $true) & (breln1compE = $true) & (in__Cong = $true) & (breln1compI = $true) & (disjointsetsI1 = $true) & (demorgan2 = $true) & (image1Equiv = $true) & (brelnall1 = $true) & (breln1invprop = $true) & (binunionTIRcontra = $true) & (subsetI2 = $true) & (dpsetconstrSub = $true) & (symdiffI1 = $true) & (eqinunit = $true) & (binintersectI = $true) & (subsetE = $true) & (ubforcartprodlem2 = $true) & (foundationAx = $true) & (cartprodfstin = $true) & (cartprodsndpairEq = $true) & (choice2fnsingleton = $true) & (cartprodsndin = $true) & (breln1unionE = $true) & (setukpairIR = $true) & (inComplementUnionImpInComplement1 = $true) & (woz2Ex = $true) & (secondinupair = $true) & (setadjoinIR = $true) & (quantDeMorgan2 = $true) & (complementSubsetComplementIntersect = $true) & (cartprodpairmemEL = $true) & (binunionIR = $true) & (upairsetIL = $true) & (woz2B = $true) & (singletonsuniq = $true) & (complementImpComplementIntersect = $true) & (emptysetsubset = $true) & (setukpairinjR = $true) & (brelnall2 = $true) & (theeq = $true) & (doubleComplementSub1 = $true) & (iftrue = $true) & (omegaIndAx = $true) & (subsetE2 = $true) & (emptyinPowerset = $true) & (powersetI = $true) & (setukpairIL = $true) & (binintersectTERcontra = $true) & (funcext = $true) & (breln1unionCommutes = $true) & (app = $true) & (binintersectSubset3 = $true) & (funcGraphProp3 = $true) & (cartprodpairin = $true) & (exuE3e = $true) & (demorgan1b = $true) & (breln1unionIL = $true) & (contrasubsetT1 = $true) & (ap2apEq1 = $true) & (doubleComplementEq = $true) & (dsetconstr__Cong = $true) & (dpsetconstrER = $true) & (doubleComplementSub2 = $true) & (notinsingleton = $true) & (subsetRefl = $true) & (ap2p = $true) & (binunionE = $true) & (setunionsingleton = $true) & (binintersectSubset1 = $true) & (doubleComplementE1 = $true) & (complementTE1 = $true) & (powersetAx = $true) & (dpsetconstrEL2 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ (funcSet @ X0 @ X2))) & ((in @ X1 @ (surjFuncSet @ X0 @ X2)) = $true)) & (sepSubset = $true) & (powersetsubset = $true) & (setadjoinOr = $true) & (complementTnotintersectT = $true) & (contrasubsetT = $true) & (complementT_lem = $true) & (woz13rule3 = $true) & (inCongP = $true) & (breln1all1 = $true) & (breln1unionEcases = $true) & (subsetTrans = $true) & (setminusERneg = $true) & (prop2setI = $true) & (inIntersectImpInUnion = $true) & (upairinpowunion = $true) & (iffalse = $true) & (noeltsimpempty = $true) & (subsetTI = $true) & (sepInPowerset = $true) & (emptysetAx = $true) & (setminusEL = $true) & (exuI2 = $true) & (setminusILneg = $true) & (setextT = $true) & (binunionTEcontra = $true) & (image1E = $true) & (woz1_2 = $true) & (funcinfuncset = $true) & (eqimpsubset1 = $true) & (setadjoinE = $true) & (binintersectLsub = $true) & (ex1E2 = $true) & (binintersectER = $true) & (woz13rule2 = $true) & (setOfPairsIsBReln1 = $true) & (binintersectTELcontra = $true) & (binunionTE = $true) & (woz1_1 = $true) & (quantDeMorgan1 = $true) & (ex1E1 = $true) & (emptyinunitempty = $true) & (demorgan2b2 = $true) & (notequalI1 = $true) & (intersectInPowersetIntersectUnions = $true) & (powersetE1 = $true) & (kfstpairEq = $true) & (singletonsswitch = $true) & (nonemptyImpWitness = $true) & (singletoninpowunion = $true) & (breln1compprop = $true) & (eqbreln = $true) & (cartprodmempair1 = $true) & (doubleComplementI1 = $true) & (lam2p = $true) & (binintersectSubset4 = $true) & (ksndsingleton = $true) & (kpairiskpair = $true) & (setextsub = $true) & (setunionAx = $true) & (theprop = $true) & (contraSubsetComplement = $true) & (funcext2 = $true) & (breln1compEex = $true) & (beta2 = $true) & (quantDeMorgan4 = $true) & (breln1SetBreln1 = $true) & (ksndpairEq = $true) & (demorgan2a = $true) & (omegaSAx = $true) & (setminusIRneg = $true) & (setminusSubset2 = $true) & (inIntersectImpInIntersectUnions = $true) & (funcGraphProp1 = $true) & (exu__Cong = $true) & (inIntersectImpInUnion2 = $true) & (exuE2 = $true) & (iffalseProp2 = $true) & (setminusT_lem = $true) & (setukpairinjL1 = $true) & (emptysetE = $true) & (lamProp = $true) & (powersetTI1 = $true) & (breln1invE = $true) & (prop2setE = $true) & (kpairp = $true) & (demorgan2a1 = $true) & (emptyInPowerset = $true) & (funcGraphProp2 = $true) & (setunionE2 = $true) & (injFuncSetFuncInj = $true) & (woz1_5 = $true) & (symdiffE = $true) & (setminusI = $true) & (infuncsetfunc = $true) & (binunionIL = $true) & (exuE3u = $true) & (cartprodpairmemER = $true) & (upairsetIR = $true) & (setunionE = $true) & (subbreln = $true) & (dpsetconstrERa = $true) & (setbeta = $true) & (demorgan2a2 = $true) & (cartprodmempaircEq = $true) & (complementUnionInPowersetComplement = $true) & (subset2powerset = $true) & (ex1I2 = $true) & (nonemptyI = $true) & (dpsetconstrEL1 = $true) & (dsetconstrEL = $true) & (binintersectEL = $true) & (nonemptyE1 = $true) & (binintersectT_lem = $true) & (injFuncSetFuncIn = $true) & (powerset__Cong = $true) & (breln1all2 = $true) & (subsetI1 = $true) & (singletonsubset = $true) & (ubforcartprodlem1 = $true) & (dsetconstrER = $true) & (lamp = $true) & (setadjoinSub = $true) & (notsubsetI = $true) & (setextAx = $true) & (emptysetimpfalse = $true) & (singletoninpowerset = $true) & (complementTcontraSubset = $true) & (inPowerset = $true) & (exuI1 = $true) & (iftrueProp2 = $true) & (iftrueProp1 = $true) & (notdexE = $true) & (powersetT_lem = $true) & (powersetI1 = $true) & (descrp = $true) & (setukpairinjR11 = $true) & (setminusSubset1 = $true) & (omega0Ax = $true) & (funcImageSingleton = $true) & (beta1 = $true) & (subPowSU = $true) & (binunionRsub = $true) & (symdiffIneg1 = $true) & (prop2set2propI = $true) & (breln1unionI = $true) & (iftrueorfalse = $true) & (binunionT_lem = $true) & (complementInPowersetComplementIntersect = $true) & (vacuousDall = $true) & (emptyI = $true) & (descr__Cong = $true) & (woz1_4 = $true) & (apProp = $true) & (inComplementUnionImpNotIn1 = $true) & (setukpairinjR12 = $true) & (image1Ex1 = $true) & (kfstsingleton = $true) & (singletonprop = $true) & (upairequniteq = $true) & (binintersectSubset2 = $true) & (image1Ex = $true) & (binunionTILcontra = $true) & (exuE1 = $true) & (upairsetE = $true) & (setadjoinSub2 = $true) & (funcGraphProp4 = $true) & (image1I = $true) & (cartprodfstpairEq = $true) & (quantDeMorgan3 = $true)),
% 7.07/2.24 inference(flattening,[status(thm)],[f1339])).
% 7.07/2.24 thf(f1339,plain,(
% 7.07/2.24 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ (funcSet @ X0 @ X2))) & ((in @ X1 @ (surjFuncSet @ X0 @ X2)) = $true)) & (injFuncSetFuncInj = $true)) & (injFuncSetFuncIn = $true)) & (injFuncInInjFuncSet = $true)) & (image1I = $true)) & (image1E = $true)) & (image1Equiv = $true)) & (image1Ex1 = $true)) & (image1Ex = $true)) & (woz2B = $true)) & (woz2A = $true)) & (woz2W = $true)) & (woz2Ex = $true)) & (breln1unionCommutes = $true)) & (breln1unionEcases = $true)) & (breln1unionE = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 7.07/2.24 inference(ennf_transformation,[status(thm)],[f699])).
% 7.07/2.24 thf(f699,plain,(
% 7.07/2.24 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ((breln1unionE = $true) => ((breln1unionEcases = $true) => ((breln1unionCommutes = $true) => ((woz2Ex = $true) => ((woz2W = $true) => ((woz2A = $true) => ((woz2B = $true) => ((image1Ex = $true) => ((image1Ex1 = $true) => ((image1Equiv = $true) => ((image1E = $true) => ((image1I = $true) => ((injFuncInInjFuncSet = $true) => ((injFuncSetFuncIn = $true) => ((injFuncSetFuncInj = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (surjFuncSet @ X0 @ X2)) = $true) => ($true = (in @ X1 @ (funcSet @ X0 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.07/2.24 inference(fool_elimination,[status(thm)],[f698])).
% 7.07/2.24 thf(f698,plain,(
% 7.07/2.24 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (surjFuncSet @ X0 @ X2)) => (in @ X1 @ (funcSet @ X0 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.07/2.24 inference(rectify,[status(thm)],[f323])).
% 7.07/2.24 thf(f323,negated_conjecture,(
% 7.07/2.24 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => ! [X3 : $i,X12 : $i,X4 : $i] : ((in @ X12 @ (surjFuncSet @ X3 @ X4)) => (in @ X12 @ (funcSet @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.07/2.24 inference(negated_conjecture,[status(cth)],[f322])).
% 7.07/2.24 thf(f322,conjecture,(
% 7.07/2.24 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => ! [X3 : $i,X12 : $i,X4 : $i] : ((in @ X12 @ (surjFuncSet @ X3 @ X4)) => (in @ X12 @ (funcSet @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.07/2.24 file('/export/starexec/sandbox/tmp/tmp.uXiymMPKZd/DTF2THF_27071.p',surjFuncSetFuncIn)).
% 7.07/2.24 thf(f9445,plain,(
% 7.07/2.24 ((in @ sK619 @ (funcSet @ sK618 @ sK620)) = $true)),
% 7.07/2.24 inference(trivial_inequality_removal,[status(thm)],[f9435])).
% 7.07/2.24 thf(f9435,plain,(
% 7.07/2.24 ($true != $true) | ((in @ sK619 @ (funcSet @ sK618 @ sK620)) = $true)),
% 7.07/2.24 inference(superposition,[status(thm)],[f6615,f6871])).
% 7.07/2.24 thf(f6871,plain,(
% 7.07/2.24 ($true = (in @ sK619 @ (dsetconstr @ (funcSet @ sK618 @ sK620) @ (surjective @ sK618 @ sK620))))),
% 7.07/2.24 inference(beta_eta_normalization,[status(thm)],[f5901])).
% 7.07/2.24 thf(f5901,plain,(
% 7.07/2.24 ((in @ sK619 @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (surjective @ Y0 @ Y1 @ Y2)))))) @ sK618 @ sK620)) = $true)),
% 7.07/2.24 inference(definition_unfolding,[status(thm)],[f4344,f3397])).
% 7.07/2.24 thf(f3397,plain,(
% 7.07/2.24 (surjFuncSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (surjective @ Y0 @ Y1 @ Y2)))))))),
% 7.07/2.24 inference(cnf_transformation,[status(thm)],[f739])).
% 7.07/2.24 thf(f739,plain,(
% 7.07/2.24 (surjFuncSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (surjective @ Y0 @ Y1 @ Y2)))))))),
% 7.07/2.24 inference(fool_elimination,[status(thm)],[f738])).
% 7.07/2.24 thf(f738,plain,(
% 7.07/2.24 (surjFuncSet = (^[X0 : $i, X1 : $i] : (dsetconstr @ (funcSet @ X0 @ X1) @ (^[X2 : $i] : (surjective @ X0 @ X1 @ X2)))))),
% 7.07/2.24 inference(rectify,[status(thm)],[f321])).
% 7.07/2.24 thf(f321,axiom,(
% 7.07/2.24 (surjFuncSet = (^[X3 : $i, X4 : $i] : (dsetconstr @ (funcSet @ X3 @ X4) @ (^[X12 : $i] : (surjective @ X3 @ X4 @ X12)))))),
% 7.07/2.24 file('/export/starexec/sandbox/tmp/tmp.uXiymMPKZd/DTF2THF_27071.p',surjFuncSet)).
% 7.07/2.24 thf(f4344,plain,(
% 7.07/2.24 ((in @ sK619 @ (surjFuncSet @ sK618 @ sK620)) = $true)),
% 7.07/2.24 inference(cnf_transformation,[status(thm)],[f2458])).
% 7.07/2.24 thf(f6615,plain,(
% 7.07/2.24 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (($true != (in @ X1 @ (dsetconstr @ X2 @ X0))) | ((in @ X1 @ X2) = $true)) )),
% 7.07/2.24 inference(beta_eta_normalization,[status(thm)],[f6614])).
% 7.07/2.24 thf(f6614,plain,(
% 7.07/2.24 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (((in @ X1 @ X2) = $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true)) )),
% 7.07/2.24 inference(trivial_inequality_removal,[status(thm)],[f6189])).
% 7.07/2.24 thf(f6189,plain,(
% 7.07/2.24 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (($true != $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true) | ((in @ X1 @ X2) = $true)) )),
% 7.07/2.24 inference(definition_unfolding,[status(thm)],[f4907,f4234])).
% 7.07/2.24 thf(f4234,plain,(
% 7.07/2.24 (dsetconstrEL = $true)),
% 7.07/2.24 inference(cnf_transformation,[status(thm)],[f2458])).
% 7.07/2.24 thf(f4907,plain,(
% 7.07/2.24 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (((in @ X1 @ X2) = $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true) | (dsetconstrEL != $true)) )),
% 7.07/2.24 inference(cnf_transformation,[status(thm)],[f2931])).
% 7.07/2.24 thf(f2931,plain,(
% 7.07/2.24 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true)) | (dsetconstrEL != $true)) & ((dsetconstrEL = $true) | (((in @ sK885 @ sK886) != $true) & ((in @ sK885 @ (dsetconstr @ sK886 @ (^[Y0 : $i]: (sK884 @ Y0)))) = $true)))),
% 7.07/2.24 inference(skolemisation,[status(esa),new_symbols(skolem,[sK884,sK885,sK886])],[f2929,f2930])).
% 7.07/2.24 thf(f2930,plain,(
% 7.07/2.24 ? [X3 : $i > $o,X4 : $i,X5 : $i] : (((in @ X4 @ X5) != $true) & ($true = (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0)))))) => (((in @ sK885 @ sK886) != $true) & ((in @ sK885 @ (dsetconstr @ sK886 @ (^[Y0 : $i]: (sK884 @ Y0)))) = $true))),
% 7.07/2.24 introduced(definition,[],[choice_axiom])).
% 7.07/2.24 thf(f2929,plain,(
% 7.07/2.24 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true)) | (dsetconstrEL != $true)) & ((dsetconstrEL = $true) | ? [X3 : $i > $o,X4 : $i,X5 : $i] : (((in @ X4 @ X5) != $true) & ($true = (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0)))))))),
% 7.07/2.24 inference(rectify,[status(thm)],[f2928])).
% 7.07/2.24 thf(f2928,plain,(
% 7.07/2.24 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true)) | (dsetconstrEL != $true)) & ((dsetconstrEL = $true) | ? [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) & ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) = $true)))),
% 7.07/2.24 inference(nnf_transformation,[status(thm)],[f1074])).
% 7.07/2.24 thf(f1074,plain,(
% 7.07/2.24 ! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true)) <=> (dsetconstrEL = $true)),
% 7.07/2.24 inference(ennf_transformation,[status(thm)],[f617])).
% 7.07/2.24 thf(f617,plain,(
% 7.07/2.24 (dsetconstrEL = $true) <=> ! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) = $true) => ((in @ X1 @ X2) = $true))),
% 7.07/2.24 inference(fool_elimination,[status(thm)],[f616])).
% 7.07/2.24 thf(f616,plain,(
% 7.07/2.24 (dsetconstrEL = ! [X0 : $i > $o,X1 : $i,X2 : $i] : ((in @ X1 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X0 @ X3)))) => (in @ X1 @ X2)))),
% 7.07/2.24 inference(rectify,[status(thm)],[f15])).
% 7.07/2.24 thf(f15,axiom,(
% 7.07/2.24 (dsetconstrEL = ! [X0 : $i > $o,X1 : $i,X3 : $i] : ((in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))) => (in @ X1 @ X3)))),
% 7.07/2.24 file('/export/starexec/sandbox/tmp/tmp.uXiymMPKZd/DTF2THF_27071.p',dsetconstrEL)).
% 7.07/2.24 % SZS output end Proof for DTF2THF_27071
% 7.07/2.24 % (27220)------------------------------
% 7.07/2.24 % (27220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.07/2.24 % (27220)Termination reason: Refutation
% 7.07/2.24
% 7.07/2.24 % (27220)Memory used [KB]: 13176
% 7.07/2.24 % (27220)Time elapsed: 0.296 s
% 7.07/2.24 % (27220)Instructions burned: 872 (million)
% 7.07/2.24 % (27220)------------------------------
% 7.07/2.24 % (27220)------------------------------
% 7.07/2.24 % (27171)Success in time 0.694 s
%------------------------------------------------------------------------------