%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU769^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n006.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 09:37:02 AM UTC 2026
% Result : Theorem 9.35s 2.83s
% Output : Refutation 9.35s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 6
% Syntax : Number of formulae : 39 ( 20 unt; 0 typ; 0 def)
% Number of atoms : 4518 (1211 equ; 0 cnn)
% Maximal formula atoms : 287 ( 115 avg)
% Number of connectives : 2297 ( 30 ~; 17 |; 868 &; 231 @)
% ( 2 <=>;1149 =>; 0 <=; 0 <~>)
% Maximal formula depth : 292 ( 55 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 561 ( 558 usr; 291 con; 0-4 aty)
% Number of variables : 97 ( 41 ^; 41 !; 15 ?; 97 :)
% 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_339,type,
sP0: $i > ( $i > $o ) > $o ).
thf(func_def_340,type,
sP1: ( $i > $o ) > $i > $o ).
thf(func_def_342,type,
sP3: ( $i > $o ) > $i > $o ).
thf(func_def_344,type,
sP5: ( $i > $o ) > $o ).
thf(func_def_345,type,
sP6: $i > $o ).
thf(func_def_346,type,
sP7: $i > $o ).
thf(func_def_347,type,
sP8: $i > $i > $i > $i > $o ).
thf(func_def_348,type,
sP9: $i > $i > $i > $o ).
thf(func_def_349,type,
sP10: $i > $o ).
thf(func_def_350,type,
sP11: $i > ( $i > $o ) > $o ).
thf(func_def_351,type,
sP12: $i > ( $i > $o ) > $i > $i > $o ).
thf(func_def_352,type,
sP13: ( $i > $o ) > $o ).
thf(func_def_353,type,
sP14: $i > $i > $i > $o ).
thf(func_def_354,type,
sP15: $i > $i > $o ).
thf(func_def_355,type,
sP16: $i > $o ).
thf(func_def_356,type,
sP17: $i > $i > $o ).
thf(func_def_357,type,
sP18: $i > $i > $o ).
thf(func_def_358,type,
sP19: $i > $i > $o ).
thf(func_def_359,type,
sP20: ( $i > $o ) > $i > $o ).
thf(func_def_361,type,
sP22: $i > $i > $i > $o ).
thf(func_def_362,type,
sP23: $i > $o > $i > $i > $o ).
thf(func_def_363,type,
sP24: $i > $o > $i > $i > $o ).
thf(func_def_364,type,
sP25: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_365,type,
sP26: $i > ( $i > $i > $o ) > $o ).
thf(func_def_367,type,
sP28: $i > $i > $o ).
thf(func_def_369,type,
sP30: $i > $i > $i > $o ).
thf(func_def_370,type,
sP31: $i > $i > $o ).
thf(func_def_372,type,
sP33: ( $i > $o ) > $o ).
thf(func_def_374,type,
sP35: $i > ( $i > $o ) > $o ).
thf(func_def_375,type,
sP36: $i > $i > $i > ( $i > $o ) > $o ).
thf(func_def_376,type,
sP37: $i > ( $i > $i ) > $i > $o ).
thf(func_def_378,type,
sP39: ( $i > $o ) > $i > $o ).
thf(func_def_380,type,
sP41: $i > $i > $i > $i > $o ).
thf(func_def_381,type,
sP42: $i > $i > $i > $i > $o ).
thf(func_def_383,type,
sP44: $i > $i > $i > $o ).
thf(func_def_384,type,
sP45: $i > $i > $i > $o ).
thf(func_def_385,type,
sP46: $i > ( $i > $i ) > $i > $o ).
thf(func_def_387,type,
sP48: $i > $i > $o > $o ).
thf(func_def_388,type,
sP49: $i > $i > $i > $o ).
thf(func_def_391,type,
sP52: $i > $i > $i > $o ).
thf(func_def_392,type,
sP53: $i > $i > ( $i > $o ) > ( $i > $o ) > $o ).
thf(func_def_393,type,
sP54: $i > ( $i > $o ) > $o ).
thf(func_def_394,type,
sP55: ( $i > $o ) > $i > $i > $o ).
thf(func_def_395,type,
sP56: ( $i > $o ) > $o ).
thf(func_def_396,type,
sP57: $i > ( $i > $o ) > $o ).
thf(func_def_399,type,
sP60: $i > $i > $i > $o ).
thf(func_def_401,type,
sP62: $i > ( $i > $o ) > $o ).
thf(func_def_403,type,
sP64: ( $i > $o ) > $i > $o ).
thf(func_def_405,type,
sP66: $i > $i > $i > $i > $o ).
thf(func_def_407,type,
sP68: $o > $i > $i > $o ).
thf(func_def_408,type,
sP69: $i > $i > $i > $o ).
thf(func_def_412,type,
sP73: ( $i > $o ) > $i > $o ).
thf(func_def_413,type,
sP74: ( $i > $o ) > $i > $o ).
thf(func_def_414,type,
sP75: ( $i > $o ) > $o ).
thf(func_def_415,type,
sP76: $o > $i > $i > $o ).
thf(func_def_417,type,
sP78: $i > $i > $i > $o ).
thf(func_def_418,type,
sP79: ( $i > $o ) > $i > $i > $o ).
thf(func_def_419,type,
sP80: $i > $i > $i > $o ).
thf(func_def_420,type,
sP81: $i > $i > $o ).
thf(func_def_423,type,
sP84: $i > $i > $o > $o ).
thf(func_def_451,type,
sK112: ( $i > $o ) > $i > $i ).
thf(func_def_452,type,
sK113: ( $i > $o ) > $i > $i ).
thf(func_def_453,type,
sK114: $i > ( $i > $o ) > $i ).
thf(func_def_455,type,
sK116: $i > $o ).
thf(func_def_464,type,
sK125: $i > $i > $i ).
thf(func_def_468,type,
sK129: $i > $o ).
thf(func_def_469,type,
sK130: ( $i > $o ) > $i > $i ).
thf(func_def_470,type,
sK131: ( $i > $o ) > $i > $i ).
thf(func_def_472,type,
sK133: ( $i > $o ) > $i ).
thf(func_def_473,type,
sK134: $i > ( $i > $o ) > $i ).
thf(func_def_474,type,
sK135: $i > $o ).
thf(func_def_486,type,
sK147: $i > $i ).
thf(func_def_487,type,
sK148: $i > $i ).
thf(func_def_504,type,
sK165: $i > $i > $i > $i ).
thf(func_def_505,type,
sK166: $i > $i > $i > $i > $i ).
thf(func_def_506,type,
sK167: $i > $i > $i > $i > $i ).
thf(func_def_507,type,
sK168: $i > $i > $i > $i > $i ).
thf(func_def_512,type,
sK173: $i > $o ).
thf(func_def_530,type,
sK191: $i > $i > $i ).
thf(func_def_531,type,
sK192: $i > $i ).
thf(func_def_540,type,
sK201: $i > ( $i > $o ) > $i > $i > $i ).
thf(func_def_541,type,
sK202: $i > ( $i > $o ) > $i > $i > $i ).
thf(func_def_542,type,
sK203: $i > ( $i > $o ) > $i ).
thf(func_def_546,type,
sK207: $i > $o ).
thf(func_def_562,type,
sK223: ( $i > $o ) > $i ).
thf(func_def_563,type,
sK224: $i > ( $i > $o ) > $i ).
thf(func_def_564,type,
sK225: $i > $o ).
thf(func_def_582,type,
sK243: $i > $i > $i ).
thf(func_def_583,type,
sK244: $i > $i > $i ).
thf(func_def_584,type,
sK245: $i > $i > $i ).
thf(func_def_585,type,
sK246: $i > $i > $i ).
thf(func_def_586,type,
sK247: $i > $i > $i > $i ).
thf(func_def_587,type,
sK248: $i > $i > $i ).
thf(func_def_588,type,
sK249: $i > $i > $i ).
thf(func_def_589,type,
sK250: $i > $i > $i ).
thf(func_def_590,type,
sK251: $i > $i ).
thf(func_def_591,type,
sK252: $i > $i ).
thf(func_def_592,type,
sK253: $i > $i ).
thf(func_def_593,type,
sK254: $i > $i ).
thf(func_def_594,type,
sK255: $i > $i > $i ).
thf(func_def_595,type,
sK256: $i > $i > $i ).
thf(func_def_596,type,
sK257: $i > $i > $i ).
thf(func_def_597,type,
sK258: $i > $i > $i > $i ).
thf(func_def_598,type,
sK259: $i > $i > $i > $i ).
thf(func_def_599,type,
sK260: $i > $i ).
thf(func_def_606,type,
sK267: $i > $o ).
thf(func_def_608,type,
sK269: $i > ( $i > $o ) > $i ).
thf(func_def_609,type,
sK270: ( $i > $o ) > $i > $i ).
thf(func_def_610,type,
sK271: $i > $i > $i > $i ).
thf(func_def_611,type,
sK272: $i > $i > $i > $i > $i ).
thf(func_def_620,type,
sK281: $i > $i > $o ).
thf(func_def_621,type,
sK282: ( $i > $i > $o ) > $i > $i ).
thf(func_def_622,type,
sK283: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_623,type,
sK284: $i > ( $i > $i > $o ) > $i ).
thf(func_def_624,type,
sK285: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_646,type,
sK307: $i > $i ).
thf(func_def_654,type,
sK315: $i > $i > $i ).
thf(func_def_672,type,
sK333: $i > $i > $i > $i ).
thf(func_def_676,type,
sK337: $i > $o ).
thf(func_def_687,type,
sK348: $i > $i > $i ).
thf(func_def_688,type,
sK349: $i > $i > $i ).
thf(func_def_698,type,
sK359: $i > ( $i > $o ) > $i ).
thf(func_def_699,type,
sK360: ( $i > $o ) > $i ).
thf(func_def_700,type,
sK361: $i > $o ).
thf(func_def_714,type,
sK375: $i > $i > $i ).
thf(func_def_717,type,
sK378: $i > $i > $o ).
thf(func_def_730,type,
sK391: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_731,type,
sK392: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_732,type,
sK393: $i > ( $i > $o ) > $i ).
thf(func_def_736,type,
sK397: $i > $o ).
thf(func_def_758,type,
sK419: $i > $o ).
thf(func_def_765,type,
sK426: $i > $i ).
thf(func_def_767,type,
sK428: $i > ( $i > $i ) > $i > $i ).
thf(func_def_768,type,
sK429: $i > ( $i > $i ) > $i > $i ).
thf(func_def_779,type,
sK440: $i > $o ).
thf(func_def_781,type,
sK442: $i > ( $i > $o ) > $i ).
thf(func_def_782,type,
sK443: ( $i > $o ) > $i > $i ).
thf(func_def_793,type,
sK454: $i > $i > $i > $i > $i ).
thf(func_def_794,type,
sK455: $i > $i > $i > $i > $i ).
thf(func_def_795,type,
sK456: $i > $i > $i > $i > $i ).
thf(func_def_796,type,
sK457: $i > $i > $i > $i > $i ).
thf(func_def_801,type,
sK462: $i > $o ).
thf(func_def_802,type,
sK463: $i > $o ).
thf(func_def_803,type,
sK464: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_804,type,
sK465: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_828,type,
sK489: $i > $i > $i > $i ).
thf(func_def_829,type,
sK490: $i > $i > $i > $i ).
thf(func_def_840,type,
sK501: $i > $i ).
thf(func_def_841,type,
sK502: ( $i > $i ) > $i > $i > $i ).
thf(func_def_842,type,
sK503: $i > ( $i > $i ) > $i > $i ).
thf(func_def_847,type,
sK508: $i > $o ).
thf(func_def_848,type,
sK509: $i > $i > $i > $o ).
thf(func_def_856,type,
sK517: $i > $i ).
thf(func_def_859,type,
sK520: $i > $i > ( $i > $i ) > $i ).
thf(func_def_868,type,
sK529: $i > $o ).
thf(func_def_869,type,
sK530: ( $i > $o ) > $i ).
thf(func_def_876,type,
sK537: $i > $o ).
thf(func_def_879,type,
sK540: $i > $i > $o ).
thf(func_def_914,type,
sK575: $i > $o ).
thf(func_def_929,type,
sK590: $i > $i > $i ).
thf(func_def_930,type,
sK591: $i > $i > $i > $i ).
thf(func_def_931,type,
sK592: $i > $i > $i > $i ).
thf(func_def_936,type,
sK597: $i > $o ).
thf(func_def_946,type,
sK607: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_947,type,
sK608: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_950,type,
sK611: $i > $o ).
thf(func_def_951,type,
sK612: $i > $o ).
thf(func_def_952,type,
sK613: $i > $o ).
thf(func_def_974,type,
sK635: ( $i > $o ) > $i > $i > $i ).
thf(func_def_975,type,
sK636: ( $i > $o ) > $i > $i > $i ).
thf(func_def_976,type,
sK637: $i > ( $i > $o ) > $i ).
thf(func_def_979,type,
sK640: $i > $o ).
thf(func_def_980,type,
sK641: $i > ( $i > $o ) > $i ).
thf(func_def_981,type,
sK642: ( $i > $o ) > $i ).
thf(func_def_982,type,
sK643: $i > $o ).
thf(func_def_983,type,
sK644: $i > $o ).
thf(func_def_985,type,
sK646: $i > ( $i > $o ) > $i ).
thf(func_def_986,type,
sK647: $i > ( $i > $o ) > $i ).
thf(func_def_1006,type,
sK667: $i > $i > $o ).
thf(func_def_1015,type,
sK676: $i > $o ).
thf(func_def_1016,type,
sK677: $i > $i > $o ).
thf(func_def_1028,type,
sK689: $i > $i ).
thf(func_def_1046,type,
sK707: $i > $i ).
thf(func_def_1048,type,
sK709: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1055,type,
sK716: $i > $o ).
thf(func_def_1064,type,
sK725: $i > $o ).
thf(func_def_1066,type,
sK727: $i > ( $i > $o ) > $i ).
thf(func_def_1067,type,
sK728: $i > ( $i > $o ) > $i ).
thf(func_def_1072,type,
sK733: $i > $o ).
thf(func_def_1074,type,
sK735: $i > ( $i > $o ) > $i ).
thf(func_def_1075,type,
sK736: ( $i > $o ) > $i > $i ).
thf(func_def_1076,type,
sK737: $i > $i > $i ).
thf(func_def_1088,type,
sK749: $i > $i ).
thf(func_def_1095,type,
sK756: $i > $o ).
thf(func_def_1117,type,
sK778: $i > $i > $o ).
thf(func_def_1120,type,
sK781: $i > $i > $i > $i > $i ).
thf(func_def_1121,type,
sK782: $i > $i > $i > $i > $i ).
thf(func_def_1126,type,
sK787: $i > $o ).
thf(func_def_1127,type,
sK788: $i > $o ).
thf(func_def_1128,type,
sK789: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1129,type,
sK790: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1143,type,
sK804: $i > $i > $i > $o ).
thf(func_def_1144,type,
sK805: $i > $i > $i > $i ).
thf(func_def_1158,type,
sK819: $i > $i ).
thf(func_def_1159,type,
sK820: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1197,type,
sK858: $i > $i > $o ).
thf(func_def_1202,type,
sK863: $i > $i ).
thf(func_def_1205,type,
sK866: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1215,type,
sK876: $i > $i ).
thf(func_def_1218,type,
sK879: $i > $i > $i ).
thf(func_def_1219,type,
sK880: ( $i > $o ) > $i > $i ).
thf(func_def_1220,type,
sK881: ( $i > $o ) > $i > $i ).
thf(func_def_1221,type,
sK882: ( $i > $o ) > $i > $i ).
thf(func_def_1222,type,
sK883: $i > $o ).
thf(func_def_1224,type,
sK885: ( $i > $o ) > $i ).
thf(func_def_1225,type,
sK886: ( $i > $o ) > $i ).
thf(func_def_1226,type,
sK887: $i > $o ).
thf(func_def_1232,type,
sK893: $i > $o ).
thf(func_def_1260,type,
sK921: $i > $i > $i > $i ).
thf(func_def_1268,type,
sK929: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1269,type,
sK930: $i > $o ).
thf(func_def_1277,type,
sK938: $i > $i > $i > $i ).
thf(func_def_1278,type,
sK939: $i > $i > $i > $i > $i ).
thf(func_def_1286,type,
sK947: $i > $i > $i ).
thf(func_def_1313,type,
sK974: $i > $o ).
thf(func_def_1327,type,
sK988: $i > $i > $i ).
thf(func_def_1328,type,
sK989: $i > $i > $o > $i ).
thf(func_def_1334,type,
sK995: $i > $i > $o ).
thf(func_def_1337,type,
sK998: $i > $o ).
thf(func_def_1339,type,
sK1000: $i > ( $i > $o ) > $i ).
thf(func_def_1341,type,
ph1002:
!>[X0: $tType] : X0 ).
thf(f9615,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f9560,f5904]) ).
thf(f5904,plain,
( ( breln @ sK295 @ sK295 @ sK294 )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f4890]) ).
thf(f4890,plain,
( ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 )
@ sK295
@ sK294 )
!= $true ),
inference(definition_unfolding,[status(thm)],[f3346,f3736]) ).
thf(f3736,plain,
( breln1
= ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 ) ) ),
inference(cnf_transformation,[status(thm)],[f812]) ).
thf(f812,plain,
( breln1
= ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 ) ) ),
inference(fool_elimination,[status(thm)],[f811]) ).
thf(f811,plain,
( breln1
= ( ^ [X0: $i,X1: $i] : ( breln @ X0 @ X0 @ X1 ) ) ),
inference(rectify,[status(thm)],[f289]) ).
thf(f289,axiom,
( breln1
= ( ^ [X3: $i,X13: $i] : ( breln @ X3 @ X3 @ X13 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2THF_16347.p',breln1) ).
thf(f3346,plain,
( ( breln1 @ sK295 @ sK294 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1794]) ).
thf(f1794,plain,
( ( binunionLsub = $true )
& ( setunionsingleton1 = $true )
& ( replAx = $true )
& ( woz13rule0 = $true )
& ( eta2 = $true )
& ( setextT = $true )
& ( secondinupair = $true )
& ( setukpairinjR = $true )
& ( ap2p = $true )
& ( exuE1 = $true )
& ( quantDeMorgan1 = $true )
& ( symdiffI2 = $true )
& ( exuI3 = $true )
& ( setunionE = $true )
& ( subsetRefl = $true )
& ( doubleComplementE1 = $true )
& ( vacuousDall = $true )
& ( disjointsetsI1 = $true )
& ( iffalse = $true )
& ( binintersectRsub = $true )
& ( setukpairinjR11 = $true )
& ( setminusSubset1 = $true )
& ( setukpairinjR2 = $true )
& ( exu__Cong = $true )
& ( funcGraphProp1 = $true )
& ( binintersectEL = $true )
& ( setminusILneg = $true )
& ( doubleComplementSub2 = $true )
& ( subsetTrans = $true )
& ( complementUnionInPowersetComplement = $true )
& ( subbreln = $true )
& ( symdiffIneg1 = $true )
& ( complementTE1 = $true )
& ( setunionsingleton = $true )
& ( eqimpsubset1 = $true )
& ( kpairp = $true )
& ( complementT_lem = $true )
& ( prop2setI = $true )
& ( binintersectTERcontra = $true )
& ( emptyinPowerset = $true )
& ( setadjoinSub = $true )
& ( contraSubsetComplement = $true )
& ( woz13rule1 = $true )
& ( powersetI1 = $true )
& ( ap2apEq1 = $true )
& ( setminusT_lem = $true )
& ( brelnall1 = $true )
& ( breln1all2 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( cartprodsndpairEq = $true )
& ( theeq = $true )
& ( upairset2E = $true )
& ( ifSingleton = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a1 = $true )
& ( dpsetconstrER = $true )
& ( inIntersectImpInUnion2 = $true )
& ( singletonsuniq = $true )
& ( woz1_1 = $true )
& ( sepInPowerset = $true )
& ( demorgan1 = $true )
& ( upairsetIR = $true )
& ( setukpairinjR1 = $true )
& ( contrasubsetT3 = $true )
& ( setadjoinIR = $true )
& ( doubleComplementSub1 = $true )
& ( emptyInPowerset = $true )
& ( powersetI = $true )
& ( upairset2IR = $true )
& ( emptyinunitempty = $true )
& ( complementTnotintersectT = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyI1 = $true )
& ( upairequniteq = $true )
& ( exuE3e = $true )
& ( notinsingleton = $true )
& ( ex1E1 = $true )
& ( setunionI = $true )
& ( ubforcartprodlem1 = $true )
& ( dpsetconstrSub = $true )
& ( exuEu = $true )
& ( emptysetE = $true )
& ( singletonsubset = $true )
& ( complementTcontraSubset = $true )
& ( cartprodmempaircEq = $true )
& ( setadjoin__Cong = $true )
& ( binunionEcases = $true )
& ( doubleComplementEq = $true )
& ( lam2p = $true )
& ( foundationAx = $true )
& ( funcImageSingleton = $true )
& ( demorgan2a = $true )
& ( binunionIR = $true )
& ( notdexE = $true )
& ( binunionE = $true )
& ( upairinpowunion = $true )
& ( eqbreln = $true )
& ( ksndsingleton = $true )
& ( binintersectSubset5 = $true )
& ( lamProp = $true )
& ( ifp = $true )
& ( symdiffIneg2 = $true )
& ( theprop = $true )
& ( lamp = $true )
& ( binintersectSubset3 = $true )
& ( in__Cong = $true )
& ( quantDeMorgan4 = $true )
& ( descr__Cong = $true )
& ( iftrueorfalse = $true )
& ( subsetE2 = $true )
& ( powersetTI1 = $true )
& ( complementImpComplementIntersect = $true )
& ( omega0Ax = $true )
& ( binintersectTELcontra = $true )
& ( wellorderingAx = $true )
& ( subset2powerset = $true )
& ( dsetconstr__Cong = $true )
& ( cartprodpairin = $true )
& ( binintersectER = $true )
& ( binintersectLsub = $true )
& ( binintersectSubset4 = $true )
& ( binunionTEcontra = $true )
& ( dpsetconstrI = $true )
& ( subsetI2 = $true )
& ( setunionAx = $true )
& ( beta2 = $true )
& ( contrasubsetT2 = $true )
& ( binunionTIRcontra = $true )
& ( ex1I = $true )
& ( eta1 = $true )
& ( setukpairIL = $true )
& ( nonemptyI = $true )
& ( subsetI1 = $true )
& ( emptyE1 = $true )
& ( cartprodmempair = $true )
& ( nonemptyImpWitness = $true )
& ( funcGraphProp2 = $true )
& ( symdiffI1 = $true )
& ( notinemptyset = $true )
& ( sepSubset = $true )
& ( eqimpsubset2 = $true )
& ( kfstpairEq = $true )
& ( funcGraphProp3 = $true )
& ( inIntersectImpInUnion = $true )
& ( notequalI1 = $true )
& ( setunion__Cong = $true )
& ( demorgan2 = $true )
& ( setminusER = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset1 = $true )
& ( kpairiskpair = $true )
& ( omegaSAx = $true )
& ( setext = $true )
& ( powersetE1 = $true )
& ( setminusIRneg = $true )
& ( ubforcartprodlem2 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( funcext2 = $true )
& ( bs114d = $true )
& ( cartprodmempair1 = $true )
& ( uniqinunit = $true )
& ( dpsetconstrEL2 = $true )
& ( dsetconstrEL = $true )
& ( cartprodpairmemER = $true )
& ( eqinunit = $true )
& ( woz13rule2 = $true )
& ( setminusI = $true )
& ( powerset__Cong = $true )
& ( powersetE = $true )
& ( emptysetAx = $true )
& ( dpsetconstrERa = $true )
& ( dsetconstrER = $true )
& ( setukpairinjL1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( setbeta = $true )
& ( subsetE = $true )
& ( setOfPairsIsBReln = $true )
& ( omega__Cong = $true )
& ( setminusEL = $true )
& ( iftrueProp1 = $true )
& ( omegaIndAx = $true )
& ( doubleComplementI1 = $true )
& ( setextsub = $true )
& ( subsetTI = $true )
& ( complementTI1 = $true )
& ( cartprodfstin = $true )
& ( funcGraphProp4 = $true )
& ( singletonsswitch = $true )
& ( setminusSubset2 = $true )
& ( symdiffE = $true )
& ( upairsubunion = $true )
& ( setadjoinIL = $true )
& ( setukpairinjL = $true )
& ( inPowerset = $true )
& ( cartprodfstpairEq = $true )
& ( singletoninpowunion = $true )
& ( woz1_3 = $true )
& ( demorgan2a2 = $true )
& ( funcextLem = $true )
& ( upairsetIL = $true )
& ( demorgan1b = $true )
& ( woz1_4 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( funcext = $true )
& ( contrasubsetT1 = $true )
& ( iftrueProp2 = $true )
& ( setukpairinjR12 = $true )
& ( subPowSU = $true )
& ( setminusERneg = $true )
& ( binunionTE = $true )
& ( notsubsetI = $true )
& ( infuncsetfunc = $true )
& ( setadjoinE = $true )
& ( setunionE2 = $true )
& ( binunionT_lem = $true )
& ( demorgan2b = $true )
& ( exuE3u = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( notequalI2 = $true )
& ( woz13rule3 = $true )
& ( subsetemptysetimpeq = $true )
& ( app = $true )
& ( lam2lamEq = $true )
& ( emptyI = $true )
& ( exuI2 = $true )
& ( powersetTE1 = $true )
& ( emptysetsubset = $true )
& ( notdallE = $true )
& ( kfstsingleton = $true )
& ( powersetT_lem = $true )
& ( setadjoinAx = $true )
& ( woz13rule4 = $true )
& ( woz1_5 = $true )
& ( contrasubsetT = $true )
& ( setminusLsub = $true )
& ( dpsetconstrEL1 = $true )
& ( noeltsimpempty = $true )
& ( setukpairIR = $true )
& ( cartprodpairsurjEq = $true )
& ( upairsetE = $true )
& ( setminusELneg = $true )
& ( setukpairinjL2 = $true )
& ( singletoninpowerset = $true )
& ( demorgan1a = $true )
& ( binunionIL = $true )
& ( dsetconstrI = $true )
& ( ksndpairEq = $true )
& ( beta1 = $true )
& ( setadjoinSub2 = $true )
& ( inCongP = $true )
& ( iffalseProp1 = $true )
& ( emptysetimpfalse = $true )
& ( descrp = $true )
& ( ubforcartprodlem3 = $true )
& ( funcinfuncset = $true )
& ( singletonprop = $true )
& ( prop2set2propI = $true )
& ( iftrue = $true )
& ( kpairsurjEq = $true )
& ( binunionTILcontra = $true )
& ( ex1I2 = $true )
& ( cartprodsndin = $true )
& ( nonemptyE1 = $true )
& ( cartprodpairmemEL = $true )
& ( emptyset__Cong = $true )
& ( binunionRsub = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( setextAx = $true )
& ( iffalseProp2 = $true )
& ( ap2apEq2 = $true )
& ( powersetsubset = $true )
& ( binintersectI = $true )
& ( binintersectSubset2 = $true )
& ( exuE2 = $true )
& ( prop2setE = $true )
& ( apProp = $true )
& ( powersetAx = $true )
& ( setunionsingleton2 = $true )
& ( ( in @ sK294 @ ( breln1Set @ sK295 ) )
= $true )
& ( ( breln1 @ sK295 @ sK294 )
!= $true )
& ( woz1_2 = $true )
& ( binintersectT_lem = $true )
& ( quantDeMorgan3 = $true )
& ( setadjoinOr = $true )
& ( exuI1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK294,sK295])],[f1182,f1793]) ).
thf(f1793,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( breln1Set @ X1 ) )
= $true )
& ( ( breln1 @ X1 @ X0 )
!= $true ) )
=> ( ( ( in @ sK294 @ ( breln1Set @ sK295 ) )
= $true )
& ( ( breln1 @ sK295 @ sK294 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1182,plain,
( ( binunionLsub = $true )
& ( setunionsingleton1 = $true )
& ( replAx = $true )
& ( woz13rule0 = $true )
& ( eta2 = $true )
& ( setextT = $true )
& ( secondinupair = $true )
& ( setukpairinjR = $true )
& ( ap2p = $true )
& ( exuE1 = $true )
& ( quantDeMorgan1 = $true )
& ( symdiffI2 = $true )
& ( exuI3 = $true )
& ( setunionE = $true )
& ( subsetRefl = $true )
& ( doubleComplementE1 = $true )
& ( vacuousDall = $true )
& ( disjointsetsI1 = $true )
& ( iffalse = $true )
& ( binintersectRsub = $true )
& ( setukpairinjR11 = $true )
& ( setminusSubset1 = $true )
& ( setukpairinjR2 = $true )
& ( exu__Cong = $true )
& ( funcGraphProp1 = $true )
& ( binintersectEL = $true )
& ( setminusILneg = $true )
& ( doubleComplementSub2 = $true )
& ( subsetTrans = $true )
& ( complementUnionInPowersetComplement = $true )
& ( subbreln = $true )
& ( symdiffIneg1 = $true )
& ( complementTE1 = $true )
& ( setunionsingleton = $true )
& ( eqimpsubset1 = $true )
& ( kpairp = $true )
& ( complementT_lem = $true )
& ( prop2setI = $true )
& ( binintersectTERcontra = $true )
& ( emptyinPowerset = $true )
& ( setadjoinSub = $true )
& ( contraSubsetComplement = $true )
& ( woz13rule1 = $true )
& ( powersetI1 = $true )
& ( ap2apEq1 = $true )
& ( setminusT_lem = $true )
& ( brelnall1 = $true )
& ( breln1all2 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( cartprodsndpairEq = $true )
& ( theeq = $true )
& ( upairset2E = $true )
& ( ifSingleton = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a1 = $true )
& ( dpsetconstrER = $true )
& ( inIntersectImpInUnion2 = $true )
& ( singletonsuniq = $true )
& ( woz1_1 = $true )
& ( sepInPowerset = $true )
& ( demorgan1 = $true )
& ( upairsetIR = $true )
& ( setukpairinjR1 = $true )
& ( contrasubsetT3 = $true )
& ( setadjoinIR = $true )
& ( doubleComplementSub1 = $true )
& ( emptyInPowerset = $true )
& ( powersetI = $true )
& ( upairset2IR = $true )
& ( emptyinunitempty = $true )
& ( complementTnotintersectT = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyI1 = $true )
& ( upairequniteq = $true )
& ( exuE3e = $true )
& ( notinsingleton = $true )
& ( ex1E1 = $true )
& ( setunionI = $true )
& ( ubforcartprodlem1 = $true )
& ( dpsetconstrSub = $true )
& ( exuEu = $true )
& ( emptysetE = $true )
& ( singletonsubset = $true )
& ( complementTcontraSubset = $true )
& ( cartprodmempaircEq = $true )
& ( setadjoin__Cong = $true )
& ( binunionEcases = $true )
& ( doubleComplementEq = $true )
& ( lam2p = $true )
& ( foundationAx = $true )
& ( funcImageSingleton = $true )
& ( demorgan2a = $true )
& ( binunionIR = $true )
& ( notdexE = $true )
& ( binunionE = $true )
& ( upairinpowunion = $true )
& ( eqbreln = $true )
& ( ksndsingleton = $true )
& ( binintersectSubset5 = $true )
& ( lamProp = $true )
& ( ifp = $true )
& ( symdiffIneg2 = $true )
& ( theprop = $true )
& ( lamp = $true )
& ( binintersectSubset3 = $true )
& ( in__Cong = $true )
& ( quantDeMorgan4 = $true )
& ( descr__Cong = $true )
& ( iftrueorfalse = $true )
& ( subsetE2 = $true )
& ( powersetTI1 = $true )
& ( complementImpComplementIntersect = $true )
& ( omega0Ax = $true )
& ( binintersectTELcontra = $true )
& ( wellorderingAx = $true )
& ( subset2powerset = $true )
& ( dsetconstr__Cong = $true )
& ( cartprodpairin = $true )
& ( binintersectER = $true )
& ( binintersectLsub = $true )
& ( binintersectSubset4 = $true )
& ( binunionTEcontra = $true )
& ( dpsetconstrI = $true )
& ( subsetI2 = $true )
& ( setunionAx = $true )
& ( beta2 = $true )
& ( contrasubsetT2 = $true )
& ( binunionTIRcontra = $true )
& ( ex1I = $true )
& ( eta1 = $true )
& ( setukpairIL = $true )
& ( nonemptyI = $true )
& ( subsetI1 = $true )
& ( emptyE1 = $true )
& ( cartprodmempair = $true )
& ( nonemptyImpWitness = $true )
& ( funcGraphProp2 = $true )
& ( symdiffI1 = $true )
& ( notinemptyset = $true )
& ( sepSubset = $true )
& ( eqimpsubset2 = $true )
& ( kfstpairEq = $true )
& ( funcGraphProp3 = $true )
& ( inIntersectImpInUnion = $true )
& ( notequalI1 = $true )
& ( setunion__Cong = $true )
& ( demorgan2 = $true )
& ( setminusER = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset1 = $true )
& ( kpairiskpair = $true )
& ( omegaSAx = $true )
& ( setext = $true )
& ( powersetE1 = $true )
& ( setminusIRneg = $true )
& ( ubforcartprodlem2 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( funcext2 = $true )
& ( bs114d = $true )
& ( cartprodmempair1 = $true )
& ( uniqinunit = $true )
& ( dpsetconstrEL2 = $true )
& ( dsetconstrEL = $true )
& ( cartprodpairmemER = $true )
& ( eqinunit = $true )
& ( woz13rule2 = $true )
& ( setminusI = $true )
& ( powerset__Cong = $true )
& ( powersetE = $true )
& ( emptysetAx = $true )
& ( dpsetconstrERa = $true )
& ( dsetconstrER = $true )
& ( setukpairinjL1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( setbeta = $true )
& ( subsetE = $true )
& ( setOfPairsIsBReln = $true )
& ( omega__Cong = $true )
& ( setminusEL = $true )
& ( iftrueProp1 = $true )
& ( omegaIndAx = $true )
& ( doubleComplementI1 = $true )
& ( setextsub = $true )
& ( subsetTI = $true )
& ( complementTI1 = $true )
& ( cartprodfstin = $true )
& ( funcGraphProp4 = $true )
& ( singletonsswitch = $true )
& ( setminusSubset2 = $true )
& ( symdiffE = $true )
& ( upairsubunion = $true )
& ( setadjoinIL = $true )
& ( setukpairinjL = $true )
& ( inPowerset = $true )
& ( cartprodfstpairEq = $true )
& ( singletoninpowunion = $true )
& ( woz1_3 = $true )
& ( demorgan2a2 = $true )
& ( funcextLem = $true )
& ( upairsetIL = $true )
& ( demorgan1b = $true )
& ( woz1_4 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( funcext = $true )
& ( contrasubsetT1 = $true )
& ( iftrueProp2 = $true )
& ( setukpairinjR12 = $true )
& ( subPowSU = $true )
& ( setminusERneg = $true )
& ( binunionTE = $true )
& ( notsubsetI = $true )
& ( infuncsetfunc = $true )
& ( setadjoinE = $true )
& ( setunionE2 = $true )
& ( binunionT_lem = $true )
& ( demorgan2b = $true )
& ( exuE3u = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( notequalI2 = $true )
& ( woz13rule3 = $true )
& ( subsetemptysetimpeq = $true )
& ( app = $true )
& ( lam2lamEq = $true )
& ( emptyI = $true )
& ( exuI2 = $true )
& ( powersetTE1 = $true )
& ( emptysetsubset = $true )
& ( notdallE = $true )
& ( kfstsingleton = $true )
& ( powersetT_lem = $true )
& ( setadjoinAx = $true )
& ( woz13rule4 = $true )
& ( woz1_5 = $true )
& ( contrasubsetT = $true )
& ( setminusLsub = $true )
& ( dpsetconstrEL1 = $true )
& ( noeltsimpempty = $true )
& ( setukpairIR = $true )
& ( cartprodpairsurjEq = $true )
& ( upairsetE = $true )
& ( setminusELneg = $true )
& ( setukpairinjL2 = $true )
& ( singletoninpowerset = $true )
& ( demorgan1a = $true )
& ( binunionIL = $true )
& ( dsetconstrI = $true )
& ( ksndpairEq = $true )
& ( beta1 = $true )
& ( setadjoinSub2 = $true )
& ( inCongP = $true )
& ( iffalseProp1 = $true )
& ( emptysetimpfalse = $true )
& ( descrp = $true )
& ( ubforcartprodlem3 = $true )
& ( funcinfuncset = $true )
& ( singletonprop = $true )
& ( prop2set2propI = $true )
& ( iftrue = $true )
& ( kpairsurjEq = $true )
& ( binunionTILcontra = $true )
& ( ex1I2 = $true )
& ( cartprodsndin = $true )
& ( nonemptyE1 = $true )
& ( cartprodpairmemEL = $true )
& ( emptyset__Cong = $true )
& ( binunionRsub = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( setextAx = $true )
& ( iffalseProp2 = $true )
& ( ap2apEq2 = $true )
& ( powersetsubset = $true )
& ( binintersectI = $true )
& ( binintersectSubset2 = $true )
& ( exuE2 = $true )
& ( prop2setE = $true )
& ( apProp = $true )
& ( powersetAx = $true )
& ( setunionsingleton2 = $true )
& ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( breln1Set @ X1 ) )
= $true )
& ( ( breln1 @ X1 @ X0 )
!= $true ) )
& ( woz1_2 = $true )
& ( binintersectT_lem = $true )
& ( quantDeMorgan3 = $true )
& ( setadjoinOr = $true )
& ( exuI1 = $true ) ),
inference(flattening,[status(thm)],[f1181]) ).
thf(f1181,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( breln1Set @ X1 ) )
= $true )
& ( ( breln1 @ X1 @ X0 )
!= $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)],[f459]) ).
thf(f459,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 )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( breln1Set @ X1 ) )
= $true )
=> ( ( breln1 @ X1 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f458]) ).
thf(f458,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
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( breln1Set @ X1 ) )
=> ( breln1 @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f293]) ).
thf(f293,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
=> ! [X13: $i,X3: $i] :
( ( in @ X13 @ ( breln1Set @ X3 ) )
=> ( breln1 @ X3 @ X13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f292]) ).
thf(f292,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
=> ! [X13: $i,X3: $i] :
( ( in @ X13 @ ( breln1Set @ X3 ) )
=> ( breln1 @ X3 @ X13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2THF_16347.p',breln1SetBreln1) ).
thf(f9560,plain,
( ( breln @ sK295 @ sK295 @ sK294 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f9556]) ).
thf(f9556,plain,
( ( $true != $true )
| ( ( breln @ sK295 @ sK295 @ sK294 )
= $true ) ),
inference(superposition,[status(thm)],[f5915,f5951]) ).
thf(f5951,plain,
( ( in @ sK294 @ ( dsetconstr @ ( powerset @ ( cartprod @ sK295 @ sK295 ) ) @ ( breln @ sK295 @ sK295 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f4889]) ).
thf(f4889,plain,
( ( in @ sK294
@ ( ^ [Y0: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ Y0 @ Y0 ) )
@ ^ [Y1: $i] :
( ^ [Y2: $i,Y3: $i] : ( breln @ Y2 @ Y2 @ Y3 )
@ Y0
@ Y1 ) )
@ sK295 ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f3347,f4667]) ).
thf(f4667,plain,
( breln1Set
= ( ^ [Y0: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ Y0 @ Y0 ) )
@ ^ [Y1: $i] :
( ^ [Y2: $i,Y3: $i] : ( breln @ Y2 @ Y2 @ Y3 )
@ Y0
@ Y1 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f3194,f3736]) ).
thf(f3194,plain,
( breln1Set
= ( ^ [Y0: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ Y0 @ Y0 ) )
@ ^ [Y1: $i] : ( breln1 @ Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f568]) ).
thf(f568,plain,
( breln1Set
= ( ^ [Y0: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ Y0 @ Y0 ) )
@ ^ [Y1: $i] : ( breln1 @ Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f567]) ).
thf(f567,plain,
( breln1Set
= ( ^ [X0: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ X0 @ X0 ) )
@ ^ [X1: $i] : ( breln1 @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f291]) ).
thf(f291,axiom,
( breln1Set
= ( ^ [X3: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ X3 @ X3 ) )
@ ^ [X13: $i] : ( breln1 @ X3 @ X13 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2THF_16347.p',breln1Set) ).
thf(f3347,plain,
( ( in @ sK294 @ ( breln1Set @ sK295 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1794]) ).
thf(f5915,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( ( in @ X1 @ ( dsetconstr @ X2 @ X0 ) )
!= $true )
| ( ( X0 @ X1 )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f5914]) ).
thf(f5914,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( ( X0 @ X1 )
= $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f5260]) ).
thf(f5260,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( ( X0 @ X1 )
= $true )
| ( $true != $true )
| ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f4103,f3456]) ).
thf(f3456,plain,
dsetconstrER = $true,
inference(cnf_transformation,[status(thm)],[f1794]) ).
thf(f4103,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true )
| ( ( X0 @ X1 )
= $true )
| ( dsetconstrER != $true ) ),
inference(cnf_transformation,[status(thm)],[f2345]) ).
thf(f2345,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true )
| ( ( X0 @ X1 )
= $true ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ( ( ( in @ sK614
@ ( dsetconstr @ sK615
@ ^ [Y0: $i] : ( sK613 @ Y0 ) ) )
= $true )
& ( ( sK613 @ sK614 )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK613,sK614,sK615])],[f2343,f2344]) ).
thf(f2344,plain,
( ? [X3: $i > $o,X4: $i,X5: $i] :
( ( ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) )
= $true )
& ( ( X3 @ X4 )
!= $true ) )
=> ( ( ( in @ sK614
@ ( dsetconstr @ sK615
@ ^ [Y0: $i] : ( sK613 @ Y0 ) ) )
= $true )
& ( ( sK613 @ sK614 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2343,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true )
| ( ( X0 @ X1 )
= $true ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ? [X3: $i > $o,X4: $i,X5: $i] :
( ( ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) )
= $true )
& ( ( X3 @ X4 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f2342]) ).
thf(f2342,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true )
| ( ( X0 @ X1 )
= $true ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
= $true )
& ( ( X0 @ X1 )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1204]) ).
thf(f1204,plain,
( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true )
| ( ( X0 @ X1 )
= $true ) )
<=> ( dsetconstrER = $true ) ),
inference(ennf_transformation,[status(thm)],[f457]) ).
thf(f457,plain,
( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
= $true )
=> ( ( X0 @ X1 )
= $true ) )
<=> ( dsetconstrER = $true ) ),
inference(fool_elimination,[status(thm)],[f456]) ).
thf(f456,plain,
( dsetconstrER
= ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X0 @ X3 ) ) )
=> ( X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f16]) ).
thf(f16,axiom,
( dsetconstrER
= ( ! [X0: $i > $o,X1: $i,X3: $i] :
( ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) )
=> ( X0 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2THF_16347.p',dsetconstrER) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU769^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.35 % Computer : n006.cluster.edu
% 0.16/0.35 % Model : x86_64 x86_64
% 0.16/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35 % Memory : 8042.1875MB
% 0.16/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Tue Feb 24 12:33:15 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.20/0.45 ---- Original DTF file ---
% 0.20/0.45 thf(spec,logic,$$dhol).
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 % File : SEU769^1 : TPTP v9.2.1. Released v3.7.0.
% 0.20/0.45 % Domain : Set Theory
% 0.20/0.45 % Problem : Binary Relations on a Set
% 0.20/0.45 % Version : Especial.
% 0.20/0.45 % English : (! A:i.! R:i.in R (breln1Set A) -> breln1 A R)
% 0.20/0.45
% 0.20/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.20/0.45 % Source : [Bro08]
% 0.20/0.45 % Names : ZFC271g [Bro08]
% 0.20/0.45
% 0.20/0.45 % Status : Theorem
% 0.20/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0, 0.14 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, 0.40 v6.2.0, 0.43 v6.1.0, 0.71 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.20/0.45 % Syntax : Number of formulae : 617 ( 291 unt; 325 typ; 291 def)
% 0.20/0.45 % Number of atoms : 1736 ( 420 equ; 0 cnn)
% 0.20/0.45 % Maximal formula atoms : 287 ( 5 avg)
% 0.20/0.45 % Number of connectives : 3715 ( 73 ~; 12 |; 44 &;2694 @)
% 0.20/0.45 % ( 14 <=>; 878 =>; 0 <=; 0 <~>)
% 0.20/0.45 % Maximal formula depth : 291 ( 2 avg)
% 0.20/0.45 % Number of types : 2 ( 0 usr)
% 0.20/0.45 % Number of type conns : 136 ( 136 >; 0 *; 0 +; 0 <<)
% 0.20/0.45 % Number of symbols : 328 ( 325 usr; 289 con; 0-4 aty)
% 0.20/0.45 % Number of variables : 995 ( 78 ^; 882 !; 35 ?; 995 :)
% 0.20/0.45 % SPC : TH0_THM_EQU_NAR
% 0.20/0.45
% 0.20/0.45 % Comments : http://mathgate.info/detsetitem.php?id=375
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 thf(in_type,type,
% 0.20/0.45 in: $i > $i > $o ).
% 0.20/0.45
% 0.20/0.45 thf(exu_type,type,
% 0.20/0.45 exu: ( $i > $o ) > $o ).
% 0.20/0.45
% 0.20/0.45 thf(exu,definition,
% 0.20/0.45 ( exu
% 0.20/0.45 = ( ^ [Xphi: $i > $o] :
% 0.20/0.45 ? [Xx: $i] :
% 0.20/0.45 ( ( Xphi @ Xx )
% 0.20/0.45 & ! [Xy: $i] :
% 0.20/0.45 ( ( Xphi @ Xy )
% 0.20/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setextAx_type,type,
% 0.20/0.45 setextAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setextAx,definition,
% 0.20/0.45 ( setextAx
% 0.20/0.45 = ( ! [A: $i,B: $i] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 <=> ( in @ Xx @ B ) )
% 0.20/0.45 => ( A = B ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptyset_type,type,
% 0.20/0.45 emptyset: $i ).
% 0.20/0.45
% 0.20/0.45 thf(emptysetAx_type,type,
% 0.20/0.45 emptysetAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptysetAx,definition,
% 0.20/0.45 ( emptysetAx
% 0.20/0.45 = ( ! [Xx: $i] :
% 0.20/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoin_type,type,
% 0.20/0.45 setadjoin: $i > $i > $i ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinAx_type,type,
% 0.20/0.45 setadjoinAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinAx,definition,
% 0.20/0.45 ( setadjoinAx
% 0.20/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.45 <=> ( ( Xy = Xx )
% 0.20/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(powerset_type,type,
% 0.20/0.45 powerset: $i > $i ).
% 0.20/0.45
% 0.20/0.45 thf(powersetAx_type,type,
% 0.20/0.45 powersetAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(powersetAx,definition,
% 0.20/0.45 ( powersetAx
% 0.20/0.45 = ( ! [A: $i,B: $i] :
% 0.20/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.45 <=> ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setunion_type,type,
% 0.20/0.45 setunion: $i > $i ).
% 0.20/0.45
% 0.20/0.45 thf(setunionAx_type,type,
% 0.20/0.45 setunionAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setunionAx,definition,
% 0.20/0.45 ( setunionAx
% 0.20/0.45 = ( ! [A: $i,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.45 <=> ? [B: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(omega_type,type,
% 0.20/0.45 omega: $i ).
% 0.20/0.45
% 0.20/0.45 thf(omega0Ax_type,type,
% 0.20/0.45 omega0Ax: $o ).
% 0.20/0.45
% 0.20/0.45 thf(omega0Ax,definition,
% 0.20/0.45 ( omega0Ax
% 0.20/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(omegaSAx_type,type,
% 0.20/0.45 omegaSAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(omegaSAx,definition,
% 0.20/0.45 ( omegaSAx
% 0.20/0.45 = ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ omega )
% 0.20/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(omegaIndAx_type,type,
% 0.20/0.45 omegaIndAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(omegaIndAx,definition,
% 0.20/0.45 ( omegaIndAx
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ( ( in @ emptyset @ A )
% 0.20/0.45 & ! [Xx: $i] :
% 0.20/0.45 ( ( ( in @ Xx @ omega )
% 0.20/0.45 & ( in @ Xx @ A ) )
% 0.20/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.20/0.45 => ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ omega )
% 0.20/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(replAx_type,type,
% 0.20/0.45 replAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(replAx,definition,
% 0.20/0.45 ( replAx
% 0.20/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( exu
% 0.20/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.20/0.45 => ? [B: $i] :
% 0.20/0.45 ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 <=> ? [Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ A )
% 0.20/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(foundationAx_type,type,
% 0.20/0.45 foundationAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(foundationAx,definition,
% 0.20/0.45 ( foundationAx
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.45 => ? [B: $i] :
% 0.20/0.45 ( ( in @ B @ A )
% 0.20/0.45 & ~ ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(wellorderingAx_type,type,
% 0.20/0.45 wellorderingAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(wellorderingAx,definition,
% 0.20/0.45 ( wellorderingAx
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ? [B: $i] :
% 0.20/0.45 ( ! [C: $i] :
% 0.20/0.45 ( ( in @ C @ B )
% 0.20/0.45 => ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ C )
% 0.20/0.45 => ( in @ Xx @ A ) ) )
% 0.20/0.45 & ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( ( in @ Xx @ A )
% 0.20/0.45 & ( in @ Xy @ A ) )
% 0.20/0.45 => ( ! [C: $i] :
% 0.20/0.45 ( ( in @ C @ B )
% 0.20/0.45 => ( ( in @ Xx @ C )
% 0.20/0.45 <=> ( in @ Xy @ C ) ) )
% 0.20/0.45 => ( Xx = Xy ) ) )
% 0.20/0.45 & ! [C: $i,D: $i] :
% 0.20/0.45 ( ( ( in @ C @ B )
% 0.20/0.45 & ( in @ D @ B ) )
% 0.20/0.45 => ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ C )
% 0.20/0.45 => ( in @ Xx @ D ) )
% 0.20/0.45 | ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ D )
% 0.20/0.45 => ( in @ Xx @ C ) ) ) )
% 0.20/0.45 & ! [C: $i] :
% 0.20/0.45 ( ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ C )
% 0.20/0.45 => ( in @ Xx @ A ) )
% 0.20/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.20/0.45 => ? [D: $i,Xx: $i] :
% 0.20/0.45 ( ( in @ D @ B )
% 0.20/0.45 & ( in @ Xx @ C )
% 0.20/0.45 & ~ ? [Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ D )
% 0.20/0.45 & ( in @ Xy @ C ) )
% 0.20/0.45 & ! [E: $i] :
% 0.20/0.45 ( ( in @ E @ B )
% 0.29/0.45 => ( ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ E )
% 0.29/0.45 => ( in @ Xy @ D ) )
% 0.29/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(descr_type,type,
% 0.29/0.45 descr: ( $i > $o ) > $i ).
% 0.29/0.45
% 0.29/0.45 thf(descrp_type,type,
% 0.29/0.45 descrp: $o ).
% 0.29/0.45
% 0.29/0.45 thf(descrp,definition,
% 0.29/0.45 ( descrp
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ( Xphi
% 0.29/0.45 @ ( descr
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstr_type,type,
% 0.29/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstrI_type,type,
% 0.29/0.45 dsetconstrI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstrI,definition,
% 0.29/0.45 ( dsetconstrI
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 => ( in @ Xx
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstrEL_type,type,
% 0.29/0.45 dsetconstrEL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstrEL,definition,
% 0.29/0.45 ( dsetconstrEL
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstrER_type,type,
% 0.29/0.45 dsetconstrER: $o ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstrER,definition,
% 0.29/0.45 ( dsetconstrER
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuE1_type,type,
% 0.29/0.45 exuE1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuE1,definition,
% 0.29/0.45 ( exuE1
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 & ! [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(prop2set_type,type,
% 0.29/0.45 prop2set: $o > $i ).
% 0.29/0.45
% 0.29/0.45 thf(prop2setE_type,type,
% 0.29/0.45 prop2setE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(prop2setE,definition,
% 0.29/0.45 ( prop2setE
% 0.29/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.29/0.45 => Xphi ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetE_type,type,
% 0.29/0.45 emptysetE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetE,definition,
% 0.29/0.45 ( emptysetE
% 0.29/0.45 = ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ emptyset )
% 0.29/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetimpfalse_type,type,
% 0.29/0.45 emptysetimpfalse: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetimpfalse,definition,
% 0.29/0.45 ( emptysetimpfalse
% 0.29/0.45 = ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ emptyset )
% 0.29/0.45 => $false ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notinemptyset_type,type,
% 0.29/0.45 notinemptyset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notinemptyset,definition,
% 0.29/0.45 ( notinemptyset
% 0.29/0.45 = ( ! [Xx: $i] :
% 0.29/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuE3e_type,type,
% 0.29/0.45 exuE3e: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuE3e,definition,
% 0.29/0.45 ( exuE3e
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setext_type,type,
% 0.29/0.45 setext: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setext,definition,
% 0.29/0.45 ( setext
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ B ) )
% 0.29/0.45 => ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ A ) )
% 0.29/0.45 => ( A = B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyI_type,type,
% 0.29/0.45 emptyI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyI,definition,
% 0.29/0.45 ( emptyI
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ~ ( in @ Xx @ A )
% 0.29/0.45 => ( A = emptyset ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(noeltsimpempty_type,type,
% 0.29/0.45 noeltsimpempty: $o ).
% 0.29/0.45
% 0.29/0.45 thf(noeltsimpempty,definition,
% 0.29/0.45 ( noeltsimpempty
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ~ ( in @ Xx @ A )
% 0.29/0.45 => ( A = emptyset ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setbeta_type,type,
% 0.29/0.45 setbeta: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setbeta,definition,
% 0.29/0.45 ( setbeta
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonempty_type,type,
% 0.29/0.45 nonempty: $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonempty,definition,
% 0.29/0.45 ( nonempty
% 0.29/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyE1_type,type,
% 0.29/0.45 nonemptyE1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyE1,definition,
% 0.29/0.45 ( nonemptyE1
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ( nonempty @ A )
% 0.29/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyI_type,type,
% 0.29/0.45 nonemptyI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyI,definition,
% 0.29/0.45 ( nonemptyI
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 => ( nonempty
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyI1_type,type,
% 0.29/0.45 nonemptyI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyI1,definition,
% 0.29/0.45 ( nonemptyI1
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinIL_type,type,
% 0.29/0.45 setadjoinIL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinIL,definition,
% 0.29/0.45 ( setadjoinIL
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyinunitempty_type,type,
% 0.29/0.45 emptyinunitempty: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyinunitempty,definition,
% 0.29/0.45 ( emptyinunitempty
% 0.29/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinIR_type,type,
% 0.29/0.45 setadjoinIR: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinIR,definition,
% 0.29/0.45 ( setadjoinIR
% 0.29/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinE_type,type,
% 0.29/0.45 setadjoinE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinE,definition,
% 0.29/0.45 ( setadjoinE
% 0.29/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.45 => ! [Xphi: $o] :
% 0.29/0.45 ( ( ( Xy = Xx )
% 0.29/0.45 => Xphi )
% 0.29/0.45 => ( ( ( in @ Xy @ A )
% 0.29/0.45 => Xphi )
% 0.29/0.45 => Xphi ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinOr_type,type,
% 0.29/0.45 setadjoinOr: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinOr,definition,
% 0.29/0.45 ( setadjoinOr
% 0.29/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.45 => ( ( Xy = Xx )
% 0.29/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setoftrueEq_type,type,
% 0.29/0.45 setoftrueEq: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setoftrueEq,definition,
% 0.29/0.45 ( setoftrueEq
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xx: $i] : $true )
% 0.29/0.45 = A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powersetI_type,type,
% 0.29/0.45 powersetI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powersetI,definition,
% 0.29/0.45 ( powersetI
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ A ) )
% 0.29/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyinPowerset_type,type,
% 0.29/0.45 emptyinPowerset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyinPowerset,definition,
% 0.29/0.45 ( emptyinPowerset
% 0.29/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyInPowerset_type,type,
% 0.29/0.45 emptyInPowerset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyInPowerset,definition,
% 0.29/0.45 ( emptyInPowerset
% 0.29/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powersetE_type,type,
% 0.29/0.45 powersetE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powersetE,definition,
% 0.29/0.45 ( powersetE
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.45 => ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setunionI_type,type,
% 0.29/0.45 setunionI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setunionI,definition,
% 0.29/0.45 ( setunionI
% 0.29/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( ( in @ B @ A )
% 0.29/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setunionE_type,type,
% 0.29/0.45 setunionE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setunionE,definition,
% 0.29/0.45 ( setunionE
% 0.29/0.45 = ( ! [A: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.45 => ! [Xphi: $o] :
% 0.29/0.45 ( ! [B: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( ( in @ B @ A )
% 0.29/0.45 => Xphi ) )
% 0.29/0.45 => Xphi ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subPowSU_type,type,
% 0.29/0.45 subPowSU: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subPowSU,definition,
% 0.29/0.45 ( subPowSU
% 0.29/0.45 = ( ! [A: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuE2_type,type,
% 0.29/0.45 exuE2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuE2,definition,
% 0.29/0.45 ( exuE2
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ! [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xy )
% 0.29/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyImpWitness_type,type,
% 0.29/0.45 nonemptyImpWitness: $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyImpWitness,definition,
% 0.29/0.45 ( nonemptyImpWitness
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ( nonempty @ A )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & $true ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(uniqinunit_type,type,
% 0.29/0.45 uniqinunit: $o ).
% 0.29/0.45
% 0.29/0.45 thf(uniqinunit,definition,
% 0.29/0.45 ( uniqinunit
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notinsingleton_type,type,
% 0.29/0.45 notinsingleton: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notinsingleton,definition,
% 0.29/0.45 ( notinsingleton
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx != Xy )
% 0.29/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(eqinunit_type,type,
% 0.29/0.45 eqinunit: $o ).
% 0.29/0.45
% 0.29/0.45 thf(eqinunit,definition,
% 0.29/0.45 ( eqinunit
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(singletonsswitch_type,type,
% 0.29/0.45 singletonsswitch: $o ).
% 0.29/0.45
% 0.29/0.45 thf(singletonsswitch,definition,
% 0.29/0.45 ( singletonsswitch
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetE_type,type,
% 0.29/0.45 upairsetE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetE,definition,
% 0.29/0.45 ( upairsetE
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.45 => ( ( Xz = Xx )
% 0.29/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetIL_type,type,
% 0.29/0.45 upairsetIL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetIL,definition,
% 0.29/0.45 ( upairsetIL
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetIR_type,type,
% 0.29/0.45 upairsetIR: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetIR,definition,
% 0.29/0.45 ( upairsetIR
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyE1_type,type,
% 0.29/0.45 emptyE1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyE1,definition,
% 0.29/0.45 ( emptyE1
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( Xphi @ Xx ) )
% 0.29/0.45 => ( ( ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 = emptyset )
% 0.29/0.45 => $false ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(vacuousDall_type,type,
% 0.29/0.45 vacuousDall: $o ).
% 0.29/0.45
% 0.29/0.45 thf(vacuousDall,definition,
% 0.29/0.45 ( vacuousDall
% 0.29/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ emptyset )
% 0.29/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan1_type,type,
% 0.29/0.45 quantDeMorgan1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan1,definition,
% 0.29/0.45 ( quantDeMorgan1
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ~ ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan2_type,type,
% 0.29/0.45 quantDeMorgan2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan2,definition,
% 0.29/0.45 ( quantDeMorgan2
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ~ ( Xphi @ Xx ) )
% 0.29/0.45 => ~ ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan3_type,type,
% 0.29/0.45 quantDeMorgan3: $o ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan3,definition,
% 0.29/0.45 ( quantDeMorgan3
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ~ ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( Xphi @ Xx ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan4_type,type,
% 0.29/0.45 quantDeMorgan4: $o ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan4,definition,
% 0.29/0.45 ( quantDeMorgan4
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ~ ( Xphi @ Xx ) )
% 0.29/0.45 => ~ ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(prop2setI_type,type,
% 0.29/0.45 prop2setI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(prop2setI,definition,
% 0.29/0.45 ( prop2setI
% 0.29/0.45 = ( ! [Xphi: $o] :
% 0.29/0.45 ( Xphi
% 0.29/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(set2prop_type,type,
% 0.29/0.45 set2prop: $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(prop2set2propI_type,type,
% 0.29/0.45 prop2set2propI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(prop2set2propI,definition,
% 0.29/0.45 ( prop2set2propI
% 0.29/0.45 = ( ! [Xphi: $o] :
% 0.29/0.45 ( Xphi
% 0.29/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notdexE_type,type,
% 0.29/0.45 notdexE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notdexE,definition,
% 0.29/0.45 ( notdexE
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ~ ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( Xphi @ Xx ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notdallE_type,type,
% 0.29/0.45 notdallE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notdallE,definition,
% 0.29/0.45 ( notdallE
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ~ ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuI1_type,type,
% 0.29/0.45 exuI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuI1,definition,
% 0.29/0.45 ( exuI1
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 & ! [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) )
% 0.29/0.45 => ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuI3_type,type,
% 0.29/0.45 exuI3: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuI3,definition,
% 0.29/0.45 ( exuI3
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.29/0.45 => ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 => ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) )
% 0.29/0.45 => ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuI2_type,type,
% 0.29/0.45 exuI2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuI2,definition,
% 0.29/0.45 ( exuI2
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] :
% 0.29/0.45 ! [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xy )
% 0.29/0.45 <=> ( Xy = Xx ) )
% 0.29/0.45 => ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(inCongP_type,type,
% 0.29/0.45 inCongP: $o ).
% 0.29/0.45
% 0.29/0.45 thf(inCongP,definition,
% 0.29/0.45 ( inCongP
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(in__Cong_type,type,
% 0.29/0.45 in__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(in__Cong,definition,
% 0.29/0.45 ( in__Cong
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( ( in @ Xx @ A )
% 0.29/0.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuE3u_type,type,
% 0.29/0.45 exuE3u: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuE3u,definition,
% 0.29/0.45 ( exuE3u
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 => ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exu__Cong_type,type,
% 0.29/0.45 exu__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exu__Cong,definition,
% 0.29/0.45 ( exu__Cong
% 0.29/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.45 => ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 <=> ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyset__Cong_type,type,
% 0.29/0.45 emptyset__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyset__Cong,definition,
% 0.29/0.45 ( emptyset__Cong
% 0.29/0.45 = ( emptyset = emptyset ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoin__Cong_type,type,
% 0.29/0.45 setadjoin__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoin__Cong,definition,
% 0.29/0.45 ( setadjoin__Cong
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ! [Xz: $i,Xu: $i] :
% 0.29/0.45 ( ( Xz = Xu )
% 0.29/0.45 => ( ( setadjoin @ Xx @ Xz )
% 0.29/0.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powerset__Cong_type,type,
% 0.29/0.45 powerset__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powerset__Cong,definition,
% 0.29/0.45 ( powerset__Cong
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ( powerset @ A )
% 0.29/0.45 = ( powerset @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setunion__Cong_type,type,
% 0.29/0.45 setunion__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setunion__Cong,definition,
% 0.29/0.45 ( setunion__Cong
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ( setunion @ A )
% 0.29/0.45 = ( setunion @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(omega__Cong_type,type,
% 0.29/0.45 omega__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(omega__Cong,definition,
% 0.29/0.45 ( omega__Cong
% 0.29/0.45 = ( omega = omega ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuEu_type,type,
% 0.29/0.45 exuEu: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuEu,definition,
% 0.29/0.45 ( exuEu
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 => ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(descr__Cong_type,type,
% 0.29/0.45 descr__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(descr__Cong,definition,
% 0.29/0.45 ( descr__Cong
% 0.29/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.45 => ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.29/0.45 => ( ( descr
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 = ( descr
% 0.29/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstr__Cong_type,type,
% 0.29/0.45 dsetconstr__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstr__Cong,definition,
% 0.29/0.45 ( dsetconstr__Cong
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( ( Xx = Xy )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.29/0.45 => ( ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 = ( dsetconstr @ B
% 0.29/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subset_type,type,
% 0.29/0.45 subset: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(disjoint_type,type,
% 0.29/0.45 disjoint: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(setsmeet_type,type,
% 0.29/0.45 setsmeet: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetI1_type,type,
% 0.29/0.45 subsetI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetI1,definition,
% 0.29/0.45 ( subsetI1
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ B ) )
% 0.29/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(eqimpsubset2_type,type,
% 0.29/0.45 eqimpsubset2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(eqimpsubset2,definition,
% 0.29/0.45 ( eqimpsubset2
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(eqimpsubset1_type,type,
% 0.29/0.45 eqimpsubset1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(eqimpsubset1,definition,
% 0.29/0.45 ( eqimpsubset1
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subsetI2_type,type,
% 0.29/0.45 subsetI2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetI2,definition,
% 0.29/0.45 ( subsetI2
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ B ) )
% 0.29/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetsubset_type,type,
% 0.29/0.45 emptysetsubset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetsubset,definition,
% 0.29/0.45 ( emptysetsubset
% 0.29/0.45 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subsetE_type,type,
% 0.29/0.45 subsetE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetE,definition,
% 0.29/0.45 ( subsetE
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subsetE2_type,type,
% 0.29/0.45 subsetE2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetE2,definition,
% 0.29/0.45 ( subsetE2
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( ~ ( in @ Xx @ B )
% 0.29/0.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notsubsetI_type,type,
% 0.29/0.45 notsubsetI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notsubsetI,definition,
% 0.29/0.45 ( notsubsetI
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ~ ( in @ Xx @ B )
% 0.29/0.45 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notequalI1_type,type,
% 0.29/0.45 notequalI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notequalI1,definition,
% 0.29/0.45 ( notequalI1
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ~ ( subset @ A @ B )
% 0.29/0.45 => ( A != B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notequalI2_type,type,
% 0.29/0.45 notequalI2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notequalI2,definition,
% 0.29/0.45 ( notequalI2
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ~ ( in @ Xx @ B )
% 0.29/0.45 => ( A != B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subsetRefl_type,type,
% 0.29/0.45 subsetRefl: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetRefl,definition,
% 0.29/0.45 ( subsetRefl
% 0.29/0.45 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subsetTrans_type,type,
% 0.29/0.45 subsetTrans: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetTrans,definition,
% 0.29/0.45 ( subsetTrans
% 0.29/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( ( subset @ B @ C )
% 0.29/0.45 => ( subset @ A @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinSub_type,type,
% 0.29/0.45 setadjoinSub: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinSub,definition,
% 0.29/0.45 ( setadjoinSub
% 0.29/0.45 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinSub2_type,type,
% 0.29/0.45 setadjoinSub2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinSub2,definition,
% 0.29/0.45 ( setadjoinSub2
% 0.29/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subset2powerset_type,type,
% 0.29/0.45 subset2powerset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subset2powerset,definition,
% 0.29/0.45 ( subset2powerset
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setextsub_type,type,
% 0.29/0.45 setextsub: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setextsub,definition,
% 0.29/0.45 ( setextsub
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( ( subset @ B @ A )
% 0.29/0.45 => ( A = B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subsetemptysetimpeq_type,type,
% 0.29/0.45 subsetemptysetimpeq: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetemptysetimpeq,definition,
% 0.29/0.45 ( subsetemptysetimpeq
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ( subset @ A @ emptyset )
% 0.29/0.45 => ( A = emptyset ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powersetI1_type,type,
% 0.29/0.45 powersetI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powersetI1,definition,
% 0.29/0.45 ( powersetI1
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( subset @ B @ A )
% 0.29/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powersetE1_type,type,
% 0.29/0.45 powersetE1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powersetE1,definition,
% 0.29/0.45 ( powersetE1
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(inPowerset_type,type,
% 0.29/0.45 inPowerset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(inPowerset,definition,
% 0.29/0.45 ( inPowerset
% 0.29/0.45 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powersetsubset_type,type,
% 0.29/0.45 powersetsubset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powersetsubset,definition,
% 0.29/0.45 ( powersetsubset
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sepInPowerset_type,type,
% 0.29/0.45 sepInPowerset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(sepInPowerset,definition,
% 0.29/0.45 ( sepInPowerset
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( in
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 @ ( powerset @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(sepSubset_type,type,
% 0.29/0.45 sepSubset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(sepSubset,definition,
% 0.29/0.45 ( sepSubset
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( subset
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binunion_type,type,
% 0.29/0.45 binunion: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(binunionIL_type,type,
% 0.29/0.45 binunionIL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binunionIL,definition,
% 0.29/0.45 ( binunionIL
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairset2IR_type,type,
% 0.29/0.45 upairset2IR: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairset2IR,definition,
% 0.29/0.45 ( upairset2IR
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binunionIR_type,type,
% 0.29/0.45 binunionIR: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binunionIR,definition,
% 0.29/0.45 ( binunionIR
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binunionEcases_type,type,
% 0.29/0.45 binunionEcases: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binunionEcases,definition,
% 0.29/0.45 ( binunionEcases
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.29/0.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.29/0.45 => ( ( ( in @ Xx @ A )
% 0.29/0.45 => Xphi )
% 0.29/0.45 => ( ( ( in @ Xx @ B )
% 0.29/0.45 => Xphi )
% 0.29/0.45 => Xphi ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binunionE_type,type,
% 0.29/0.45 binunionE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binunionE,definition,
% 0.29/0.45 ( binunionE
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.29/0.45 => ( ( in @ Xx @ A )
% 0.29/0.45 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binunionLsub_type,type,
% 0.29/0.45 binunionLsub: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binunionLsub,definition,
% 0.29/0.45 ( binunionLsub
% 0.29/0.45 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binunionRsub_type,type,
% 0.29/0.45 binunionRsub: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binunionRsub,definition,
% 0.29/0.45 ( binunionRsub
% 0.29/0.45 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersect_type,type,
% 0.29/0.45 binintersect: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectI_type,type,
% 0.29/0.45 binintersectI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectI,definition,
% 0.29/0.45 ( binintersectI
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset5_type,type,
% 0.29/0.45 binintersectSubset5: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset5,definition,
% 0.29/0.45 ( binintersectSubset5
% 0.29/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.45 ( ( subset @ C @ A )
% 0.29/0.45 => ( ( subset @ C @ B )
% 0.29/0.45 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectEL_type,type,
% 0.29/0.45 binintersectEL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectEL,definition,
% 0.29/0.45 ( binintersectEL
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.29/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectLsub_type,type,
% 0.29/0.45 binintersectLsub: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectLsub,definition,
% 0.29/0.45 ( binintersectLsub
% 0.29/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset2_type,type,
% 0.29/0.45 binintersectSubset2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset2,definition,
% 0.29/0.45 ( binintersectSubset2
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( ( binintersect @ A @ B )
% 0.29/0.45 = A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset3_type,type,
% 0.29/0.45 binintersectSubset3: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset3,definition,
% 0.29/0.45 ( binintersectSubset3
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( ( binintersect @ A @ B )
% 0.29/0.45 = B )
% 0.29/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectER_type,type,
% 0.29/0.45 binintersectER: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectER,definition,
% 0.29/0.45 ( binintersectER
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.29/0.45 => ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(disjointsetsI1_type,type,
% 0.29/0.45 disjointsetsI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(disjointsetsI1,definition,
% 0.29/0.45 ( disjointsetsI1
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ~ ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( in @ Xx @ B ) )
% 0.29/0.45 => ( ( binintersect @ A @ B )
% 0.29/0.45 = emptyset ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectRsub_type,type,
% 0.29/0.45 binintersectRsub: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectRsub,definition,
% 0.29/0.45 ( binintersectRsub
% 0.29/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset4_type,type,
% 0.29/0.45 binintersectSubset4: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset4,definition,
% 0.29/0.45 ( binintersectSubset4
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( subset @ B @ A )
% 0.29/0.45 => ( ( binintersect @ A @ B )
% 0.29/0.45 = B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset1_type,type,
% 0.29/0.45 binintersectSubset1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(binintersectSubset1,definition,
% 0.29/0.45 ( binintersectSubset1
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( ( binintersect @ A @ B )
% 0.29/0.45 = A )
% 0.29/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(bs114d_type,type,
% 0.29/0.45 bs114d: $o ).
% 0.29/0.45
% 0.29/0.45 thf(bs114d,definition,
% 0.29/0.45 ( bs114d
% 0.29/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.45 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.29/0.45 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(regular_type,type,
% 0.29/0.45 regular: $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminus_type,type,
% 0.29/0.45 setminus: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(setminusI_type,type,
% 0.29/0.45 setminusI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusI,definition,
% 0.29/0.45 ( setminusI
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ~ ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusEL_type,type,
% 0.29/0.45 setminusEL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusEL,definition,
% 0.29/0.45 ( setminusEL
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusER_type,type,
% 0.29/0.45 setminusER: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusER,definition,
% 0.29/0.45 ( setminusER
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.45 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusSubset2_type,type,
% 0.29/0.45 setminusSubset2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusSubset2,definition,
% 0.29/0.45 ( setminusSubset2
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( subset @ A @ B )
% 0.29/0.45 => ( ( setminus @ A @ B )
% 0.29/0.45 = emptyset ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusERneg_type,type,
% 0.29/0.45 setminusERneg: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusERneg,definition,
% 0.29/0.45 ( setminusERneg
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.45 => ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusELneg_type,type,
% 0.29/0.45 setminusELneg: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusELneg,definition,
% 0.29/0.45 ( setminusELneg
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.45 => ( ~ ( in @ Xx @ B )
% 0.29/0.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusILneg_type,type,
% 0.29/0.45 setminusILneg: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusILneg,definition,
% 0.29/0.45 ( setminusILneg
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ~ ( in @ Xx @ A )
% 0.29/0.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusIRneg_type,type,
% 0.29/0.45 setminusIRneg: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusIRneg,definition,
% 0.29/0.45 ( setminusIRneg
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusLsub_type,type,
% 0.29/0.45 setminusLsub: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusLsub,definition,
% 0.29/0.45 ( setminusLsub
% 0.29/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setminusSubset1_type,type,
% 0.29/0.45 setminusSubset1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setminusSubset1,definition,
% 0.29/0.45 ( setminusSubset1
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( ( setminus @ A @ B )
% 0.29/0.45 = emptyset )
% 0.29/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(symdiff_type,type,
% 0.29/0.45 symdiff: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffE_type,type,
% 0.29/0.45 symdiffE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffE,definition,
% 0.29/0.45 ( symdiffE
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.29/0.45 => ! [Xphi: $o] :
% 0.29/0.45 ( ( ( in @ Xx @ A )
% 0.29/0.45 => ( ~ ( in @ Xx @ B )
% 0.29/0.45 => Xphi ) )
% 0.29/0.45 => ( ( ~ ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ B )
% 0.29/0.45 => Xphi ) )
% 0.29/0.45 => Xphi ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffI1_type,type,
% 0.29/0.45 symdiffI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffI1,definition,
% 0.29/0.45 ( symdiffI1
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ~ ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffI2_type,type,
% 0.29/0.45 symdiffI2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffI2,definition,
% 0.29/0.45 ( symdiffI2
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ~ ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffIneg1_type,type,
% 0.29/0.45 symdiffIneg1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffIneg1,definition,
% 0.29/0.45 ( symdiffIneg1
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx @ B )
% 0.29/0.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffIneg2_type,type,
% 0.29/0.45 symdiffIneg2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(symdiffIneg2,definition,
% 0.29/0.45 ( symdiffIneg2
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ~ ( in @ Xx @ A )
% 0.29/0.45 => ( ~ ( in @ Xx @ B )
% 0.29/0.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(iskpair_type,type,
% 0.29/0.45 iskpair: $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(secondinupair_type,type,
% 0.29/0.45 secondinupair: $o ).
% 0.29/0.45
% 0.29/0.45 thf(secondinupair,definition,
% 0.29/0.45 ( secondinupair
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setukpairIL_type,type,
% 0.29/0.45 setukpairIL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setukpairIL,definition,
% 0.29/0.45 ( setukpairIL
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setukpairIR_type,type,
% 0.29/0.45 setukpairIR: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setukpairIR,definition,
% 0.29/0.45 ( setukpairIR
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(kpairiskpair_type,type,
% 0.29/0.45 kpairiskpair: $o ).
% 0.29/0.45
% 0.29/0.45 thf(kpairiskpair,definition,
% 0.29/0.45 ( kpairiskpair
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(kpair_type,type,
% 0.29/0.45 kpair: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(kpairp_type,type,
% 0.29/0.45 kpairp: $o ).
% 0.29/0.45
% 0.29/0.45 thf(kpairp,definition,
% 0.29/0.45 ( kpairp
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(cartprod_type,type,
% 0.29/0.45 cartprod: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(singletonsubset_type,type,
% 0.29/0.45 singletonsubset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(singletonsubset,definition,
% 0.29/0.45 ( singletonsubset
% 0.29/0.45 = ( ! [A: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(singletoninpowerset_type,type,
% 0.29/0.45 singletoninpowerset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(singletoninpowerset,definition,
% 0.29/0.45 ( singletoninpowerset
% 0.29/0.45 = ( ! [A: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(singletoninpowunion_type,type,
% 0.29/0.45 singletoninpowunion: $o ).
% 0.29/0.45
% 0.29/0.45 thf(singletoninpowunion,definition,
% 0.29/0.45 ( singletoninpowunion
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairset2E_type,type,
% 0.29/0.45 upairset2E: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairset2E,definition,
% 0.29/0.45 ( upairset2E
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.45 => ( ( Xz = Xx )
% 0.29/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairsubunion_type,type,
% 0.29/0.45 upairsubunion: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairsubunion,definition,
% 0.29/0.45 ( upairsubunion
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairinpowunion_type,type,
% 0.29/0.45 upairinpowunion: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairinpowunion,definition,
% 0.29/0.45 ( upairinpowunion
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(ubforcartprodlem1_type,type,
% 0.29/0.45 ubforcartprodlem1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(ubforcartprodlem1,definition,
% 0.29/0.45 ( ubforcartprodlem1
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(ubforcartprodlem2_type,type,
% 0.29/0.45 ubforcartprodlem2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(ubforcartprodlem2,definition,
% 0.29/0.45 ( ubforcartprodlem2
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(ubforcartprodlem3_type,type,
% 0.29/0.45 ubforcartprodlem3: $o ).
% 0.29/0.45
% 0.29/0.45 thf(ubforcartprodlem3,definition,
% 0.29/0.45 ( ubforcartprodlem3
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(cartprodpairin_type,type,
% 0.29/0.45 cartprodpairin: $o ).
% 0.29/0.45
% 0.29/0.45 thf(cartprodpairin,definition,
% 0.29/0.45 ( cartprodpairin
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(cartprodmempair1_type,type,
% 0.29/0.45 cartprodmempair1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodmempair1,definition,
% 0.29/0.46 ( cartprodmempair1
% 0.29/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.46 => ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ? [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 & ( Xu
% 0.29/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodmempair_type,type,
% 0.29/0.46 cartprodmempair: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodmempair,definition,
% 0.29/0.46 ( cartprodmempair
% 0.29/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunionE2_type,type,
% 0.29/0.46 setunionE2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunionE2,definition,
% 0.29/0.46 ( setunionE2
% 0.29/0.46 = ( ! [A: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.46 => ? [X: $i] :
% 0.29/0.46 ( ( in @ X @ A )
% 0.29/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunionsingleton1_type,type,
% 0.29/0.46 setunionsingleton1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunionsingleton1,definition,
% 0.29/0.46 ( setunionsingleton1
% 0.29/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunionsingleton2_type,type,
% 0.29/0.46 setunionsingleton2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunionsingleton2,definition,
% 0.29/0.46 ( setunionsingleton2
% 0.29/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunionsingleton_type,type,
% 0.29/0.46 setunionsingleton: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunionsingleton,definition,
% 0.29/0.46 ( setunionsingleton
% 0.29/0.46 = ( ! [Xx: $i] :
% 0.29/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.29/0.46 = Xx ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(singleton_type,type,
% 0.29/0.46 singleton: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(singletonprop_type,type,
% 0.29/0.46 singletonprop: $o ).
% 0.29/0.46
% 0.29/0.46 thf(singletonprop,definition,
% 0.29/0.46 ( singletonprop
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 => ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) )
% 0.29/0.46 => ( ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ( Xphi @ Xx ) )
% 0.29/0.46 => ( singleton
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ex1_type,type,
% 0.29/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.29/0.46
% 0.29/0.46 thf(ex1,definition,
% 0.29/0.46 ( ex1
% 0.29/0.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( singleton
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ex1E1_type,type,
% 0.29/0.46 ex1E1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ex1E1,definition,
% 0.29/0.46 ( ex1E1
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ( ex1 @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ex1I_type,type,
% 0.29/0.46 ex1I: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ex1I,definition,
% 0.29/0.46 ( ex1I
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 => ( ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xy = Xx ) ) )
% 0.29/0.46 => ( ex1 @ A
% 0.29/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ex1I2_type,type,
% 0.29/0.46 ex1I2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ex1I2,definition,
% 0.29/0.46 ( ex1I2
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 => ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) )
% 0.29/0.46 => ( ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ( Xphi @ Xx ) )
% 0.29/0.46 => ( ex1 @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(singletonsuniq_type,type,
% 0.29/0.46 singletonsuniq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(singletonsuniq,definition,
% 0.29/0.46 ( singletonsuniq
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.29/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(atmost1p_type,type,
% 0.29/0.46 atmost1p: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(atleast2p_type,type,
% 0.29/0.46 atleast2p: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(atmost2p_type,type,
% 0.29/0.46 atmost2p: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(upairsetp_type,type,
% 0.29/0.46 upairsetp: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjL1_type,type,
% 0.29/0.46 setukpairinjL1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjL1,definition,
% 0.29/0.46 ( setukpairinjL1
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.29/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(kfstsingleton_type,type,
% 0.29/0.46 kfstsingleton: $o ).
% 0.29/0.46
% 0.29/0.46 thf(kfstsingleton,definition,
% 0.29/0.46 ( kfstsingleton
% 0.29/0.46 = ( ! [Xu: $i] :
% 0.29/0.46 ( ( iskpair @ Xu )
% 0.29/0.46 => ( singleton
% 0.29/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.29/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(theprop_type,type,
% 0.29/0.46 theprop: $o ).
% 0.29/0.46
% 0.29/0.46 thf(theprop,definition,
% 0.29/0.46 ( theprop
% 0.29/0.46 = ( ! [X: $i] :
% 0.29/0.46 ( ( singleton @ X )
% 0.29/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(kfst_type,type,
% 0.29/0.46 kfst: $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(kfstpairEq_type,type,
% 0.29/0.46 kfstpairEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(kfstpairEq,definition,
% 0.29/0.46 ( kfstpairEq
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.29/0.46 = Xx ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodfstin_type,type,
% 0.29/0.46 cartprodfstin: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodfstin,definition,
% 0.29/0.46 ( cartprodfstin
% 0.29/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjL2_type,type,
% 0.29/0.46 setukpairinjL2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjL2,definition,
% 0.29/0.46 ( setukpairinjL2
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.29/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.29/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjL_type,type,
% 0.29/0.46 setukpairinjL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjL,definition,
% 0.29/0.46 ( setukpairinjL
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.29/0.46 = ( kpair @ Xz @ Xu ) )
% 0.29/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR11_type,type,
% 0.29/0.46 setukpairinjR11: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR11,definition,
% 0.29/0.46 ( setukpairinjR11
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx = Xy )
% 0.29/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR12_type,type,
% 0.29/0.46 setukpairinjR12: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR12,definition,
% 0.29/0.46 ( setukpairinjR12
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx = Xy )
% 0.29/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.29/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR1_type,type,
% 0.29/0.46 setukpairinjR1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR1,definition,
% 0.29/0.46 ( setukpairinjR1
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.29/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.29/0.46 => ( ( Xz = Xu )
% 0.29/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(upairequniteq_type,type,
% 0.29/0.46 upairequniteq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(upairequniteq,definition,
% 0.29/0.46 ( upairequniteq
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR2_type,type,
% 0.29/0.46 setukpairinjR2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR2,definition,
% 0.29/0.46 ( setukpairinjR2
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.29/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.29/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR_type,type,
% 0.29/0.46 setukpairinjR: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairinjR,definition,
% 0.29/0.46 ( setukpairinjR
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.29/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.29/0.46 = ( kpair @ Xz @ Xu ) )
% 0.29/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ksndsingleton_type,type,
% 0.29/0.46 ksndsingleton: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ksndsingleton,definition,
% 0.29/0.46 ( ksndsingleton
% 0.29/0.46 = ( ! [Xu: $i] :
% 0.29/0.46 ( ( iskpair @ Xu )
% 0.29/0.46 => ( singleton
% 0.29/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.29/0.46 @ ^ [Xx: $i] :
% 0.29/0.46 ( Xu
% 0.29/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ksnd_type,type,
% 0.29/0.46 ksnd: $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(ksndpairEq_type,type,
% 0.29/0.46 ksndpairEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ksndpairEq,definition,
% 0.29/0.46 ( ksndpairEq
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.29/0.46 = Xy ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(kpairsurjEq_type,type,
% 0.29/0.46 kpairsurjEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(kpairsurjEq,definition,
% 0.29/0.46 ( kpairsurjEq
% 0.29/0.46 = ( ! [Xu: $i] :
% 0.29/0.46 ( ( iskpair @ Xu )
% 0.29/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.29/0.46 = Xu ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodsndin_type,type,
% 0.29/0.46 cartprodsndin: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodsndin,definition,
% 0.29/0.46 ( cartprodsndin
% 0.29/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodpairmemEL_type,type,
% 0.29/0.46 cartprodpairmemEL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodpairmemEL,definition,
% 0.29/0.46 ( cartprodpairmemEL
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodpairmemER_type,type,
% 0.29/0.46 cartprodpairmemER: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodpairmemER,definition,
% 0.29/0.46 ( cartprodpairmemER
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.29/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodmempaircEq_type,type,
% 0.29/0.46 cartprodmempaircEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodmempaircEq,definition,
% 0.29/0.46 ( cartprodmempaircEq
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( kpair @ Xx @ Xy )
% 0.29/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodfstpairEq_type,type,
% 0.29/0.46 cartprodfstpairEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodfstpairEq,definition,
% 0.29/0.46 ( cartprodfstpairEq
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.29/0.46 = Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodsndpairEq_type,type,
% 0.29/0.46 cartprodsndpairEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodsndpairEq,definition,
% 0.29/0.46 ( cartprodsndpairEq
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.29/0.46 = Xy ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodpairsurjEq_type,type,
% 0.29/0.46 cartprodpairsurjEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(cartprodpairsurjEq,definition,
% 0.29/0.46 ( cartprodpairsurjEq
% 0.29/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.29/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.29/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.29/0.46 = Xu ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(breln_type,type,
% 0.29/0.46 breln: $i > $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(breln,definition,
% 0.29/0.46 ( breln
% 0.29/0.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstr_type,type,
% 0.29/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrI_type,type,
% 0.29/0.46 dpsetconstrI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrI,definition,
% 0.29/0.46 ( dpsetconstrI
% 0.29/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.29/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.46 @ ( dpsetconstr @ A @ B
% 0.29/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrSub_type,type,
% 0.29/0.46 dpsetconstrSub: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrSub,definition,
% 0.29/0.46 ( dpsetconstrSub
% 0.29/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.29/0.46 ( subset
% 0.29/0.46 @ ( dpsetconstr @ A @ B
% 0.29/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.29/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setOfPairsIsBReln_type,type,
% 0.29/0.46 setOfPairsIsBReln: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setOfPairsIsBReln,definition,
% 0.29/0.46 ( setOfPairsIsBReln
% 0.29/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.29/0.46 ( breln @ A @ B
% 0.29/0.46 @ ( dpsetconstr @ A @ B
% 0.29/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrERa_type,type,
% 0.29/0.46 dpsetconstrERa: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrERa,definition,
% 0.29/0.46 ( dpsetconstrERa
% 0.29/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.46 @ ( dpsetconstr @ A @ B
% 0.29/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.29/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrEL1_type,type,
% 0.29/0.46 dpsetconstrEL1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrEL1,definition,
% 0.29/0.46 ( dpsetconstrEL1
% 0.29/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.46 @ ( dpsetconstr @ A @ B
% 0.29/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrEL2_type,type,
% 0.29/0.46 dpsetconstrEL2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrEL2,definition,
% 0.29/0.46 ( dpsetconstrEL2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.46 @ ( dpsetconstr @ A @ B
% 0.29/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.29/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrER_type,type,
% 0.29/0.46 dpsetconstrER: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dpsetconstrER,definition,
% 0.29/0.46 ( dpsetconstrER
% 0.29/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.29/0.46 @ ( dpsetconstr @ A @ B
% 0.29/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.29/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(func_type,type,
% 0.29/0.46 func: $i > $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcSet_type,type,
% 0.29/0.46 funcSet: $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(funcImageSingleton_type,type,
% 0.29/0.46 funcImageSingleton: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcImageSingleton,definition,
% 0.29/0.46 ( funcImageSingleton
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( singleton
% 0.29/0.46 @ ( dsetconstr @ B
% 0.29/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(apProp_type,type,
% 0.29/0.46 apProp: $o ).
% 0.29/0.46
% 0.29/0.46 thf(apProp,definition,
% 0.29/0.46 ( apProp
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in
% 0.29/0.46 @ ( setunion
% 0.29/0.46 @ ( dsetconstr @ B
% 0.29/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.29/0.46 @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ap_type,type,
% 0.29/0.46 ap: $i > $i > $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(app_type,type,
% 0.29/0.46 app: $o ).
% 0.29/0.46
% 0.29/0.46 thf(app,definition,
% 0.29/0.46 ( app
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(infuncsetfunc_type,type,
% 0.29/0.46 infuncsetfunc: $o ).
% 0.29/0.46
% 0.29/0.46 thf(infuncsetfunc,definition,
% 0.29/0.46 ( infuncsetfunc
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ap2p_type,type,
% 0.29/0.46 ap2p: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ap2p,definition,
% 0.29/0.46 ( ap2p
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(funcinfuncset_type,type,
% 0.29/0.46 funcinfuncset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcinfuncset,definition,
% 0.29/0.46 ( funcinfuncset
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(lamProp_type,type,
% 0.29/0.46 lamProp: $o ).
% 0.29/0.46
% 0.29/0.46 thf(lamProp,definition,
% 0.29/0.46 ( lamProp
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.46 => ( func @ A @ B
% 0.29/0.46 @ ( dpsetconstr @ A @ B
% 0.29/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xf @ Xx )
% 0.29/0.46 = Xy ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(lam_type,type,
% 0.29/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.29/0.46
% 0.29/0.46 thf(lamp_type,type,
% 0.29/0.46 lamp: $o ).
% 0.29/0.46
% 0.29/0.46 thf(lamp,definition,
% 0.29/0.46 ( lamp
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.46 => ( func @ A @ B
% 0.29/0.46 @ ( lam @ A @ B
% 0.29/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(lam2p_type,type,
% 0.29/0.46 lam2p: $o ).
% 0.29/0.46
% 0.29/0.46 thf(lam2p,definition,
% 0.29/0.46 ( lam2p
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.46 => ( in
% 0.29/0.46 @ ( lam @ A @ B
% 0.29/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.29/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(brelnall1_type,type,
% 0.29/0.46 brelnall1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(brelnall1,definition,
% 0.29/0.46 ( brelnall1
% 0.29/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.29/0.46 ( ( breln @ A @ B @ R )
% 0.29/0.46 => ! [Xphi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ R )
% 0.29/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(brelnall2_type,type,
% 0.29/0.46 brelnall2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(brelnall2,definition,
% 0.29/0.46 ( brelnall2
% 0.29/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.29/0.46 ( ( breln @ A @ B @ R )
% 0.29/0.46 => ! [Xphi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ R )
% 0.29/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ex1E2_type,type,
% 0.29/0.46 ex1E2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ex1E2,definition,
% 0.29/0.46 ( ex1E2
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ( ex1 @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 => ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(funcGraphProp1_type,type,
% 0.29/0.46 funcGraphProp1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcGraphProp1,definition,
% 0.29/0.46 ( funcGraphProp1
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(funcGraphProp3_type,type,
% 0.29/0.46 funcGraphProp3: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcGraphProp3,definition,
% 0.29/0.46 ( funcGraphProp3
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(funcGraphProp2_type,type,
% 0.29/0.46 funcGraphProp2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcGraphProp2,definition,
% 0.29/0.46 ( funcGraphProp2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.29/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(funcextLem_type,type,
% 0.29/0.46 funcextLem: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcextLem,definition,
% 0.29/0.46 ( funcextLem
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ! [Xg: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xg )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.29/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(funcGraphProp4_type,type,
% 0.29/0.46 funcGraphProp4: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcGraphProp4,definition,
% 0.29/0.46 ( funcGraphProp4
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.29/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subbreln_type,type,
% 0.29/0.46 subbreln: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subbreln,definition,
% 0.29/0.46 ( subbreln
% 0.29/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.29/0.46 ( ( breln @ A @ B @ R )
% 0.29/0.46 => ! [S: $i] :
% 0.29/0.46 ( ( breln @ A @ B @ S )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.29/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(eqbreln_type,type,
% 0.29/0.46 eqbreln: $o ).
% 0.29/0.46
% 0.29/0.46 thf(eqbreln,definition,
% 0.29/0.46 ( eqbreln
% 0.29/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.29/0.46 ( ( breln @ A @ B @ R )
% 0.29/0.46 => ! [S: $i] :
% 0.29/0.46 ( ( breln @ A @ B @ S )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.29/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.29/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(funcext_type,type,
% 0.29/0.46 funcext: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcext,definition,
% 0.29/0.46 ( funcext
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ! [Xg: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xg )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.29/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(funcext2_type,type,
% 0.29/0.46 funcext2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(funcext2,definition,
% 0.29/0.46 ( funcext2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.46 => ! [Xg: $i] :
% 0.29/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.29/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ap2apEq1_type,type,
% 0.29/0.46 ap2apEq1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ap2apEq1,definition,
% 0.29/0.46 ( ap2apEq1
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ap2apEq2_type,type,
% 0.29/0.46 ap2apEq2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ap2apEq2,definition,
% 0.29/0.46 ( ap2apEq2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.29/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(beta1_type,type,
% 0.29/0.46 beta1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(beta1,definition,
% 0.29/0.46 ( beta1
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( ap @ A @ B
% 0.29/0.46 @ ( lam @ A @ B
% 0.29/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.29/0.46 @ Xx )
% 0.29/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(eta1_type,type,
% 0.29/0.46 eta1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(eta1,definition,
% 0.29/0.46 ( eta1
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( func @ A @ B @ Xf )
% 0.29/0.46 => ( ( lam @ A @ B
% 0.29/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.29/0.46 = Xf ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(lam2lamEq_type,type,
% 0.29/0.46 lam2lamEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(lam2lamEq,definition,
% 0.29/0.46 ( lam2lamEq
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.46 => ( ( lam @ A @ B
% 0.29/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.29/0.46 = ( lam @ A @ B
% 0.29/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(beta2_type,type,
% 0.29/0.46 beta2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(beta2,definition,
% 0.29/0.46 ( beta2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( ap @ A @ B
% 0.29/0.46 @ ( lam @ A @ B
% 0.29/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.29/0.46 @ Xx )
% 0.29/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(eta2_type,type,
% 0.29/0.46 eta2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(eta2,definition,
% 0.29/0.46 ( eta2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.29/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.29/0.46 => ( ( lam @ A @ B
% 0.29/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.29/0.46 = Xf ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(iffalseProp1_type,type,
% 0.29/0.46 iffalseProp1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(iffalseProp1,definition,
% 0.29/0.46 ( iffalseProp1
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( ~ Xphi
% 0.29/0.46 => ( in @ Xy
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xz: $i] :
% 0.29/0.46 ( ( Xphi
% 0.29/0.46 & ( Xz = Xx ) )
% 0.29/0.46 | ( ~ Xphi
% 0.29/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(iffalseProp2_type,type,
% 0.29/0.46 iffalseProp2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(iffalseProp2,definition,
% 0.29/0.46 ( iffalseProp2
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( ~ Xphi
% 0.29/0.46 => ( ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xz: $i] :
% 0.29/0.46 ( ( Xphi
% 0.29/0.46 & ( Xz = Xx ) )
% 0.29/0.46 | ( ~ Xphi
% 0.29/0.46 & ( Xz = Xy ) ) ) )
% 0.29/0.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(iftrueProp1_type,type,
% 0.29/0.46 iftrueProp1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(iftrueProp1,definition,
% 0.29/0.46 ( iftrueProp1
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( Xphi
% 0.29/0.46 => ( in @ Xx
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xz: $i] :
% 0.29/0.46 ( ( Xphi
% 0.29/0.46 & ( Xz = Xx ) )
% 0.29/0.46 | ( ~ Xphi
% 0.29/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(iftrueProp2_type,type,
% 0.29/0.46 iftrueProp2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(iftrueProp2,definition,
% 0.29/0.46 ( iftrueProp2
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( Xphi
% 0.29/0.46 => ( ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xz: $i] :
% 0.29/0.46 ( ( Xphi
% 0.29/0.46 & ( Xz = Xx ) )
% 0.29/0.46 | ( ~ Xphi
% 0.29/0.46 & ( Xz = Xy ) ) ) )
% 0.29/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(ifSingleton_type,type,
% 0.29/0.46 ifSingleton: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ifSingleton,definition,
% 0.29/0.46 ( ifSingleton
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( singleton
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xz: $i] :
% 0.29/0.46 ( ( Xphi
% 0.29/0.46 & ( Xz = Xx ) )
% 0.29/0.46 | ( ~ Xphi
% 0.29/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(if_type,type,
% 0.29/0.46 if: $i > $o > $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(ifp_type,type,
% 0.29/0.46 ifp: $o ).
% 0.29/0.46
% 0.29/0.46 thf(ifp,definition,
% 0.29/0.46 ( ifp
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(theeq_type,type,
% 0.29/0.46 theeq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(theeq,definition,
% 0.29/0.46 ( theeq
% 0.29/0.46 = ( ! [X: $i] :
% 0.29/0.46 ( ( singleton @ X )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ X )
% 0.29/0.46 => ( ( setunion @ X )
% 0.29/0.46 = Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(iftrue_type,type,
% 0.29/0.46 iftrue: $o ).
% 0.29/0.46
% 0.29/0.46 thf(iftrue,definition,
% 0.29/0.46 ( iftrue
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( Xphi
% 0.29/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.29/0.46 = Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(iffalse_type,type,
% 0.29/0.46 iffalse: $o ).
% 0.29/0.46
% 0.29/0.46 thf(iffalse,definition,
% 0.29/0.46 ( iffalse
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( ~ Xphi
% 0.29/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.29/0.46 = Xy ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(iftrueorfalse_type,type,
% 0.29/0.46 iftrueorfalse: $o ).
% 0.29/0.46
% 0.29/0.46 thf(iftrueorfalse,definition,
% 0.29/0.46 ( iftrueorfalse
% 0.29/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectT_lem_type,type,
% 0.29/0.46 binintersectT_lem: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectT_lem,definition,
% 0.29/0.46 ( binintersectT_lem
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionT_lem_type,type,
% 0.29/0.46 binunionT_lem: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionT_lem,definition,
% 0.29/0.46 ( binunionT_lem
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powersetT_lem_type,type,
% 0.29/0.46 powersetT_lem: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetT_lem,definition,
% 0.29/0.46 ( powersetT_lem
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusT_lem_type,type,
% 0.29/0.46 setminusT_lem: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusT_lem,definition,
% 0.29/0.46 ( setminusT_lem
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementT_lem_type,type,
% 0.29/0.46 complementT_lem: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementT_lem,definition,
% 0.29/0.46 ( complementT_lem
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setextT_type,type,
% 0.29/0.46 setextT: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setextT,definition,
% 0.29/0.46 ( setextT
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ X )
% 0.29/0.46 => ( in @ Xx @ Y ) ) )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ Y )
% 0.29/0.46 => ( in @ Xx @ X ) ) )
% 0.29/0.46 => ( X = Y ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subsetTI_type,type,
% 0.29/0.46 subsetTI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetTI,definition,
% 0.29/0.46 ( subsetTI
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ X )
% 0.29/0.46 => ( in @ Xx @ Y ) ) )
% 0.29/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powersetTI1_type,type,
% 0.29/0.46 powersetTI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetTI1,definition,
% 0.29/0.46 ( powersetTI1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ X )
% 0.29/0.46 => ( in @ Xx @ Y ) ) )
% 0.29/0.46 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powersetTE1_type,type,
% 0.29/0.46 powersetTE1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetTE1,definition,
% 0.29/0.46 ( powersetTE1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ X @ ( powerset @ Y ) )
% 0.29/0.46 => ( ( in @ Xx @ X )
% 0.29/0.46 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementTI1_type,type,
% 0.29/0.46 complementTI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementTI1,definition,
% 0.29/0.46 ( complementTI1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ X )
% 0.29/0.46 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementTE1_type,type,
% 0.29/0.46 complementTE1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementTE1,definition,
% 0.29/0.46 ( complementTE1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.29/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectTELcontra_type,type,
% 0.29/0.46 binintersectTELcontra: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectTELcontra,definition,
% 0.29/0.46 ( binintersectTELcontra
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ X )
% 0.29/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectTERcontra_type,type,
% 0.29/0.46 binintersectTERcontra: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectTERcontra,definition,
% 0.29/0.46 ( binintersectTERcontra
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ Y )
% 0.29/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(contrasubsetT_type,type,
% 0.29/0.46 contrasubsetT: $o ).
% 0.29/0.46
% 0.29/0.46 thf(contrasubsetT,definition,
% 0.29/0.46 ( contrasubsetT
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.29/0.46 => ( ( in @ Xx @ Y )
% 0.29/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(contrasubsetT1_type,type,
% 0.29/0.46 contrasubsetT1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(contrasubsetT1,definition,
% 0.29/0.46 ( contrasubsetT1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( subset @ X @ Y )
% 0.29/0.46 => ( ~ ( in @ Xx @ Y )
% 0.29/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(contrasubsetT2_type,type,
% 0.29/0.46 contrasubsetT2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(contrasubsetT2,definition,
% 0.29/0.46 ( contrasubsetT2
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ X @ Y )
% 0.29/0.46 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(contrasubsetT3_type,type,
% 0.29/0.46 contrasubsetT3: $o ).
% 0.29/0.46
% 0.29/0.46 thf(contrasubsetT3,definition,
% 0.29/0.46 ( contrasubsetT3
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.29/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementI1_type,type,
% 0.29/0.46 doubleComplementI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementI1,definition,
% 0.29/0.46 ( doubleComplementI1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ X )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementE1_type,type,
% 0.29/0.46 doubleComplementE1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementE1,definition,
% 0.29/0.46 ( doubleComplementE1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.29/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementSub1_type,type,
% 0.29/0.46 doubleComplementSub1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementSub1,definition,
% 0.29/0.46 ( doubleComplementSub1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementSub2_type,type,
% 0.29/0.46 doubleComplementSub2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementSub2,definition,
% 0.29/0.46 ( doubleComplementSub2
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementEq_type,type,
% 0.29/0.46 doubleComplementEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(doubleComplementEq,definition,
% 0.29/0.46 ( doubleComplementEq
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ( X
% 0.29/0.46 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementTnotintersectT_type,type,
% 0.29/0.46 complementTnotintersectT: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementTnotintersectT,definition,
% 0.29/0.46 ( complementTnotintersectT
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.29/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementImpComplementIntersect_type,type,
% 0.29/0.46 complementImpComplementIntersect: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementImpComplementIntersect,definition,
% 0.29/0.46 ( complementImpComplementIntersect
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementSubsetComplementIntersect_type,type,
% 0.29/0.46 complementSubsetComplementIntersect: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementSubsetComplementIntersect,definition,
% 0.29/0.46 ( complementSubsetComplementIntersect
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementInPowersetComplementIntersect_type,type,
% 0.29/0.46 complementInPowersetComplementIntersect: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementInPowersetComplementIntersect,definition,
% 0.29/0.46 ( complementInPowersetComplementIntersect
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(contraSubsetComplement_type,type,
% 0.29/0.46 contraSubsetComplement: $o ).
% 0.29/0.46
% 0.29/0.46 thf(contraSubsetComplement,definition,
% 0.29/0.46 ( contraSubsetComplement
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ Y )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementTcontraSubset_type,type,
% 0.29/0.46 complementTcontraSubset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementTcontraSubset,definition,
% 0.29/0.46 ( complementTcontraSubset
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.29/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionTILcontra_type,type,
% 0.29/0.46 binunionTILcontra: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionTILcontra,definition,
% 0.29/0.46 ( binunionTILcontra
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.29/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionTIRcontra_type,type,
% 0.29/0.46 binunionTIRcontra: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionTIRcontra,definition,
% 0.29/0.46 ( binunionTIRcontra
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.29/0.46 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(inIntersectImpInUnion_type,type,
% 0.29/0.46 inIntersectImpInUnion: $o ).
% 0.29/0.46
% 0.29/0.46 thf(inIntersectImpInUnion,definition,
% 0.29/0.46 ( inIntersectImpInUnion
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.29/0.46 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(inIntersectImpInUnion2_type,type,
% 0.29/0.46 inIntersectImpInUnion2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(inIntersectImpInUnion2,definition,
% 0.29/0.46 ( inIntersectImpInUnion2
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.29/0.46 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(inIntersectImpInIntersectUnions_type,type,
% 0.29/0.46 inIntersectImpInIntersectUnions: $o ).
% 0.29/0.46
% 0.29/0.46 thf(inIntersectImpInIntersectUnions,definition,
% 0.29/0.46 ( inIntersectImpInIntersectUnions
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.29/0.46 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(intersectInPowersetIntersectUnions_type,type,
% 0.29/0.46 intersectInPowersetIntersectUnions: $o ).
% 0.29/0.46
% 0.29/0.46 thf(intersectInPowersetIntersectUnions,definition,
% 0.29/0.46 ( intersectInPowersetIntersectUnions
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(inComplementUnionImpNotIn1_type,type,
% 0.29/0.46 inComplementUnionImpNotIn1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(inComplementUnionImpNotIn1,definition,
% 0.29/0.46 ( inComplementUnionImpNotIn1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(inComplementUnionImpInComplement1_type,type,
% 0.29/0.46 inComplementUnionImpInComplement1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(inComplementUnionImpInComplement1,definition,
% 0.29/0.46 ( inComplementUnionImpInComplement1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionTE_type,type,
% 0.29/0.46 binunionTE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionTE,definition,
% 0.29/0.46 ( binunionTE
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.29/0.46 => ( ( ( in @ Xx @ X )
% 0.29/0.46 => Xphi )
% 0.29/0.46 => ( ( ( in @ Xx @ Y )
% 0.29/0.46 => Xphi )
% 0.29/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionTEcontra_type,type,
% 0.29/0.46 binunionTEcontra: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionTEcontra,definition,
% 0.29/0.46 ( binunionTEcontra
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ X )
% 0.29/0.46 => ( ~ ( in @ Xx @ Y )
% 0.29/0.46 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2a1_type,type,
% 0.29/0.46 demorgan2a1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2a1,definition,
% 0.29/0.46 ( demorgan2a1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(complementUnionInPowersetComplement_type,type,
% 0.29/0.46 complementUnionInPowersetComplement: $o ).
% 0.29/0.46
% 0.29/0.46 thf(complementUnionInPowersetComplement,definition,
% 0.29/0.46 ( complementUnionInPowersetComplement
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2a2_type,type,
% 0.29/0.46 demorgan2a2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2a2,definition,
% 0.29/0.46 ( demorgan2a2
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan1a_type,type,
% 0.29/0.46 demorgan1a: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan1a,definition,
% 0.29/0.46 ( demorgan1a
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.29/0.46 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan1b_type,type,
% 0.29/0.46 demorgan1b: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan1b,definition,
% 0.29/0.46 ( demorgan1b
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan1_type,type,
% 0.29/0.46 demorgan1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan1,definition,
% 0.29/0.46 ( demorgan1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.29/0.46 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2a_type,type,
% 0.29/0.46 demorgan2a: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2a,definition,
% 0.29/0.46 ( demorgan2a
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.29/0.46 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2b2_type,type,
% 0.29/0.46 demorgan2b2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2b2,definition,
% 0.29/0.46 ( demorgan2b2
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.29/0.46 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2b_type,type,
% 0.29/0.46 demorgan2b: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2b,definition,
% 0.29/0.46 ( demorgan2b
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2_type,type,
% 0.29/0.46 demorgan2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(demorgan2,definition,
% 0.29/0.46 ( demorgan2
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.29/0.46 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule0_type,type,
% 0.29/0.46 woz13rule0: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule0,definition,
% 0.29/0.46 ( woz13rule0
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule1_type,type,
% 0.29/0.46 woz13rule1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule1,definition,
% 0.29/0.46 ( woz13rule1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ X @ Z )
% 0.29/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule2_type,type,
% 0.29/0.46 woz13rule2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule2,definition,
% 0.29/0.46 ( woz13rule2
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ Y @ Z )
% 0.29/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule3_type,type,
% 0.29/0.46 woz13rule3: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule3,definition,
% 0.29/0.46 ( woz13rule3
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ X @ Y )
% 0.29/0.46 => ( ( subset @ X @ Z )
% 0.29/0.46 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule4_type,type,
% 0.29/0.46 woz13rule4: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz13rule4,definition,
% 0.29/0.46 ( woz13rule4
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ! [W: $i] :
% 0.29/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ X @ Z )
% 0.29/0.46 => ( ( subset @ Y @ W )
% 0.29/0.46 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_1_type,type,
% 0.29/0.46 woz1_1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_1,definition,
% 0.29/0.46 ( woz1_1
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_2_type,type,
% 0.29/0.46 woz1_2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_2,definition,
% 0.29/0.46 ( woz1_2
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ! [W: $i] :
% 0.29/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.29/0.46 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.29/0.46 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_3_type,type,
% 0.29/0.46 woz1_3: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_3,definition,
% 0.29/0.46 ( woz1_3
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ! [Z: $i] :
% 0.29/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_4_type,type,
% 0.29/0.46 woz1_4: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_4,definition,
% 0.29/0.46 ( woz1_4
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.29/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_5_type,type,
% 0.29/0.46 woz1_5: $o ).
% 0.29/0.46
% 0.29/0.46 thf(woz1_5,definition,
% 0.29/0.46 ( woz1_5
% 0.29/0.46 = ( ! [A: $i,X: $i] :
% 0.29/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.29/0.46 => ! [Y: $i] :
% 0.29/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.29/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(breln1_type,type,
% 0.29/0.46 breln1: $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(breln1,definition,
% 0.29/0.46 ( breln1
% 0.29/0.46 = ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(breln1all2_type,type,
% 0.29/0.46 breln1all2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(breln1all2,definition,
% 0.29/0.46 ( breln1all2
% 0.29/0.46 = ( ! [A: $i,R: $i] :
% 0.29/0.46 ( ( breln1 @ A @ R )
% 0.29/0.46 => ! [Xphi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ R )
% 0.29/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(breln1Set_type,type,
% 0.29/0.46 breln1Set: $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(breln1Set,definition,
% 0.29/0.46 ( breln1Set
% 0.29/0.46 = ( ^ [A: $i] :
% 0.29/0.46 ( dsetconstr @ ( powerset @ ( cartprod @ A @ A ) )
% 0.29/0.46 @ ^ [R: $i] : ( breln1 @ A @ R ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(breln1SetBreln1,conjecture,
% 0.29/0.46 ( setextAx
% 0.29/0.46 => ( emptysetAx
% 0.29/0.46 => ( setadjoinAx
% 0.29/0.46 => ( powersetAx
% 0.29/0.46 => ( setunionAx
% 0.29/0.46 => ( omega0Ax
% 0.29/0.46 => ( omegaSAx
% 0.29/0.46 => ( omegaIndAx
% 0.29/0.46 => ( replAx
% 0.29/0.46 => ( foundationAx
% 0.29/0.46 => ( wellorderingAx
% 0.29/0.46 => ( descrp
% 0.29/0.46 => ( dsetconstrI
% 0.29/0.46 => ( dsetconstrEL
% 0.29/0.46 => ( dsetconstrER
% 0.29/0.46 => ( exuE1
% 0.29/0.46 => ( prop2setE
% 0.29/0.46 => ( emptysetE
% 0.29/0.46 => ( emptysetimpfalse
% 0.29/0.46 => ( notinemptyset
% 0.29/0.46 => ( exuE3e
% 0.29/0.46 => ( setext
% 0.29/0.46 => ( emptyI
% 0.29/0.46 => ( noeltsimpempty
% 0.29/0.46 => ( setbeta
% 0.29/0.46 => ( nonemptyE1
% 0.29/0.46 => ( nonemptyI
% 0.29/0.46 => ( nonemptyI1
% 0.29/0.46 => ( setadjoinIL
% 0.29/0.46 => ( emptyinunitempty
% 0.29/0.46 => ( setadjoinIR
% 0.29/0.46 => ( setadjoinE
% 0.29/0.46 => ( setadjoinOr
% 0.29/0.46 => ( setoftrueEq
% 0.29/0.46 => ( powersetI
% 0.29/0.46 => ( emptyinPowerset
% 0.29/0.46 => ( emptyInPowerset
% 0.29/0.46 => ( powersetE
% 0.29/0.46 => ( setunionI
% 0.29/0.46 => ( setunionE
% 0.29/0.46 => ( subPowSU
% 0.29/0.46 => ( exuE2
% 0.29/0.46 => ( nonemptyImpWitness
% 0.29/0.46 => ( uniqinunit
% 0.29/0.46 => ( notinsingleton
% 0.29/0.46 => ( eqinunit
% 0.29/0.46 => ( singletonsswitch
% 0.29/0.46 => ( upairsetE
% 0.29/0.46 => ( upairsetIL
% 0.29/0.46 => ( upairsetIR
% 0.29/0.46 => ( emptyE1
% 0.29/0.46 => ( vacuousDall
% 0.29/0.46 => ( quantDeMorgan1
% 0.29/0.46 => ( quantDeMorgan2
% 0.29/0.46 => ( quantDeMorgan3
% 0.29/0.46 => ( quantDeMorgan4
% 0.29/0.46 => ( prop2setI
% 0.29/0.46 => ( prop2set2propI
% 0.29/0.46 => ( notdexE
% 0.29/0.46 => ( notdallE
% 0.29/0.46 => ( exuI1
% 0.29/0.46 => ( exuI3
% 0.29/0.46 => ( exuI2
% 0.29/0.46 => ( inCongP
% 0.29/0.46 => ( in__Cong
% 0.29/0.46 => ( exuE3u
% 0.29/0.46 => ( exu__Cong
% 0.29/0.46 => ( emptyset__Cong
% 0.29/0.46 => ( setadjoin__Cong
% 0.29/0.46 => ( powerset__Cong
% 0.29/0.46 => ( setunion__Cong
% 0.29/0.46 => ( omega__Cong
% 0.29/0.46 => ( exuEu
% 0.29/0.46 => ( descr__Cong
% 0.29/0.46 => ( dsetconstr__Cong
% 0.29/0.46 => ( subsetI1
% 0.29/0.46 => ( eqimpsubset2
% 0.29/0.46 => ( eqimpsubset1
% 0.29/0.46 => ( subsetI2
% 0.29/0.46 => ( emptysetsubset
% 0.29/0.46 => ( subsetE
% 0.29/0.46 => ( subsetE2
% 0.29/0.46 => ( notsubsetI
% 0.29/0.46 => ( notequalI1
% 0.29/0.46 => ( notequalI2
% 0.29/0.46 => ( subsetRefl
% 0.29/0.46 => ( subsetTrans
% 0.29/0.46 => ( setadjoinSub
% 0.29/0.46 => ( setadjoinSub2
% 0.29/0.46 => ( subset2powerset
% 0.29/0.46 => ( setextsub
% 0.29/0.46 => ( subsetemptysetimpeq
% 0.29/0.46 => ( powersetI1
% 0.29/0.46 => ( powersetE1
% 0.29/0.46 => ( inPowerset
% 0.29/0.46 => ( powersetsubset
% 0.29/0.46 => ( sepInPowerset
% 0.29/0.46 => ( sepSubset
% 0.29/0.46 => ( binunionIL
% 0.29/0.46 => ( upairset2IR
% 0.29/0.46 => ( binunionIR
% 0.29/0.46 => ( binunionEcases
% 0.29/0.46 => ( binunionE
% 0.29/0.46 => ( binunionLsub
% 0.29/0.46 => ( binunionRsub
% 0.29/0.46 => ( binintersectI
% 0.29/0.46 => ( binintersectSubset5
% 0.29/0.46 => ( binintersectEL
% 0.29/0.46 => ( binintersectLsub
% 0.29/0.46 => ( binintersectSubset2
% 0.29/0.46 => ( binintersectSubset3
% 0.29/0.46 => ( binintersectER
% 0.29/0.46 => ( disjointsetsI1
% 0.29/0.46 => ( binintersectRsub
% 0.29/0.46 => ( binintersectSubset4
% 0.29/0.46 => ( binintersectSubset1
% 0.29/0.46 => ( bs114d
% 0.29/0.46 => ( setminusI
% 0.29/0.46 => ( setminusEL
% 0.29/0.46 => ( setminusER
% 0.29/0.46 => ( setminusSubset2
% 0.29/0.46 => ( setminusERneg
% 0.29/0.46 => ( setminusELneg
% 0.29/0.46 => ( setminusILneg
% 0.29/0.46 => ( setminusIRneg
% 0.29/0.46 => ( setminusLsub
% 0.29/0.46 => ( setminusSubset1
% 0.29/0.46 => ( symdiffE
% 0.29/0.46 => ( symdiffI1
% 0.29/0.46 => ( symdiffI2
% 0.29/0.46 => ( symdiffIneg1
% 0.29/0.46 => ( symdiffIneg2
% 0.29/0.46 => ( secondinupair
% 0.29/0.46 => ( setukpairIL
% 0.29/0.46 => ( setukpairIR
% 0.29/0.46 => ( kpairiskpair
% 0.29/0.46 => ( kpairp
% 0.29/0.46 => ( singletonsubset
% 0.29/0.46 => ( singletoninpowerset
% 0.29/0.46 => ( singletoninpowunion
% 0.29/0.46 => ( upairset2E
% 0.29/0.46 => ( upairsubunion
% 0.29/0.46 => ( upairinpowunion
% 0.29/0.46 => ( ubforcartprodlem1
% 0.29/0.46 => ( ubforcartprodlem2
% 0.29/0.46 => ( ubforcartprodlem3
% 0.29/0.46 => ( cartprodpairin
% 0.29/0.46 => ( cartprodmempair1
% 0.29/0.46 => ( cartprodmempair
% 0.29/0.46 => ( setunionE2
% 0.29/0.46 => ( setunionsingleton1
% 0.29/0.46 => ( setunionsingleton2
% 0.29/0.46 => ( setunionsingleton
% 0.29/0.46 => ( singletonprop
% 0.29/0.46 => ( ex1E1
% 0.29/0.46 => ( ex1I
% 0.29/0.46 => ( ex1I2
% 0.29/0.46 => ( singletonsuniq
% 0.29/0.46 => ( setukpairinjL1
% 0.29/0.46 => ( kfstsingleton
% 0.29/0.46 => ( theprop
% 0.29/0.46 => ( kfstpairEq
% 0.29/0.46 => ( cartprodfstin
% 0.29/0.46 => ( setukpairinjL2
% 0.29/0.46 => ( setukpairinjL
% 0.29/0.46 => ( setukpairinjR11
% 0.29/0.46 => ( setukpairinjR12
% 0.29/0.46 => ( setukpairinjR1
% 0.29/0.46 => ( upairequniteq
% 0.29/0.46 => ( setukpairinjR2
% 0.29/0.46 => ( setukpairinjR
% 0.29/0.46 => ( ksndsingleton
% 0.29/0.46 => ( ksndpairEq
% 0.29/0.46 => ( kpairsurjEq
% 0.29/0.46 => ( cartprodsndin
% 0.29/0.46 => ( cartprodpairmemEL
% 0.29/0.46 => ( cartprodpairmemER
% 0.29/0.46 => ( cartprodmempaircEq
% 0.29/0.46 => ( cartprodfstpairEq
% 0.29/0.46 => ( cartprodsndpairEq
% 0.29/0.46 => ( cartprodpairsurjEq
% 0.29/0.46 => ( dpsetconstrI
% 0.29/0.46 => ( dpsetconstrSub
% 0.29/0.46 => ( setOfPairsIsBReln
% 0.29/0.46 => ( dpsetconstrERa
% 0.29/0.46 => ( dpsetconstrEL1
% 0.29/0.46 => ( dpsetconstrEL2
% 0.29/0.46 => ( dpsetconstrER
% 0.29/0.46 => ( funcImageSingleton
% 0.29/0.46 => ( apProp
% 0.29/0.46 => ( app
% 0.29/0.46 => ( infuncsetfunc
% 0.29/0.46 => ( ap2p
% 0.29/0.46 => ( funcinfuncset
% 0.29/0.46 => ( lamProp
% 0.29/0.46 => ( lamp
% 0.29/0.46 => ( lam2p
% 0.29/0.46 => ( brelnall1
% 0.29/0.46 => ( brelnall2
% 0.29/0.46 => ( ex1E2
% 0.29/0.46 => ( funcGraphProp1
% 0.29/0.46 => ( funcGraphProp3
% 0.29/0.46 => ( funcGraphProp2
% 0.29/0.46 => ( funcextLem
% 0.29/0.46 => ( funcGraphProp4
% 0.29/0.46 => ( subbreln
% 0.29/0.46 => ( eqbreln
% 0.29/0.46 => ( funcext
% 0.29/0.46 => ( funcext2
% 0.29/0.46 => ( ap2apEq1
% 0.29/0.46 => ( ap2apEq2
% 0.29/0.46 => ( beta1
% 0.29/0.46 => ( eta1
% 0.29/0.46 => ( lam2lamEq
% 0.29/0.46 => ( beta2
% 0.29/0.46 => ( eta2
% 0.29/0.46 => ( iffalseProp1
% 0.29/0.46 => ( iffalseProp2
% 0.29/0.46 => ( iftrueProp1
% 0.29/0.46 => ( iftrueProp2
% 0.29/0.46 => ( ifSingleton
% 0.29/0.46 => ( ifp
% 0.29/0.46 => ( theeq
% 0.29/0.46 => ( iftrue
% 0.29/0.46 => ( iffalse
% 0.29/0.46 => ( iftrueorfalse
% 0.29/0.46 => ( binintersectT_lem
% 0.29/0.46 => ( binunionT_lem
% 0.29/0.46 => ( powersetT_lem
% 0.29/0.46 => ( setminusT_lem
% 0.29/0.46 => ( complementT_lem
% 0.29/0.46 => ( setextT
% 0.29/0.46 => ( subsetTI
% 0.29/0.46 => ( powersetTI1
% 0.29/0.46 => ( powersetTE1
% 0.29/0.46 => ( complementTI1
% 0.29/0.46 => ( complementTE1
% 0.29/0.46 => ( binintersectTELcontra
% 0.29/0.46 => ( binintersectTERcontra
% 0.29/0.46 => ( contrasubsetT
% 0.29/0.46 => ( contrasubsetT1
% 0.29/0.46 => ( contrasubsetT2
% 0.29/0.46 => ( contrasubsetT3
% 0.29/0.46 => ( doubleComplementI1
% 0.29/0.46 => ( doubleComplementE1
% 0.29/0.46 => ( doubleComplementSub1
% 0.29/0.46 => ( doubleComplementSub2
% 0.29/0.46 => ( doubleComplementEq
% 0.29/0.46 => ( complementTnotintersectT
% 0.29/0.46 => ( complementImpComplementIntersect
% 0.29/0.46 => ( complementSubsetComplementIntersect
% 0.29/0.46 => ( complementInPowersetComplementIntersect
% 0.29/0.46 => ( contraSubsetComplement
% 0.29/0.46 => ( complementTcontraSubset
% 0.29/0.46 => ( binunionTILcontra
% 0.29/0.46 => ( binunionTIRcontra
% 0.29/0.46 => ( inIntersectImpInUnion
% 0.29/0.46 => ( inIntersectImpInUnion2
% 0.29/0.46 => ( inIntersectImpInIntersectUnions
% 0.29/0.46 => ( intersectInPowersetIntersectUnions
% 0.29/0.46 => ( inComplementUnionImpNotIn1
% 0.29/0.46 => ( inComplementUnionImpInComplement1
% 0.29/0.46 => ( binunionTE
% 0.29/0.46 => ( binunionTEcontra
% 0.29/0.46 => ( demorgan2a1
% 0.29/0.46 => ( complementUnionInPowersetComplement
% 0.29/0.46 => ( demorgan2a2
% 0.29/0.46 => ( demorgan1a
% 0.29/0.46 => ( demorgan1b
% 0.29/0.46 => ( demorgan1
% 0.29/0.46 => ( demorgan2a
% 0.29/0.46 => ( demorgan2b2
% 0.29/0.46 => ( demorgan2b
% 0.29/0.46 => ( demorgan2
% 0.29/0.46 => ( woz13rule0
% 0.29/0.46 => ( woz13rule1
% 0.29/0.46 => ( woz13rule2
% 0.29/0.46 => ( woz13rule3
% 0.29/0.46 => ( woz13rule4
% 0.29/0.46 => ( woz1_1
% 0.29/0.46 => ( woz1_2
% 0.29/0.46 => ( woz1_3
% 0.29/0.46 => ( woz1_4
% 0.29/0.46 => ( woz1_5
% 0.29/0.46 => ( breln1all2
% 0.29/0.46 => ! [A: $i,R: $i] :
% 0.29/0.46 ( ( in @ R @ ( breln1Set @ A ) )
% 0.29/0.46 => ( breln1 @ A @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 3.38/1.71 ---- Embedded in THF ---
% 3.38/1.71 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.38/1.71 %%% Generated on Tue Feb 24 12:33:16 EST 2026
% 3.38/1.71 %%% using '$$dhol' embedding, version 1.3.0.
% 3.38/1.71 %%% Logic specification used:
% 3.38/1.71 %%% thf(spec, logic, $$dhol).
% 3.38/1.71
% 3.38/1.71 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2DTF16347.p
% See solution above
% 3.38/1.71 ------------------------
% 3.38/1.78 ---- Cleaned THF ---
% 3.38/1.78 thf(in_type,type,
% 3.38/1.78 in: $i > $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(exu_type,type,
% 3.38/1.78 exu: ( $i > $o ) > $o ).
% 3.38/1.78
% 3.38/1.78 thf(exu,definition,
% 3.38/1.78 ( exu
% 3.38/1.78 = ( ^ [Xphi: $i > $o] :
% 3.38/1.78 ? [Xx: $i] :
% 3.38/1.78 ( ( Xphi @ Xx )
% 3.38/1.78 & ! [Xy: $i] :
% 3.38/1.78 ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setextAx_type,type,
% 3.38/1.78 setextAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setextAx,definition,
% 3.38/1.78 ( setextAx
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 <=> ( in @ Xx @ B ) )
% 3.38/1.78 => ( A = B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptyset_type,type,
% 3.38/1.78 emptyset: $i ).
% 3.38/1.78
% 3.38/1.78 thf(emptysetAx_type,type,
% 3.38/1.78 emptysetAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptysetAx,definition,
% 3.38/1.78 ( emptysetAx
% 3.38/1.78 = ( ! [Xx: $i] :
% 3.38/1.78 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoin_type,type,
% 3.38/1.78 setadjoin: $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinAx_type,type,
% 3.38/1.78 setadjoinAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinAx,definition,
% 3.38/1.78 ( setadjoinAx
% 3.38/1.78 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.38/1.78 <=> ( ( Xy = Xx )
% 3.38/1.78 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powerset_type,type,
% 3.38/1.78 powerset: $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(powersetAx_type,type,
% 3.38/1.78 powersetAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetAx,definition,
% 3.38/1.78 ( powersetAx
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( in @ B @ ( powerset @ A ) )
% 3.38/1.78 <=> ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setunion_type,type,
% 3.38/1.78 setunion: $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(setunionAx_type,type,
% 3.38/1.78 setunionAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setunionAx,definition,
% 3.38/1.78 ( setunionAx
% 3.38/1.78 = ( ! [A: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( setunion @ A ) )
% 3.38/1.78 <=> ? [B: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 & ( in @ B @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(omega_type,type,
% 3.38/1.78 omega: $i ).
% 3.38/1.78
% 3.38/1.78 thf(omega0Ax_type,type,
% 3.38/1.78 omega0Ax: $o ).
% 3.38/1.78
% 3.38/1.78 thf(omega0Ax,definition,
% 3.38/1.78 ( omega0Ax
% 3.38/1.78 = ( in @ emptyset @ omega ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(omegaSAx_type,type,
% 3.38/1.78 omegaSAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(omegaSAx,definition,
% 3.38/1.78 ( omegaSAx
% 3.38/1.78 = ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ omega )
% 3.38/1.78 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(omegaIndAx_type,type,
% 3.38/1.78 omegaIndAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(omegaIndAx,definition,
% 3.38/1.78 ( omegaIndAx
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ( ( in @ emptyset @ A )
% 3.38/1.78 & ! [Xx: $i] :
% 3.38/1.78 ( ( ( in @ Xx @ omega )
% 3.38/1.78 & ( in @ Xx @ A ) )
% 3.38/1.78 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ omega )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(replAx_type,type,
% 3.38/1.78 replAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(replAx,definition,
% 3.38/1.78 ( replAx
% 3.38/1.78 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( exu
% 3.38/1.78 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.38/1.78 => ? [B: $i] :
% 3.38/1.78 ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 <=> ? [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(foundationAx_type,type,
% 3.38/1.78 foundationAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(foundationAx,definition,
% 3.38/1.78 ( foundationAx
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.38/1.78 => ? [B: $i] :
% 3.38/1.78 ( ( in @ B @ A )
% 3.38/1.78 & ~ ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(wellorderingAx_type,type,
% 3.38/1.78 wellorderingAx: $o ).
% 3.38/1.78
% 3.38/1.78 thf(wellorderingAx,definition,
% 3.38/1.78 ( wellorderingAx
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ? [B: $i] :
% 3.38/1.78 ( ! [C: $i] :
% 3.38/1.78 ( ( in @ C @ B )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ C )
% 3.38/1.78 => ( in @ Xx @ A ) ) )
% 3.38/1.78 & ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( ( in @ Xx @ A )
% 3.38/1.78 & ( in @ Xy @ A ) )
% 3.38/1.78 => ( ! [C: $i] :
% 3.38/1.78 ( ( in @ C @ B )
% 3.38/1.78 => ( ( in @ Xx @ C )
% 3.38/1.78 <=> ( in @ Xy @ C ) ) )
% 3.38/1.78 => ( Xx = Xy ) ) )
% 3.38/1.78 & ! [C: $i,D: $i] :
% 3.38/1.78 ( ( ( in @ C @ B )
% 3.38/1.78 & ( in @ D @ B ) )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ C )
% 3.38/1.78 => ( in @ Xx @ D ) )
% 3.38/1.78 | ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ D )
% 3.38/1.78 => ( in @ Xx @ C ) ) ) )
% 3.38/1.78 & ! [C: $i] :
% 3.38/1.78 ( ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ C )
% 3.38/1.78 => ( in @ Xx @ A ) )
% 3.38/1.78 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.38/1.78 => ? [D: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ D @ B )
% 3.38/1.78 & ( in @ Xx @ C )
% 3.38/1.78 & ~ ? [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ D )
% 3.38/1.78 & ( in @ Xy @ C ) )
% 3.38/1.78 & ! [E: $i] :
% 3.38/1.78 ( ( in @ E @ B )
% 3.38/1.78 => ( ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ E )
% 3.38/1.78 => ( in @ Xy @ D ) )
% 3.38/1.78 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(descr_type,type,
% 3.38/1.78 descr: ( $i > $o ) > $i ).
% 3.38/1.78
% 3.38/1.78 thf(descrp_type,type,
% 3.38/1.78 descrp: $o ).
% 3.38/1.78
% 3.38/1.78 thf(descrp,definition,
% 3.38/1.78 ( descrp
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ( Xphi
% 3.38/1.78 @ ( descr
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstr_type,type,
% 3.38/1.78 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstrI_type,type,
% 3.38/1.78 dsetconstrI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstrI,definition,
% 3.38/1.78 ( dsetconstrI
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 => ( in @ Xx
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstrEL_type,type,
% 3.38/1.78 dsetconstrEL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstrEL,definition,
% 3.38/1.78 ( dsetconstrEL
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstrER_type,type,
% 3.38/1.78 dsetconstrER: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstrER,definition,
% 3.38/1.78 ( dsetconstrER
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.38/1.78 => ( Xphi @ Xx ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exuE1_type,type,
% 3.38/1.78 exuE1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exuE1,definition,
% 3.38/1.78 ( exuE1
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ? [Xx: $i] :
% 3.38/1.78 ( ( Xphi @ Xx )
% 3.38/1.78 & ! [Xy: $i] :
% 3.38/1.78 ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(prop2set_type,type,
% 3.38/1.78 prop2set: $o > $i ).
% 3.38/1.78
% 3.38/1.78 thf(prop2setE_type,type,
% 3.38/1.78 prop2setE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(prop2setE,definition,
% 3.38/1.78 ( prop2setE
% 3.38/1.78 = ( ! [Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.38/1.78 => Xphi ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptysetE_type,type,
% 3.38/1.78 emptysetE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptysetE,definition,
% 3.38/1.78 ( emptysetE
% 3.38/1.78 = ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ emptyset )
% 3.38/1.78 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptysetimpfalse_type,type,
% 3.38/1.78 emptysetimpfalse: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptysetimpfalse,definition,
% 3.38/1.78 ( emptysetimpfalse
% 3.38/1.78 = ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ emptyset )
% 3.38/1.78 => $false ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(notinemptyset_type,type,
% 3.38/1.78 notinemptyset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(notinemptyset,definition,
% 3.38/1.78 ( notinemptyset
% 3.38/1.78 = ( ! [Xx: $i] :
% 3.38/1.78 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exuE3e_type,type,
% 3.38/1.78 exuE3e: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exuE3e,definition,
% 3.38/1.78 ( exuE3e
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setext_type,type,
% 3.38/1.78 setext: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setext,definition,
% 3.38/1.78 ( setext
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ Xx @ B ) )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ A ) )
% 3.38/1.78 => ( A = B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptyI_type,type,
% 3.38/1.78 emptyI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptyI,definition,
% 3.38/1.78 ( emptyI
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ~ ( in @ Xx @ A )
% 3.38/1.78 => ( A = emptyset ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(noeltsimpempty_type,type,
% 3.38/1.78 noeltsimpempty: $o ).
% 3.38/1.78
% 3.38/1.78 thf(noeltsimpempty,definition,
% 3.38/1.78 ( noeltsimpempty
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ~ ( in @ Xx @ A )
% 3.38/1.78 => ( A = emptyset ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setbeta_type,type,
% 3.38/1.78 setbeta: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setbeta,definition,
% 3.38/1.78 ( setbeta
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.38/1.78 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(nonempty_type,type,
% 3.38/1.78 nonempty: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(nonempty,definition,
% 3.38/1.78 ( nonempty
% 3.38/1.78 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(nonemptyE1_type,type,
% 3.38/1.78 nonemptyE1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(nonemptyE1,definition,
% 3.38/1.78 ( nonemptyE1
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ( nonempty @ A )
% 3.38/1.78 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(nonemptyI_type,type,
% 3.38/1.78 nonemptyI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(nonemptyI,definition,
% 3.38/1.78 ( nonemptyI
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 => ( nonempty
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(nonemptyI1_type,type,
% 3.38/1.78 nonemptyI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(nonemptyI1,definition,
% 3.38/1.78 ( nonemptyI1
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.38/1.78 => ( nonempty @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinIL_type,type,
% 3.38/1.78 setadjoinIL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinIL,definition,
% 3.38/1.78 ( setadjoinIL
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptyinunitempty_type,type,
% 3.38/1.78 emptyinunitempty: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptyinunitempty,definition,
% 3.38/1.78 ( emptyinunitempty
% 3.38/1.78 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinIR_type,type,
% 3.38/1.78 setadjoinIR: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinIR,definition,
% 3.38/1.78 ( setadjoinIR
% 3.38/1.78 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinE_type,type,
% 3.38/1.78 setadjoinE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinE,definition,
% 3.38/1.78 ( setadjoinE
% 3.38/1.78 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.38/1.78 => ! [Xphi: $o] :
% 3.38/1.78 ( ( ( Xy = Xx )
% 3.38/1.78 => Xphi )
% 3.38/1.78 => ( ( ( in @ Xy @ A )
% 3.38/1.78 => Xphi )
% 3.38/1.78 => Xphi ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinOr_type,type,
% 3.38/1.78 setadjoinOr: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinOr,definition,
% 3.38/1.78 ( setadjoinOr
% 3.38/1.78 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.38/1.78 => ( ( Xy = Xx )
% 3.38/1.78 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setoftrueEq_type,type,
% 3.38/1.78 setoftrueEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setoftrueEq,definition,
% 3.38/1.78 ( setoftrueEq
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xx: $i] : $true )
% 3.38/1.78 = A ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powersetI_type,type,
% 3.38/1.78 powersetI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetI,definition,
% 3.38/1.78 ( powersetI
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ A ) )
% 3.38/1.78 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptyinPowerset_type,type,
% 3.38/1.78 emptyinPowerset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptyinPowerset,definition,
% 3.38/1.78 ( emptyinPowerset
% 3.38/1.78 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptyInPowerset_type,type,
% 3.38/1.78 emptyInPowerset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptyInPowerset,definition,
% 3.38/1.78 ( emptyInPowerset
% 3.38/1.78 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powersetE_type,type,
% 3.38/1.78 powersetE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetE,definition,
% 3.38/1.78 ( powersetE
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ B @ ( powerset @ A ) )
% 3.38/1.78 => ( ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setunionI_type,type,
% 3.38/1.78 setunionI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setunionI,definition,
% 3.38/1.78 ( setunionI
% 3.38/1.78 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 => ( ( in @ B @ A )
% 3.38/1.78 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setunionE_type,type,
% 3.38/1.78 setunionE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setunionE,definition,
% 3.38/1.78 ( setunionE
% 3.38/1.78 = ( ! [A: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( setunion @ A ) )
% 3.38/1.78 => ! [Xphi: $o] :
% 3.38/1.78 ( ! [B: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 => ( ( in @ B @ A )
% 3.38/1.78 => Xphi ) )
% 3.38/1.78 => Xphi ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subPowSU_type,type,
% 3.38/1.78 subPowSU: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subPowSU,definition,
% 3.38/1.78 ( subPowSU
% 3.38/1.78 = ( ! [A: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exuE2_type,type,
% 3.38/1.78 exuE2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exuE2,definition,
% 3.38/1.78 ( exuE2
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ? [Xx: $i] :
% 3.38/1.78 ! [Xy: $i] :
% 3.38/1.78 ( ( Xphi @ Xy )
% 3.38/1.78 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(nonemptyImpWitness_type,type,
% 3.38/1.78 nonemptyImpWitness: $o ).
% 3.38/1.78
% 3.38/1.78 thf(nonemptyImpWitness,definition,
% 3.38/1.78 ( nonemptyImpWitness
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ( nonempty @ A )
% 3.38/1.78 => ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & $true ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(uniqinunit_type,type,
% 3.38/1.78 uniqinunit: $o ).
% 3.38/1.78
% 3.38/1.78 thf(uniqinunit,definition,
% 3.38/1.78 ( uniqinunit
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(notinsingleton_type,type,
% 3.38/1.78 notinsingleton: $o ).
% 3.38/1.78
% 3.38/1.78 thf(notinsingleton,definition,
% 3.38/1.78 ( notinsingleton
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx != Xy )
% 3.38/1.78 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(eqinunit_type,type,
% 3.38/1.78 eqinunit: $o ).
% 3.38/1.78
% 3.38/1.78 thf(eqinunit,definition,
% 3.38/1.78 ( eqinunit
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx = Xy )
% 3.38/1.78 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(singletonsswitch_type,type,
% 3.38/1.78 singletonsswitch: $o ).
% 3.38/1.78
% 3.38/1.78 thf(singletonsswitch,definition,
% 3.38/1.78 ( singletonsswitch
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.38/1.78 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(upairsetE_type,type,
% 3.38/1.78 upairsetE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairsetE,definition,
% 3.38/1.78 ( upairsetE
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.38/1.78 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.38/1.78 => ( ( Xz = Xx )
% 3.38/1.78 | ( Xz = Xy ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(upairsetIL_type,type,
% 3.38/1.78 upairsetIL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairsetIL,definition,
% 3.38/1.78 ( upairsetIL
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(upairsetIR_type,type,
% 3.38/1.78 upairsetIR: $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairsetIR,definition,
% 3.38/1.78 ( upairsetIR
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptyE1_type,type,
% 3.38/1.78 emptyE1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptyE1,definition,
% 3.38/1.78 ( emptyE1
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ( Xphi @ Xx ) )
% 3.38/1.78 => ( ( ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 = emptyset )
% 3.38/1.78 => $false ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(vacuousDall_type,type,
% 3.38/1.78 vacuousDall: $o ).
% 3.38/1.78
% 3.38/1.78 thf(vacuousDall,definition,
% 3.38/1.78 ( vacuousDall
% 3.38/1.78 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ emptyset )
% 3.38/1.78 => ( Xphi @ Xx ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(quantDeMorgan1_type,type,
% 3.38/1.78 quantDeMorgan1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(quantDeMorgan1,definition,
% 3.38/1.78 ( quantDeMorgan1
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ~ ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( Xphi @ Xx ) )
% 3.38/1.78 => ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(quantDeMorgan2_type,type,
% 3.38/1.78 quantDeMorgan2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(quantDeMorgan2,definition,
% 3.38/1.78 ( quantDeMorgan2
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ~ ( Xphi @ Xx ) )
% 3.38/1.78 => ~ ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(quantDeMorgan3_type,type,
% 3.38/1.78 quantDeMorgan3: $o ).
% 3.38/1.78
% 3.38/1.78 thf(quantDeMorgan3,definition,
% 3.38/1.78 ( quantDeMorgan3
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ~ ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ( Xphi @ Xx ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(quantDeMorgan4_type,type,
% 3.38/1.78 quantDeMorgan4: $o ).
% 3.38/1.78
% 3.38/1.78 thf(quantDeMorgan4,definition,
% 3.38/1.78 ( quantDeMorgan4
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ~ ( Xphi @ Xx ) )
% 3.38/1.78 => ~ ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(prop2setI_type,type,
% 3.38/1.78 prop2setI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(prop2setI,definition,
% 3.38/1.78 ( prop2setI
% 3.38/1.78 = ( ! [Xphi: $o] :
% 3.38/1.78 ( Xphi
% 3.38/1.78 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(set2prop_type,type,
% 3.38/1.78 set2prop: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(prop2set2propI_type,type,
% 3.38/1.78 prop2set2propI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(prop2set2propI,definition,
% 3.38/1.78 ( prop2set2propI
% 3.38/1.78 = ( ! [Xphi: $o] :
% 3.38/1.78 ( Xphi
% 3.38/1.78 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(notdexE_type,type,
% 3.38/1.78 notdexE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(notdexE,definition,
% 3.38/1.78 ( notdexE
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ~ ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ( Xphi @ Xx ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(notdallE_type,type,
% 3.38/1.78 notdallE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(notdallE,definition,
% 3.38/1.78 ( notdallE
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ~ ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( Xphi @ Xx ) )
% 3.38/1.78 => ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exuI1_type,type,
% 3.38/1.78 exuI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exuI1,definition,
% 3.38/1.78 ( exuI1
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ? [Xx: $i] :
% 3.38/1.78 ( ( Xphi @ Xx )
% 3.38/1.78 & ! [Xy: $i] :
% 3.38/1.78 ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) )
% 3.38/1.78 => ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exuI3_type,type,
% 3.38/1.78 exuI3: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exuI3,definition,
% 3.38/1.78 ( exuI3
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.38/1.78 => ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xphi @ Xx )
% 3.38/1.78 => ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) )
% 3.38/1.78 => ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exuI2_type,type,
% 3.38/1.78 exuI2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exuI2,definition,
% 3.38/1.78 ( exuI2
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ? [Xx: $i] :
% 3.38/1.78 ! [Xy: $i] :
% 3.38/1.78 ( ( Xphi @ Xy )
% 3.38/1.78 <=> ( Xy = Xx ) )
% 3.38/1.78 => ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(inCongP_type,type,
% 3.38/1.78 inCongP: $o ).
% 3.38/1.78
% 3.38/1.78 thf(inCongP,definition,
% 3.38/1.78 ( inCongP
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( A = B )
% 3.38/1.78 => ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx = Xy )
% 3.38/1.78 => ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(in__Cong_type,type,
% 3.38/1.78 in__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(in__Cong,definition,
% 3.38/1.78 ( in__Cong
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( A = B )
% 3.38/1.78 => ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx = Xy )
% 3.38/1.78 => ( ( in @ Xx @ A )
% 3.38/1.78 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exuE3u_type,type,
% 3.38/1.78 exuE3u: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exuE3u,definition,
% 3.38/1.78 ( exuE3u
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xphi @ Xx )
% 3.38/1.78 => ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exu__Cong_type,type,
% 3.38/1.78 exu__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exu__Cong,definition,
% 3.38/1.78 ( exu__Cong
% 3.38/1.78 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx = Xy )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 <=> ( Xpsi @ Xy ) ) )
% 3.38/1.78 => ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 <=> ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptyset__Cong_type,type,
% 3.38/1.78 emptyset__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptyset__Cong,definition,
% 3.38/1.78 ( emptyset__Cong
% 3.38/1.78 = ( emptyset = emptyset ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoin__Cong_type,type,
% 3.38/1.78 setadjoin__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoin__Cong,definition,
% 3.38/1.78 ( setadjoin__Cong
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx = Xy )
% 3.38/1.78 => ! [Xz: $i,Xu: $i] :
% 3.38/1.78 ( ( Xz = Xu )
% 3.38/1.78 => ( ( setadjoin @ Xx @ Xz )
% 3.38/1.78 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powerset__Cong_type,type,
% 3.38/1.78 powerset__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powerset__Cong,definition,
% 3.38/1.78 ( powerset__Cong
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( A = B )
% 3.38/1.78 => ( ( powerset @ A )
% 3.38/1.78 = ( powerset @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setunion__Cong_type,type,
% 3.38/1.78 setunion__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setunion__Cong,definition,
% 3.38/1.78 ( setunion__Cong
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( A = B )
% 3.38/1.78 => ( ( setunion @ A )
% 3.38/1.78 = ( setunion @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(omega__Cong_type,type,
% 3.38/1.78 omega__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(omega__Cong,definition,
% 3.38/1.78 ( omega__Cong
% 3.38/1.78 = ( omega = omega ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(exuEu_type,type,
% 3.38/1.78 exuEu: $o ).
% 3.38/1.78
% 3.38/1.78 thf(exuEu,definition,
% 3.38/1.78 ( exuEu
% 3.38/1.78 = ( ! [Xphi: $i > $o] :
% 3.38/1.78 ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xphi @ Xx )
% 3.38/1.78 => ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(descr__Cong_type,type,
% 3.38/1.78 descr__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(descr__Cong,definition,
% 3.38/1.78 ( descr__Cong
% 3.38/1.78 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx = Xy )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 <=> ( Xpsi @ Xy ) ) )
% 3.38/1.78 => ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ( ( exu
% 3.38/1.78 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.38/1.78 => ( ( descr
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 = ( descr
% 3.38/1.78 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstr__Cong_type,type,
% 3.38/1.78 dsetconstr__Cong: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dsetconstr__Cong,definition,
% 3.38/1.78 ( dsetconstr__Cong
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( A = B )
% 3.38/1.78 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( Xx = Xy )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.38/1.78 => ( ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 = ( dsetconstr @ B
% 3.38/1.78 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subset_type,type,
% 3.38/1.78 subset: $i > $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(disjoint_type,type,
% 3.38/1.78 disjoint: $i > $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(setsmeet_type,type,
% 3.38/1.78 setsmeet: $i > $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetI1_type,type,
% 3.38/1.78 subsetI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetI1,definition,
% 3.38/1.78 ( subsetI1
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ Xx @ B ) )
% 3.38/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(eqimpsubset2_type,type,
% 3.38/1.78 eqimpsubset2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(eqimpsubset2,definition,
% 3.38/1.78 ( eqimpsubset2
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( A = B )
% 3.38/1.78 => ( subset @ B @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(eqimpsubset1_type,type,
% 3.38/1.78 eqimpsubset1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(eqimpsubset1,definition,
% 3.38/1.78 ( eqimpsubset1
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( A = B )
% 3.38/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subsetI2_type,type,
% 3.38/1.78 subsetI2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetI2,definition,
% 3.38/1.78 ( subsetI2
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ Xx @ B ) )
% 3.38/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(emptysetsubset_type,type,
% 3.38/1.78 emptysetsubset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(emptysetsubset,definition,
% 3.38/1.78 ( emptysetsubset
% 3.38/1.78 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subsetE_type,type,
% 3.38/1.78 subsetE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetE,definition,
% 3.38/1.78 ( subsetE
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subsetE2_type,type,
% 3.38/1.78 subsetE2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetE2,definition,
% 3.38/1.78 ( subsetE2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( ~ ( in @ Xx @ B )
% 3.38/1.78 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(notsubsetI_type,type,
% 3.38/1.78 notsubsetI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(notsubsetI,definition,
% 3.38/1.78 ( notsubsetI
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ B )
% 3.38/1.78 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(notequalI1_type,type,
% 3.38/1.78 notequalI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(notequalI1,definition,
% 3.38/1.78 ( notequalI1
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ~ ( subset @ A @ B )
% 3.38/1.78 => ( A != B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(notequalI2_type,type,
% 3.38/1.78 notequalI2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(notequalI2,definition,
% 3.38/1.78 ( notequalI2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ B )
% 3.38/1.78 => ( A != B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subsetRefl_type,type,
% 3.38/1.78 subsetRefl: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetRefl,definition,
% 3.38/1.78 ( subsetRefl
% 3.38/1.78 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subsetTrans_type,type,
% 3.38/1.78 subsetTrans: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetTrans,definition,
% 3.38/1.78 ( subsetTrans
% 3.38/1.78 = ( ! [A: $i,B: $i,C: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( ( subset @ B @ C )
% 3.38/1.78 => ( subset @ A @ C ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinSub_type,type,
% 3.38/1.78 setadjoinSub: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinSub,definition,
% 3.38/1.78 ( setadjoinSub
% 3.38/1.78 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinSub2_type,type,
% 3.38/1.78 setadjoinSub2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setadjoinSub2,definition,
% 3.38/1.78 ( setadjoinSub2
% 3.38/1.78 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subset2powerset_type,type,
% 3.38/1.78 subset2powerset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subset2powerset,definition,
% 3.38/1.78 ( subset2powerset
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setextsub_type,type,
% 3.38/1.78 setextsub: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setextsub,definition,
% 3.38/1.78 ( setextsub
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( ( subset @ B @ A )
% 3.38/1.78 => ( A = B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subsetemptysetimpeq_type,type,
% 3.38/1.78 subsetemptysetimpeq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetemptysetimpeq,definition,
% 3.38/1.78 ( subsetemptysetimpeq
% 3.38/1.78 = ( ! [A: $i] :
% 3.38/1.78 ( ( subset @ A @ emptyset )
% 3.38/1.78 => ( A = emptyset ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powersetI1_type,type,
% 3.38/1.78 powersetI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetI1,definition,
% 3.38/1.78 ( powersetI1
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( subset @ B @ A )
% 3.38/1.78 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powersetE1_type,type,
% 3.38/1.78 powersetE1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetE1,definition,
% 3.38/1.78 ( powersetE1
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( in @ B @ ( powerset @ A ) )
% 3.38/1.78 => ( subset @ B @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(inPowerset_type,type,
% 3.38/1.78 inPowerset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(inPowerset,definition,
% 3.38/1.78 ( inPowerset
% 3.38/1.78 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powersetsubset_type,type,
% 3.38/1.78 powersetsubset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetsubset,definition,
% 3.38/1.78 ( powersetsubset
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(sepInPowerset_type,type,
% 3.38/1.78 sepInPowerset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(sepInPowerset,definition,
% 3.38/1.78 ( sepInPowerset
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( in
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 @ ( powerset @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(sepSubset_type,type,
% 3.38/1.78 sepSubset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(sepSubset,definition,
% 3.38/1.78 ( sepSubset
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( subset
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 @ A ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunion_type,type,
% 3.38/1.78 binunion: $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(binunionIL_type,type,
% 3.38/1.78 binunionIL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionIL,definition,
% 3.38/1.78 ( binunionIL
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(upairset2IR_type,type,
% 3.38/1.78 upairset2IR: $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairset2IR,definition,
% 3.38/1.78 ( upairset2IR
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionIR_type,type,
% 3.38/1.78 binunionIR: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionIR,definition,
% 3.38/1.78 ( binunionIR
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionEcases_type,type,
% 3.38/1.78 binunionEcases: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionEcases,definition,
% 3.38/1.78 ( binunionEcases
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.38/1.78 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.38/1.78 => ( ( ( in @ Xx @ A )
% 3.38/1.78 => Xphi )
% 3.38/1.78 => ( ( ( in @ Xx @ B )
% 3.38/1.78 => Xphi )
% 3.38/1.78 => Xphi ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionE_type,type,
% 3.38/1.78 binunionE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionE,definition,
% 3.38/1.78 ( binunionE
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.38/1.78 => ( ( in @ Xx @ A )
% 3.38/1.78 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionLsub_type,type,
% 3.38/1.78 binunionLsub: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionLsub,definition,
% 3.38/1.78 ( binunionLsub
% 3.38/1.78 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionRsub_type,type,
% 3.38/1.78 binunionRsub: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionRsub,definition,
% 3.38/1.78 ( binunionRsub
% 3.38/1.78 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersect_type,type,
% 3.38/1.78 binintersect: $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectI_type,type,
% 3.38/1.78 binintersectI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectI,definition,
% 3.38/1.78 ( binintersectI
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset5_type,type,
% 3.38/1.78 binintersectSubset5: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset5,definition,
% 3.38/1.78 ( binintersectSubset5
% 3.38/1.78 = ( ! [A: $i,B: $i,C: $i] :
% 3.38/1.78 ( ( subset @ C @ A )
% 3.38/1.78 => ( ( subset @ C @ B )
% 3.38/1.78 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectEL_type,type,
% 3.38/1.78 binintersectEL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectEL,definition,
% 3.38/1.78 ( binintersectEL
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectLsub_type,type,
% 3.38/1.78 binintersectLsub: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectLsub,definition,
% 3.38/1.78 ( binintersectLsub
% 3.38/1.78 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset2_type,type,
% 3.38/1.78 binintersectSubset2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset2,definition,
% 3.38/1.78 ( binintersectSubset2
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( ( binintersect @ A @ B )
% 3.38/1.78 = A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset3_type,type,
% 3.38/1.78 binintersectSubset3: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset3,definition,
% 3.38/1.78 ( binintersectSubset3
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( ( binintersect @ A @ B )
% 3.38/1.78 = B )
% 3.38/1.78 => ( subset @ B @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectER_type,type,
% 3.38/1.78 binintersectER: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectER,definition,
% 3.38/1.78 ( binintersectER
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.38/1.78 => ( in @ Xx @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(disjointsetsI1_type,type,
% 3.38/1.78 disjointsetsI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(disjointsetsI1,definition,
% 3.38/1.78 ( disjointsetsI1
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ~ ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ( in @ Xx @ B ) )
% 3.38/1.78 => ( ( binintersect @ A @ B )
% 3.38/1.78 = emptyset ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectRsub_type,type,
% 3.38/1.78 binintersectRsub: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectRsub,definition,
% 3.38/1.78 ( binintersectRsub
% 3.38/1.78 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset4_type,type,
% 3.38/1.78 binintersectSubset4: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset4,definition,
% 3.38/1.78 ( binintersectSubset4
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( subset @ B @ A )
% 3.38/1.78 => ( ( binintersect @ A @ B )
% 3.38/1.78 = B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset1_type,type,
% 3.38/1.78 binintersectSubset1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectSubset1,definition,
% 3.38/1.78 ( binintersectSubset1
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( ( binintersect @ A @ B )
% 3.38/1.78 = A )
% 3.38/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(bs114d_type,type,
% 3.38/1.78 bs114d: $o ).
% 3.38/1.78
% 3.38/1.78 thf(bs114d,definition,
% 3.38/1.78 ( bs114d
% 3.38/1.78 = ( ! [A: $i,B: $i,C: $i] :
% 3.38/1.78 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.38/1.78 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(regular_type,type,
% 3.38/1.78 regular: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminus_type,type,
% 3.38/1.78 setminus: $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(setminusI_type,type,
% 3.38/1.78 setminusI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusI,definition,
% 3.38/1.78 ( setminusI
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusEL_type,type,
% 3.38/1.78 setminusEL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusEL,definition,
% 3.38/1.78 ( setminusEL
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusER_type,type,
% 3.38/1.78 setminusER: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusER,definition,
% 3.38/1.78 ( setminusER
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.38/1.78 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusSubset2_type,type,
% 3.38/1.78 setminusSubset2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusSubset2,definition,
% 3.38/1.78 ( setminusSubset2
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( subset @ A @ B )
% 3.38/1.78 => ( ( setminus @ A @ B )
% 3.38/1.78 = emptyset ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusERneg_type,type,
% 3.38/1.78 setminusERneg: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusERneg,definition,
% 3.38/1.78 ( setminusERneg
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.38/1.78 => ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusELneg_type,type,
% 3.38/1.78 setminusELneg: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusELneg,definition,
% 3.38/1.78 ( setminusELneg
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.38/1.78 => ( ~ ( in @ Xx @ B )
% 3.38/1.78 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusILneg_type,type,
% 3.38/1.78 setminusILneg: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusILneg,definition,
% 3.38/1.78 ( setminusILneg
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ~ ( in @ Xx @ A )
% 3.38/1.78 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusIRneg_type,type,
% 3.38/1.78 setminusIRneg: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusIRneg,definition,
% 3.38/1.78 ( setminusIRneg
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ B )
% 3.38/1.78 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusLsub_type,type,
% 3.38/1.78 setminusLsub: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusLsub,definition,
% 3.38/1.78 ( setminusLsub
% 3.38/1.78 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusSubset1_type,type,
% 3.38/1.78 setminusSubset1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusSubset1,definition,
% 3.38/1.78 ( setminusSubset1
% 3.38/1.78 = ( ! [A: $i,B: $i] :
% 3.38/1.78 ( ( ( setminus @ A @ B )
% 3.38/1.78 = emptyset )
% 3.38/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(symdiff_type,type,
% 3.38/1.78 symdiff: $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffE_type,type,
% 3.38/1.78 symdiffE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffE,definition,
% 3.38/1.78 ( symdiffE
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.38/1.78 => ! [Xphi: $o] :
% 3.38/1.78 ( ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ B )
% 3.38/1.78 => Xphi ) )
% 3.38/1.78 => ( ( ~ ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ B )
% 3.38/1.78 => Xphi ) )
% 3.38/1.78 => Xphi ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffI1_type,type,
% 3.38/1.78 symdiffI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffI1,definition,
% 3.38/1.78 ( symdiffI1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffI2_type,type,
% 3.38/1.78 symdiffI2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffI2,definition,
% 3.38/1.78 ( symdiffI2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ~ ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ B )
% 3.38/1.78 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffIneg1_type,type,
% 3.38/1.78 symdiffIneg1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffIneg1,definition,
% 3.38/1.78 ( symdiffIneg1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ B )
% 3.38/1.78 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffIneg2_type,type,
% 3.38/1.78 symdiffIneg2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(symdiffIneg2,definition,
% 3.38/1.78 ( symdiffIneg2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ~ ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ B )
% 3.38/1.78 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(iskpair_type,type,
% 3.38/1.78 iskpair: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(secondinupair_type,type,
% 3.38/1.78 secondinupair: $o ).
% 3.38/1.78
% 3.38/1.78 thf(secondinupair,definition,
% 3.38/1.78 ( secondinupair
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairIL_type,type,
% 3.38/1.78 setukpairIL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairIL,definition,
% 3.38/1.78 ( setukpairIL
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairIR_type,type,
% 3.38/1.78 setukpairIR: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairIR,definition,
% 3.38/1.78 ( setukpairIR
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(kpairiskpair_type,type,
% 3.38/1.78 kpairiskpair: $o ).
% 3.38/1.78
% 3.38/1.78 thf(kpairiskpair,definition,
% 3.38/1.78 ( kpairiskpair
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(kpair_type,type,
% 3.38/1.78 kpair: $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(kpairp_type,type,
% 3.38/1.78 kpairp: $o ).
% 3.38/1.78
% 3.38/1.78 thf(kpairp,definition,
% 3.38/1.78 ( kpairp
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprod_type,type,
% 3.38/1.78 cartprod: $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(singletonsubset_type,type,
% 3.38/1.78 singletonsubset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(singletonsubset,definition,
% 3.38/1.78 ( singletonsubset
% 3.38/1.78 = ( ! [A: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(singletoninpowerset_type,type,
% 3.38/1.78 singletoninpowerset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(singletoninpowerset,definition,
% 3.38/1.78 ( singletoninpowerset
% 3.38/1.78 = ( ! [A: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(singletoninpowunion_type,type,
% 3.38/1.78 singletoninpowunion: $o ).
% 3.38/1.78
% 3.38/1.78 thf(singletoninpowunion,definition,
% 3.38/1.78 ( singletoninpowunion
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(upairset2E_type,type,
% 3.38/1.78 upairset2E: $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairset2E,definition,
% 3.38/1.78 ( upairset2E
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.38/1.78 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.38/1.78 => ( ( Xz = Xx )
% 3.38/1.78 | ( Xz = Xy ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(upairsubunion_type,type,
% 3.38/1.78 upairsubunion: $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairsubunion,definition,
% 3.38/1.78 ( upairsubunion
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(upairinpowunion_type,type,
% 3.38/1.78 upairinpowunion: $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairinpowunion,definition,
% 3.38/1.78 ( upairinpowunion
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ubforcartprodlem1_type,type,
% 3.38/1.78 ubforcartprodlem1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ubforcartprodlem1,definition,
% 3.38/1.78 ( ubforcartprodlem1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ubforcartprodlem2_type,type,
% 3.38/1.78 ubforcartprodlem2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ubforcartprodlem2,definition,
% 3.38/1.78 ( ubforcartprodlem2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ubforcartprodlem3_type,type,
% 3.38/1.78 ubforcartprodlem3: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ubforcartprodlem3,definition,
% 3.38/1.78 ( ubforcartprodlem3
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodpairin_type,type,
% 3.38/1.78 cartprodpairin: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodpairin,definition,
% 3.38/1.78 ( cartprodpairin
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodmempair1_type,type,
% 3.38/1.78 cartprodmempair1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodmempair1,definition,
% 3.38/1.78 ( cartprodmempair1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.38/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.38/1.78 => ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ? [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 & ( Xu
% 3.38/1.78 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodmempair_type,type,
% 3.38/1.78 cartprodmempair: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodmempair,definition,
% 3.38/1.78 ( cartprodmempair
% 3.38/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.38/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.38/1.78 => ( iskpair @ Xu ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setunionE2_type,type,
% 3.38/1.78 setunionE2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setunionE2,definition,
% 3.38/1.78 ( setunionE2
% 3.38/1.78 = ( ! [A: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( setunion @ A ) )
% 3.38/1.78 => ? [X: $i] :
% 3.38/1.78 ( ( in @ X @ A )
% 3.38/1.78 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setunionsingleton1_type,type,
% 3.38/1.78 setunionsingleton1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setunionsingleton1,definition,
% 3.38/1.78 ( setunionsingleton1
% 3.38/1.78 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setunionsingleton2_type,type,
% 3.38/1.78 setunionsingleton2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setunionsingleton2,definition,
% 3.38/1.78 ( setunionsingleton2
% 3.38/1.78 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setunionsingleton_type,type,
% 3.38/1.78 setunionsingleton: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setunionsingleton,definition,
% 3.38/1.78 ( setunionsingleton
% 3.38/1.78 = ( ! [Xx: $i] :
% 3.38/1.78 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.38/1.78 = Xx ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(singleton_type,type,
% 3.38/1.78 singleton: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(singletonprop_type,type,
% 3.38/1.78 singletonprop: $o ).
% 3.38/1.78
% 3.38/1.78 thf(singletonprop,definition,
% 3.38/1.78 ( singletonprop
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 => ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) )
% 3.38/1.78 => ( ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ( Xphi @ Xx ) )
% 3.38/1.78 => ( singleton
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ex1_type,type,
% 3.38/1.78 ex1: $i > ( $i > $o ) > $o ).
% 3.38/1.78
% 3.38/1.78 thf(ex1,definition,
% 3.38/1.78 ( ex1
% 3.38/1.78 = ( ^ [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( singleton
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ex1E1_type,type,
% 3.38/1.78 ex1E1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ex1E1,definition,
% 3.38/1.78 ( ex1E1
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ( ex1 @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ex1I_type,type,
% 3.38/1.78 ex1I: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ex1I,definition,
% 3.38/1.78 ( ex1I
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 => ( ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xy = Xx ) ) )
% 3.38/1.78 => ( ex1 @ A
% 3.38/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ex1I2_type,type,
% 3.38/1.78 ex1I2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ex1I2,definition,
% 3.38/1.78 ( ex1I2
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 => ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) )
% 3.38/1.78 => ( ? [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 & ( Xphi @ Xx ) )
% 3.38/1.78 => ( ex1 @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(singletonsuniq_type,type,
% 3.38/1.78 singletonsuniq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(singletonsuniq,definition,
% 3.38/1.78 ( singletonsuniq
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( ( setadjoin @ Xx @ emptyset )
% 3.38/1.78 = ( setadjoin @ Xy @ emptyset ) )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(atmost1p_type,type,
% 3.38/1.78 atmost1p: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(atleast2p_type,type,
% 3.38/1.78 atleast2p: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(atmost2p_type,type,
% 3.38/1.78 atmost2p: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairsetp_type,type,
% 3.38/1.78 upairsetp: $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjL1_type,type,
% 3.38/1.78 setukpairinjL1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjL1,definition,
% 3.38/1.78 ( setukpairinjL1
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.38/1.78 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.38/1.78 => ( Xx = Xz ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(kfstsingleton_type,type,
% 3.38/1.78 kfstsingleton: $o ).
% 3.38/1.78
% 3.38/1.78 thf(kfstsingleton,definition,
% 3.38/1.78 ( kfstsingleton
% 3.38/1.78 = ( ! [Xu: $i] :
% 3.38/1.78 ( ( iskpair @ Xu )
% 3.38/1.78 => ( singleton
% 3.38/1.78 @ ( dsetconstr @ ( setunion @ Xu )
% 3.38/1.78 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(theprop_type,type,
% 3.38/1.78 theprop: $o ).
% 3.38/1.78
% 3.38/1.78 thf(theprop,definition,
% 3.38/1.78 ( theprop
% 3.38/1.78 = ( ! [X: $i] :
% 3.38/1.78 ( ( singleton @ X )
% 3.38/1.78 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(kfst_type,type,
% 3.38/1.78 kfst: $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(kfstpairEq_type,type,
% 3.38/1.78 kfstpairEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(kfstpairEq,definition,
% 3.38/1.78 ( kfstpairEq
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.38/1.78 = Xx ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodfstin_type,type,
% 3.38/1.78 cartprodfstin: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodfstin,definition,
% 3.38/1.78 ( cartprodfstin
% 3.38/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.38/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.38/1.78 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjL2_type,type,
% 3.38/1.78 setukpairinjL2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjL2,definition,
% 3.38/1.78 ( setukpairinjL2
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.38/1.78 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.38/1.78 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.38/1.78 => ( Xx = Xz ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjL_type,type,
% 3.38/1.78 setukpairinjL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjL,definition,
% 3.38/1.78 ( setukpairinjL
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.38/1.78 ( ( ( kpair @ Xx @ Xy )
% 3.38/1.78 = ( kpair @ Xz @ Xu ) )
% 3.38/1.78 => ( Xx = Xz ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR11_type,type,
% 3.38/1.78 setukpairinjR11: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR11,definition,
% 3.38/1.78 ( setukpairinjR11
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx = Xy )
% 3.38/1.78 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.38/1.78 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR12_type,type,
% 3.38/1.78 setukpairinjR12: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR12,definition,
% 3.38/1.78 ( setukpairinjR12
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xx = Xy )
% 3.38/1.78 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.38/1.78 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR1_type,type,
% 3.38/1.78 setukpairinjR1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR1,definition,
% 3.38/1.78 ( setukpairinjR1
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.38/1.78 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.38/1.78 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.38/1.78 => ( ( Xz = Xu )
% 3.38/1.78 => ( Xy = Xu ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(upairequniteq_type,type,
% 3.38/1.78 upairequniteq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(upairequniteq,definition,
% 3.38/1.78 ( upairequniteq
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.38/1.78 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.38/1.78 = ( setadjoin @ Xz @ emptyset ) )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR2_type,type,
% 3.38/1.78 setukpairinjR2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR2,definition,
% 3.38/1.78 ( setukpairinjR2
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.38/1.78 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.38/1.78 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.38/1.78 => ( Xy = Xu ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR_type,type,
% 3.38/1.78 setukpairinjR: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setukpairinjR,definition,
% 3.38/1.78 ( setukpairinjR
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.38/1.78 ( ( ( kpair @ Xx @ Xy )
% 3.38/1.78 = ( kpair @ Xz @ Xu ) )
% 3.38/1.78 => ( Xy = Xu ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ksndsingleton_type,type,
% 3.38/1.78 ksndsingleton: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ksndsingleton,definition,
% 3.38/1.78 ( ksndsingleton
% 3.38/1.78 = ( ! [Xu: $i] :
% 3.38/1.78 ( ( iskpair @ Xu )
% 3.38/1.78 => ( singleton
% 3.38/1.78 @ ( dsetconstr @ ( setunion @ Xu )
% 3.38/1.78 @ ^ [Xx: $i] :
% 3.38/1.78 ( Xu
% 3.38/1.78 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ksnd_type,type,
% 3.38/1.78 ksnd: $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(ksndpairEq_type,type,
% 3.38/1.78 ksndpairEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ksndpairEq,definition,
% 3.38/1.78 ( ksndpairEq
% 3.38/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.38/1.78 = Xy ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(kpairsurjEq_type,type,
% 3.38/1.78 kpairsurjEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(kpairsurjEq,definition,
% 3.38/1.78 ( kpairsurjEq
% 3.38/1.78 = ( ! [Xu: $i] :
% 3.38/1.78 ( ( iskpair @ Xu )
% 3.38/1.78 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.38/1.78 = Xu ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodsndin_type,type,
% 3.38/1.78 cartprodsndin: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodsndin,definition,
% 3.38/1.78 ( cartprodsndin
% 3.38/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.38/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.38/1.78 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodpairmemEL_type,type,
% 3.38/1.78 cartprodpairmemEL: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodpairmemEL,definition,
% 3.38/1.78 ( cartprodpairmemEL
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodpairmemER_type,type,
% 3.38/1.78 cartprodpairmemER: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodpairmemER,definition,
% 3.38/1.78 ( cartprodpairmemER
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.38/1.78 => ( in @ Xy @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodmempaircEq_type,type,
% 3.38/1.78 cartprodmempaircEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodmempaircEq,definition,
% 3.38/1.78 ( cartprodmempaircEq
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( kpair @ Xx @ Xy )
% 3.38/1.78 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodfstpairEq_type,type,
% 3.38/1.78 cartprodfstpairEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodfstpairEq,definition,
% 3.38/1.78 ( cartprodfstpairEq
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.38/1.78 = Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodsndpairEq_type,type,
% 3.38/1.78 cartprodsndpairEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodsndpairEq,definition,
% 3.38/1.78 ( cartprodsndpairEq
% 3.38/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.38/1.78 = Xy ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodpairsurjEq_type,type,
% 3.38/1.78 cartprodpairsurjEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(cartprodpairsurjEq,definition,
% 3.38/1.78 ( cartprodpairsurjEq
% 3.38/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.38/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.38/1.78 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.38/1.78 = Xu ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(breln_type,type,
% 3.38/1.78 breln: $i > $i > $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(breln,definition,
% 3.38/1.78 ( breln
% 3.38/1.78 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstr_type,type,
% 3.38/1.78 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrI_type,type,
% 3.38/1.78 dpsetconstrI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrI,definition,
% 3.38/1.78 ( dpsetconstrI
% 3.38/1.78 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( Xphi @ Xx @ Xy )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ Xy )
% 3.38/1.78 @ ( dpsetconstr @ A @ B
% 3.38/1.78 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrSub_type,type,
% 3.38/1.78 dpsetconstrSub: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrSub,definition,
% 3.38/1.78 ( dpsetconstrSub
% 3.38/1.78 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.38/1.78 ( subset
% 3.38/1.78 @ ( dpsetconstr @ A @ B
% 3.38/1.78 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.38/1.78 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setOfPairsIsBReln_type,type,
% 3.38/1.78 setOfPairsIsBReln: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setOfPairsIsBReln,definition,
% 3.38/1.78 ( setOfPairsIsBReln
% 3.38/1.78 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.38/1.78 ( breln @ A @ B
% 3.38/1.78 @ ( dpsetconstr @ A @ B
% 3.38/1.78 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrERa_type,type,
% 3.38/1.78 dpsetconstrERa: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrERa,definition,
% 3.38/1.78 ( dpsetconstrERa
% 3.38/1.78 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.38/1.78 @ ( dpsetconstr @ A @ B
% 3.38/1.78 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.38/1.78 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrEL1_type,type,
% 3.38/1.78 dpsetconstrEL1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrEL1,definition,
% 3.38/1.78 ( dpsetconstrEL1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ ( kpair @ Xx @ Xy )
% 3.38/1.78 @ ( dpsetconstr @ A @ B
% 3.38/1.78 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrEL2_type,type,
% 3.38/1.78 dpsetconstrEL2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrEL2,definition,
% 3.38/1.78 ( dpsetconstrEL2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ ( kpair @ Xx @ Xy )
% 3.38/1.78 @ ( dpsetconstr @ A @ B
% 3.38/1.78 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.38/1.78 => ( in @ Xy @ B ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrER_type,type,
% 3.38/1.78 dpsetconstrER: $o ).
% 3.38/1.78
% 3.38/1.78 thf(dpsetconstrER,definition,
% 3.38/1.78 ( dpsetconstrER
% 3.38/1.78 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( in @ ( kpair @ Xx @ Xy )
% 3.38/1.78 @ ( dpsetconstr @ A @ B
% 3.38/1.78 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.38/1.78 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(func_type,type,
% 3.38/1.78 func: $i > $i > $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcSet_type,type,
% 3.38/1.78 funcSet: $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(funcImageSingleton_type,type,
% 3.38/1.78 funcImageSingleton: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcImageSingleton,definition,
% 3.38/1.78 ( funcImageSingleton
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( singleton
% 3.38/1.78 @ ( dsetconstr @ B
% 3.38/1.78 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(apProp_type,type,
% 3.38/1.78 apProp: $o ).
% 3.38/1.78
% 3.38/1.78 thf(apProp,definition,
% 3.38/1.78 ( apProp
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in
% 3.38/1.78 @ ( setunion
% 3.38/1.78 @ ( dsetconstr @ B
% 3.38/1.78 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.38/1.78 @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ap_type,type,
% 3.38/1.78 ap: $i > $i > $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(app_type,type,
% 3.38/1.78 app: $o ).
% 3.38/1.78
% 3.38/1.78 thf(app,definition,
% 3.38/1.78 ( app
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(infuncsetfunc_type,type,
% 3.38/1.78 infuncsetfunc: $o ).
% 3.38/1.78
% 3.38/1.78 thf(infuncsetfunc,definition,
% 3.38/1.78 ( infuncsetfunc
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.38/1.78 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ap2p_type,type,
% 3.38/1.78 ap2p: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ap2p,definition,
% 3.38/1.78 ( ap2p
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(funcinfuncset_type,type,
% 3.38/1.78 funcinfuncset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcinfuncset,definition,
% 3.38/1.78 ( funcinfuncset
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(lamProp_type,type,
% 3.38/1.78 lamProp: $o ).
% 3.38/1.78
% 3.38/1.78 thf(lamProp,definition,
% 3.38/1.78 ( lamProp
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.38/1.78 => ( func @ A @ B
% 3.38/1.78 @ ( dpsetconstr @ A @ B
% 3.38/1.78 @ ^ [Xx: $i,Xy: $i] :
% 3.38/1.78 ( ( Xf @ Xx )
% 3.38/1.78 = Xy ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(lam_type,type,
% 3.38/1.78 lam: $i > $i > ( $i > $i ) > $i ).
% 3.38/1.78
% 3.38/1.78 thf(lamp_type,type,
% 3.38/1.78 lamp: $o ).
% 3.38/1.78
% 3.38/1.78 thf(lamp,definition,
% 3.38/1.78 ( lamp
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.38/1.78 => ( func @ A @ B
% 3.38/1.78 @ ( lam @ A @ B
% 3.38/1.78 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(lam2p_type,type,
% 3.38/1.78 lam2p: $o ).
% 3.38/1.78
% 3.38/1.78 thf(lam2p,definition,
% 3.38/1.78 ( lam2p
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.38/1.78 => ( in
% 3.38/1.78 @ ( lam @ A @ B
% 3.38/1.78 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.38/1.78 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(brelnall1_type,type,
% 3.38/1.78 brelnall1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(brelnall1,definition,
% 3.38/1.78 ( brelnall1
% 3.38/1.78 = ( ! [A: $i,B: $i,R: $i] :
% 3.38/1.78 ( ( breln @ A @ B @ R )
% 3.38/1.78 => ! [Xphi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.38/1.78 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ R )
% 3.38/1.78 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(brelnall2_type,type,
% 3.38/1.78 brelnall2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(brelnall2,definition,
% 3.38/1.78 ( brelnall2
% 3.38/1.78 = ( ! [A: $i,B: $i,R: $i] :
% 3.38/1.78 ( ( breln @ A @ B @ R )
% 3.38/1.78 => ! [Xphi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.38/1.78 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ R )
% 3.38/1.78 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ex1E2_type,type,
% 3.38/1.78 ex1E2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ex1E2,definition,
% 3.38/1.78 ( ex1E2
% 3.38/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.38/1.78 ( ( ex1 @ A
% 3.38/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( ( Xphi @ Xx )
% 3.38/1.78 => ( ( Xphi @ Xy )
% 3.38/1.78 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(funcGraphProp1_type,type,
% 3.38/1.78 funcGraphProp1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcGraphProp1,definition,
% 3.38/1.78 ( funcGraphProp1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(funcGraphProp3_type,type,
% 3.38/1.78 funcGraphProp3: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcGraphProp3,definition,
% 3.38/1.78 ( funcGraphProp3
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(funcGraphProp2_type,type,
% 3.38/1.78 funcGraphProp2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcGraphProp2,definition,
% 3.38/1.78 ( funcGraphProp2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.38/1.78 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.38/1.78 = Xy ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(funcextLem_type,type,
% 3.38/1.78 funcextLem: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcextLem,definition,
% 3.38/1.78 ( funcextLem
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ! [Xg: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xg )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.38/1.78 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(funcGraphProp4_type,type,
% 3.38/1.78 funcGraphProp4: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcGraphProp4,definition,
% 3.38/1.78 ( funcGraphProp4
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.38/1.78 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.38/1.78 = Xy ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subbreln_type,type,
% 3.38/1.78 subbreln: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subbreln,definition,
% 3.38/1.78 ( subbreln
% 3.38/1.78 = ( ! [A: $i,B: $i,R: $i] :
% 3.38/1.78 ( ( breln @ A @ B @ R )
% 3.38/1.78 => ! [S: $i] :
% 3.38/1.78 ( ( breln @ A @ B @ S )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.38/1.78 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(eqbreln_type,type,
% 3.38/1.78 eqbreln: $o ).
% 3.38/1.78
% 3.38/1.78 thf(eqbreln,definition,
% 3.38/1.78 ( eqbreln
% 3.38/1.78 = ( ! [A: $i,B: $i,R: $i] :
% 3.38/1.78 ( ( breln @ A @ B @ R )
% 3.38/1.78 => ! [S: $i] :
% 3.38/1.78 ( ( breln @ A @ B @ S )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ B )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.38/1.78 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.38/1.78 => ( R = S ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(funcext_type,type,
% 3.38/1.78 funcext: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcext,definition,
% 3.38/1.78 ( funcext
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ! [Xg: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xg )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.38/1.78 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.38/1.78 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(funcext2_type,type,
% 3.38/1.78 funcext2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(funcext2,definition,
% 3.38/1.78 ( funcext2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.38/1.78 => ! [Xg: $i] :
% 3.38/1.78 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.38/1.78 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.38/1.78 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ap2apEq1_type,type,
% 3.38/1.78 ap2apEq1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ap2apEq1,definition,
% 3.38/1.78 ( ap2apEq1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.38/1.78 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ap2apEq2_type,type,
% 3.38/1.78 ap2apEq2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ap2apEq2,definition,
% 3.38/1.78 ( ap2apEq2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.38/1.78 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(beta1_type,type,
% 3.38/1.78 beta1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(beta1,definition,
% 3.38/1.78 ( beta1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( ap @ A @ B
% 3.38/1.78 @ ( lam @ A @ B
% 3.38/1.78 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.38/1.78 @ Xx )
% 3.38/1.78 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(eta1_type,type,
% 3.38/1.78 eta1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(eta1,definition,
% 3.38/1.78 ( eta1
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( func @ A @ B @ Xf )
% 3.38/1.78 => ( ( lam @ A @ B
% 3.38/1.78 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.38/1.78 = Xf ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(lam2lamEq_type,type,
% 3.38/1.78 lam2lamEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(lam2lamEq,definition,
% 3.38/1.78 ( lam2lamEq
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.38/1.78 => ( ( lam @ A @ B
% 3.38/1.78 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.38/1.78 = ( lam @ A @ B
% 3.38/1.78 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(beta2_type,type,
% 3.38/1.78 beta2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(beta2,definition,
% 3.38/1.78 ( beta2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( ap @ A @ B
% 3.38/1.78 @ ( lam @ A @ B
% 3.38/1.78 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.38/1.78 @ Xx )
% 3.38/1.78 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(eta2_type,type,
% 3.38/1.78 eta2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(eta2,definition,
% 3.38/1.78 ( eta2
% 3.38/1.78 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.38/1.78 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.38/1.78 => ( ( lam @ A @ B
% 3.38/1.78 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.38/1.78 = Xf ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(iffalseProp1_type,type,
% 3.38/1.78 iffalseProp1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(iffalseProp1,definition,
% 3.38/1.78 ( iffalseProp1
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( ~ Xphi
% 3.38/1.78 => ( in @ Xy
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xz: $i] :
% 3.38/1.78 ( ( Xphi
% 3.38/1.78 & ( Xz = Xx ) )
% 3.38/1.78 | ( ~ Xphi
% 3.38/1.78 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(iffalseProp2_type,type,
% 3.38/1.78 iffalseProp2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(iffalseProp2,definition,
% 3.38/1.78 ( iffalseProp2
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( ~ Xphi
% 3.38/1.78 => ( ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xz: $i] :
% 3.38/1.78 ( ( Xphi
% 3.38/1.78 & ( Xz = Xx ) )
% 3.38/1.78 | ( ~ Xphi
% 3.38/1.78 & ( Xz = Xy ) ) ) )
% 3.38/1.78 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(iftrueProp1_type,type,
% 3.38/1.78 iftrueProp1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(iftrueProp1,definition,
% 3.38/1.78 ( iftrueProp1
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( Xphi
% 3.38/1.78 => ( in @ Xx
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xz: $i] :
% 3.38/1.78 ( ( Xphi
% 3.38/1.78 & ( Xz = Xx ) )
% 3.38/1.78 | ( ~ Xphi
% 3.38/1.78 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(iftrueProp2_type,type,
% 3.38/1.78 iftrueProp2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(iftrueProp2,definition,
% 3.38/1.78 ( iftrueProp2
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( Xphi
% 3.38/1.78 => ( ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xz: $i] :
% 3.38/1.78 ( ( Xphi
% 3.38/1.78 & ( Xz = Xx ) )
% 3.38/1.78 | ( ~ Xphi
% 3.38/1.78 & ( Xz = Xy ) ) ) )
% 3.38/1.78 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(ifSingleton_type,type,
% 3.38/1.78 ifSingleton: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ifSingleton,definition,
% 3.38/1.78 ( ifSingleton
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( singleton
% 3.38/1.78 @ ( dsetconstr @ A
% 3.38/1.78 @ ^ [Xz: $i] :
% 3.38/1.78 ( ( Xphi
% 3.38/1.78 & ( Xz = Xx ) )
% 3.38/1.78 | ( ~ Xphi
% 3.38/1.78 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(if_type,type,
% 3.38/1.78 if: $i > $o > $i > $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(ifp_type,type,
% 3.38/1.78 ifp: $o ).
% 3.38/1.78
% 3.38/1.78 thf(ifp,definition,
% 3.38/1.78 ( ifp
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(theeq_type,type,
% 3.38/1.78 theeq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(theeq,definition,
% 3.38/1.78 ( theeq
% 3.38/1.78 = ( ! [X: $i] :
% 3.38/1.78 ( ( singleton @ X )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ X )
% 3.38/1.78 => ( ( setunion @ X )
% 3.38/1.78 = Xx ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(iftrue_type,type,
% 3.38/1.78 iftrue: $o ).
% 3.38/1.78
% 3.38/1.78 thf(iftrue,definition,
% 3.38/1.78 ( iftrue
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( Xphi
% 3.38/1.78 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.38/1.78 = Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(iffalse_type,type,
% 3.38/1.78 iffalse: $o ).
% 3.38/1.78
% 3.38/1.78 thf(iffalse,definition,
% 3.38/1.78 ( iffalse
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( ~ Xphi
% 3.38/1.78 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.38/1.78 = Xy ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(iftrueorfalse_type,type,
% 3.38/1.78 iftrueorfalse: $o ).
% 3.38/1.78
% 3.38/1.78 thf(iftrueorfalse,definition,
% 3.38/1.78 ( iftrueorfalse
% 3.38/1.78 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectT_lem_type,type,
% 3.38/1.78 binintersectT_lem: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectT_lem,definition,
% 3.38/1.78 ( binintersectT_lem
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionT_lem_type,type,
% 3.38/1.78 binunionT_lem: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionT_lem,definition,
% 3.38/1.78 ( binunionT_lem
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powersetT_lem_type,type,
% 3.38/1.78 powersetT_lem: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetT_lem,definition,
% 3.38/1.78 ( powersetT_lem
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setminusT_lem_type,type,
% 3.38/1.78 setminusT_lem: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setminusT_lem,definition,
% 3.38/1.78 ( setminusT_lem
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementT_lem_type,type,
% 3.38/1.78 complementT_lem: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementT_lem,definition,
% 3.38/1.78 ( complementT_lem
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(setextT_type,type,
% 3.38/1.78 setextT: $o ).
% 3.38/1.78
% 3.38/1.78 thf(setextT,definition,
% 3.38/1.78 ( setextT
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ X )
% 3.38/1.78 => ( in @ Xx @ Y ) ) )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ Y )
% 3.38/1.78 => ( in @ Xx @ X ) ) )
% 3.38/1.78 => ( X = Y ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(subsetTI_type,type,
% 3.38/1.78 subsetTI: $o ).
% 3.38/1.78
% 3.38/1.78 thf(subsetTI,definition,
% 3.38/1.78 ( subsetTI
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ X )
% 3.38/1.78 => ( in @ Xx @ Y ) ) )
% 3.38/1.78 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powersetTI1_type,type,
% 3.38/1.78 powersetTI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetTI1,definition,
% 3.38/1.78 ( powersetTI1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ X )
% 3.38/1.78 => ( in @ Xx @ Y ) ) )
% 3.38/1.78 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(powersetTE1_type,type,
% 3.38/1.78 powersetTE1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(powersetTE1,definition,
% 3.38/1.78 ( powersetTE1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ X @ ( powerset @ Y ) )
% 3.38/1.78 => ( ( in @ Xx @ X )
% 3.38/1.78 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementTI1_type,type,
% 3.38/1.78 complementTI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementTI1,definition,
% 3.38/1.78 ( complementTI1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ X )
% 3.38/1.78 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementTE1_type,type,
% 3.38/1.78 complementTE1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementTE1,definition,
% 3.38/1.78 ( complementTE1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.38/1.78 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectTELcontra_type,type,
% 3.38/1.78 binintersectTELcontra: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectTELcontra,definition,
% 3.38/1.78 ( binintersectTELcontra
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ X )
% 3.38/1.78 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectTERcontra_type,type,
% 3.38/1.78 binintersectTERcontra: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binintersectTERcontra,definition,
% 3.38/1.78 ( binintersectTERcontra
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ Y )
% 3.38/1.78 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(contrasubsetT_type,type,
% 3.38/1.78 contrasubsetT: $o ).
% 3.38/1.78
% 3.38/1.78 thf(contrasubsetT,definition,
% 3.38/1.78 ( contrasubsetT
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.38/1.78 => ( ( in @ Xx @ Y )
% 3.38/1.78 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(contrasubsetT1_type,type,
% 3.38/1.78 contrasubsetT1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(contrasubsetT1,definition,
% 3.38/1.78 ( contrasubsetT1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( subset @ X @ Y )
% 3.38/1.78 => ( ~ ( in @ Xx @ Y )
% 3.38/1.78 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(contrasubsetT2_type,type,
% 3.38/1.78 contrasubsetT2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(contrasubsetT2,definition,
% 3.38/1.78 ( contrasubsetT2
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ X @ Y )
% 3.38/1.78 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(contrasubsetT3_type,type,
% 3.38/1.78 contrasubsetT3: $o ).
% 3.38/1.78
% 3.38/1.78 thf(contrasubsetT3,definition,
% 3.38/1.78 ( contrasubsetT3
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.38/1.78 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementI1_type,type,
% 3.38/1.78 doubleComplementI1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementI1,definition,
% 3.38/1.78 ( doubleComplementI1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ X )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementE1_type,type,
% 3.38/1.78 doubleComplementE1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementE1,definition,
% 3.38/1.78 ( doubleComplementE1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.38/1.78 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementSub1_type,type,
% 3.38/1.78 doubleComplementSub1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementSub1,definition,
% 3.38/1.78 ( doubleComplementSub1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementSub2_type,type,
% 3.38/1.78 doubleComplementSub2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementSub2,definition,
% 3.38/1.78 ( doubleComplementSub2
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementEq_type,type,
% 3.38/1.78 doubleComplementEq: $o ).
% 3.38/1.78
% 3.38/1.78 thf(doubleComplementEq,definition,
% 3.38/1.78 ( doubleComplementEq
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ( X
% 3.38/1.78 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementTnotintersectT_type,type,
% 3.38/1.78 complementTnotintersectT: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementTnotintersectT,definition,
% 3.38/1.78 ( complementTnotintersectT
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.38/1.78 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementImpComplementIntersect_type,type,
% 3.38/1.78 complementImpComplementIntersect: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementImpComplementIntersect,definition,
% 3.38/1.78 ( complementImpComplementIntersect
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementSubsetComplementIntersect_type,type,
% 3.38/1.78 complementSubsetComplementIntersect: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementSubsetComplementIntersect,definition,
% 3.38/1.78 ( complementSubsetComplementIntersect
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementInPowersetComplementIntersect_type,type,
% 3.38/1.78 complementInPowersetComplementIntersect: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementInPowersetComplementIntersect,definition,
% 3.38/1.78 ( complementInPowersetComplementIntersect
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(contraSubsetComplement_type,type,
% 3.38/1.78 contraSubsetComplement: $o ).
% 3.38/1.78
% 3.38/1.78 thf(contraSubsetComplement,definition,
% 3.38/1.78 ( contraSubsetComplement
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ Y )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementTcontraSubset_type,type,
% 3.38/1.78 complementTcontraSubset: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementTcontraSubset,definition,
% 3.38/1.78 ( complementTcontraSubset
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.38/1.78 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionTILcontra_type,type,
% 3.38/1.78 binunionTILcontra: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionTILcontra,definition,
% 3.38/1.78 ( binunionTILcontra
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.38/1.78 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionTIRcontra_type,type,
% 3.38/1.78 binunionTIRcontra: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionTIRcontra,definition,
% 3.38/1.78 ( binunionTIRcontra
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.38/1.78 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(inIntersectImpInUnion_type,type,
% 3.38/1.78 inIntersectImpInUnion: $o ).
% 3.38/1.78
% 3.38/1.78 thf(inIntersectImpInUnion,definition,
% 3.38/1.78 ( inIntersectImpInUnion
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.38/1.78 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(inIntersectImpInUnion2_type,type,
% 3.38/1.78 inIntersectImpInUnion2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(inIntersectImpInUnion2,definition,
% 3.38/1.78 ( inIntersectImpInUnion2
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.38/1.78 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(inIntersectImpInIntersectUnions_type,type,
% 3.38/1.78 inIntersectImpInIntersectUnions: $o ).
% 3.38/1.78
% 3.38/1.78 thf(inIntersectImpInIntersectUnions,definition,
% 3.38/1.78 ( inIntersectImpInIntersectUnions
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.38/1.78 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(intersectInPowersetIntersectUnions_type,type,
% 3.38/1.78 intersectInPowersetIntersectUnions: $o ).
% 3.38/1.78
% 3.38/1.78 thf(intersectInPowersetIntersectUnions,definition,
% 3.38/1.78 ( intersectInPowersetIntersectUnions
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(inComplementUnionImpNotIn1_type,type,
% 3.38/1.78 inComplementUnionImpNotIn1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(inComplementUnionImpNotIn1,definition,
% 3.38/1.78 ( inComplementUnionImpNotIn1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.38/1.78 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(inComplementUnionImpInComplement1_type,type,
% 3.38/1.78 inComplementUnionImpInComplement1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(inComplementUnionImpInComplement1,definition,
% 3.38/1.78 ( inComplementUnionImpInComplement1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionTE_type,type,
% 3.38/1.78 binunionTE: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionTE,definition,
% 3.38/1.78 ( binunionTE
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xphi: $o,Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.38/1.78 => ( ( ( in @ Xx @ X )
% 3.38/1.78 => Xphi )
% 3.38/1.78 => ( ( ( in @ Xx @ Y )
% 3.38/1.78 => Xphi )
% 3.38/1.78 => Xphi ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(binunionTEcontra_type,type,
% 3.38/1.78 binunionTEcontra: $o ).
% 3.38/1.78
% 3.38/1.78 thf(binunionTEcontra,definition,
% 3.38/1.78 ( binunionTEcontra
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ~ ( in @ Xx @ X )
% 3.38/1.78 => ( ~ ( in @ Xx @ Y )
% 3.38/1.78 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2a1_type,type,
% 3.38/1.78 demorgan2a1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2a1,definition,
% 3.38/1.78 ( demorgan2a1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(complementUnionInPowersetComplement_type,type,
% 3.38/1.78 complementUnionInPowersetComplement: $o ).
% 3.38/1.78
% 3.38/1.78 thf(complementUnionInPowersetComplement,definition,
% 3.38/1.78 ( complementUnionInPowersetComplement
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2a2_type,type,
% 3.38/1.78 demorgan2a2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2a2,definition,
% 3.38/1.78 ( demorgan2a2
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan1a_type,type,
% 3.38/1.78 demorgan1a: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan1a,definition,
% 3.38/1.78 ( demorgan1a
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.38/1.78 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan1b_type,type,
% 3.38/1.78 demorgan1b: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan1b,definition,
% 3.38/1.78 ( demorgan1b
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan1_type,type,
% 3.38/1.78 demorgan1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan1,definition,
% 3.38/1.78 ( demorgan1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.38/1.78 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2a_type,type,
% 3.38/1.78 demorgan2a: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2a,definition,
% 3.38/1.78 ( demorgan2a
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.38/1.78 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2b2_type,type,
% 3.38/1.78 demorgan2b2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2b2,definition,
% 3.38/1.78 ( demorgan2b2
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.38/1.78 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2b_type,type,
% 3.38/1.78 demorgan2b: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2b,definition,
% 3.38/1.78 ( demorgan2b
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.38/1.78 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2_type,type,
% 3.38/1.78 demorgan2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(demorgan2,definition,
% 3.38/1.78 ( demorgan2
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.38/1.78 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule0_type,type,
% 3.38/1.78 woz13rule0: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule0,definition,
% 3.38/1.78 ( woz13rule0
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.38/1.78 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule1_type,type,
% 3.38/1.78 woz13rule1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule1,definition,
% 3.38/1.78 ( woz13rule1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ X @ Z )
% 3.38/1.78 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule2_type,type,
% 3.38/1.78 woz13rule2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule2,definition,
% 3.38/1.78 ( woz13rule2
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ Y @ Z )
% 3.38/1.78 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule3_type,type,
% 3.38/1.78 woz13rule3: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule3,definition,
% 3.38/1.78 ( woz13rule3
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ X @ Y )
% 3.38/1.78 => ( ( subset @ X @ Z )
% 3.38/1.78 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule4_type,type,
% 3.38/1.78 woz13rule4: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz13rule4,definition,
% 3.38/1.78 ( woz13rule4
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ! [W: $i] :
% 3.38/1.78 ( ( in @ W @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ X @ Z )
% 3.38/1.78 => ( ( subset @ Y @ W )
% 3.38/1.78 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_1_type,type,
% 3.38/1.78 woz1_1: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_1,definition,
% 3.38/1.78 ( woz1_1
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_2_type,type,
% 3.38/1.78 woz1_2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_2,definition,
% 3.38/1.78 ( woz1_2
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ! [W: $i] :
% 3.38/1.78 ( ( in @ W @ ( powerset @ A ) )
% 3.38/1.78 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.38/1.78 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_3_type,type,
% 3.38/1.78 woz1_3: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_3,definition,
% 3.38/1.78 ( woz1_3
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ! [Z: $i] :
% 3.38/1.78 ( ( in @ Z @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_4_type,type,
% 3.38/1.78 woz1_4: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_4,definition,
% 3.38/1.78 ( woz1_4
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.38/1.78 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_5_type,type,
% 3.38/1.78 woz1_5: $o ).
% 3.38/1.78
% 3.38/1.78 thf(woz1_5,definition,
% 3.38/1.78 ( woz1_5
% 3.38/1.78 = ( ! [A: $i,X: $i] :
% 3.38/1.78 ( ( in @ X @ ( powerset @ A ) )
% 3.38/1.78 => ! [Y: $i] :
% 3.38/1.78 ( ( in @ Y @ ( powerset @ A ) )
% 3.38/1.78 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(breln1_type,type,
% 3.38/1.78 breln1: $i > $i > $o ).
% 3.38/1.78
% 3.38/1.78 thf(breln1,definition,
% 3.38/1.78 ( breln1
% 3.38/1.78 = ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(breln1all2_type,type,
% 3.38/1.78 breln1all2: $o ).
% 3.38/1.78
% 3.38/1.78 thf(breln1all2,definition,
% 3.38/1.78 ( breln1all2
% 3.38/1.78 = ( ! [A: $i,R: $i] :
% 3.38/1.78 ( ( breln1 @ A @ R )
% 3.38/1.78 => ! [Xphi: $i > $o] :
% 3.38/1.78 ( ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ A )
% 3.38/1.78 => ! [Xy: $i] :
% 3.38/1.78 ( ( in @ Xy @ A )
% 3.38/1.78 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.38/1.78 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.38/1.78 => ! [Xx: $i] :
% 3.38/1.78 ( ( in @ Xx @ R )
% 3.38/1.78 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(breln1Set_type,type,
% 3.38/1.78 breln1Set: $i > $i ).
% 3.38/1.78
% 3.38/1.78 thf(breln1Set,definition,
% 3.38/1.78 ( breln1Set
% 3.38/1.78 = ( ^ [A: $i] :
% 3.38/1.78 ( dsetconstr @ ( powerset @ ( cartprod @ A @ A ) )
% 3.38/1.78 @ ^ [R: $i] : ( breln1 @ A @ R ) ) ) ) ).
% 3.38/1.78
% 3.38/1.78 thf(breln1SetBreln1,conjecture,
% 3.38/1.78 ( setextAx
% 3.38/1.78 => ( emptysetAx
% 3.38/1.78 => ( setadjoinAx
% 3.38/1.78 => ( powersetAx
% 3.38/1.78 => ( setunionAx
% 3.38/1.78 => ( omega0Ax
% 3.38/1.78 => ( omegaSAx
% 3.38/1.78 => ( omegaIndAx
% 3.38/1.78 => ( replAx
% 3.38/1.78 => ( foundationAx
% 3.38/1.78 => ( wellorderingAx
% 3.38/1.78 => ( descrp
% 3.38/1.78 => ( dsetconstrI
% 3.38/1.78 => ( dsetconstrEL
% 3.38/1.78 => ( dsetconstrER
% 3.38/1.78 => ( exuE1
% 3.38/1.78 => ( prop2setE
% 3.38/1.78 => ( emptysetE
% 3.38/1.78 => ( emptysetimpfalse
% 3.38/1.78 => ( notinemptyset
% 3.38/1.78 => ( exuE3e
% 3.38/1.78 => ( setext
% 3.38/1.78 => ( emptyI
% 3.38/1.78 => ( noeltsimpempty
% 3.38/1.78 => ( setbeta
% 3.38/1.78 => ( nonemptyE1
% 3.38/1.78 => ( nonemptyI
% 3.38/1.78 => ( nonemptyI1
% 3.38/1.78 => ( setadjoinIL
% 3.38/1.78 => ( emptyinunitempty
% 3.38/1.78 => ( setadjoinIR
% 3.38/1.78 => ( setadjoinE
% 3.38/1.78 => ( setadjoinOr
% 3.38/1.78 => ( setoftrueEq
% 3.38/1.78 => ( powersetI
% 3.38/1.78 => ( emptyinPowerset
% 3.38/1.78 => ( emptyInPowerset
% 3.38/1.78 => ( powersetE
% 3.38/1.78 => ( setunionI
% 3.38/1.78 => ( setunionE
% 3.38/1.78 => ( subPowSU
% 3.38/1.78 => ( exuE2
% 3.38/1.78 => ( nonemptyImpWitness
% 3.38/1.78 => ( uniqinunit
% 3.38/1.78 => ( notinsingleton
% 3.38/1.78 => ( eqinunit
% 3.38/1.78 => ( singletonsswitch
% 3.38/1.78 => ( upairsetE
% 3.38/1.78 => ( upairsetIL
% 3.38/1.78 => ( upairsetIR
% 3.38/1.78 => ( emptyE1
% 3.38/1.78 => ( vacuousDall
% 3.38/1.78 => ( quantDeMorgan1
% 3.38/1.78 => ( quantDeMorgan2
% 3.38/1.78 => ( quantDeMorgan3
% 3.38/1.78 => ( quantDeMorgan4
% 3.38/1.78 => ( prop2setI
% 3.38/1.78 => ( prop2set2propI
% 3.38/1.78 => ( notdexE
% 3.38/1.78 => ( notdallE
% 3.38/1.78 => ( exuI1
% 3.38/1.78 => ( exuI3
% 3.38/1.78 => ( exuI2
% 3.38/1.78 => ( inCongP
% 3.38/1.78 => ( in__Cong
% 3.38/1.78 => ( exuE3u
% 3.38/1.78 => ( exu__Cong
% 3.38/1.78 => ( emptyset__Cong
% 3.38/1.78 => ( setadjoin__Cong
% 3.38/1.78 => ( powerset__Cong
% 3.38/1.78 => ( setunion__Cong
% 3.38/1.78 => ( omega__Cong
% 3.38/1.78 => ( exuEu
% 3.38/1.78 => ( descr__Cong
% 3.38/1.78 => ( dsetconstr__Cong
% 3.38/1.78 => ( subsetI1
% 3.38/1.78 => ( eqimpsubset2
% 3.38/1.78 => ( eqimpsubset1
% 3.38/1.78 => ( subsetI2
% 3.38/1.78 => ( emptysetsubset
% 3.38/1.78 => ( subsetE
% 3.38/1.78 => ( subsetE2
% 3.38/1.78 => ( notsubsetI
% 3.38/1.78 => ( notequalI1
% 3.38/1.78 => ( notequalI2
% 3.38/1.78 => ( subsetRefl
% 3.38/1.78 => ( subsetTrans
% 3.38/1.78 => ( setadjoinSub
% 3.38/1.78 => ( setadjoinSub2
% 3.38/1.78 => ( subset2powerset
% 3.38/1.78 => ( setextsub
% 3.38/1.78 => ( subsetemptysetimpeq
% 3.38/1.78 => ( powersetI1
% 3.38/1.78 => ( powersetE1
% 3.38/1.78 => ( inPowerset
% 3.38/1.78 => ( powersetsubset
% 3.38/1.78 => ( sepInPowerset
% 3.38/1.78 => ( sepSubset
% 3.38/1.78 => ( binunionIL
% 3.38/1.78 => ( upairset2IR
% 3.38/1.78 => ( binunionIR
% 3.38/1.78 => ( binunionEcases
% 3.38/1.78 => ( binunionE
% 3.38/1.78 => ( binunionLsub
% 3.38/1.78 => ( binunionRsub
% 3.38/1.78 => ( binintersectI
% 3.38/1.78 => ( binintersectSubset5
% 3.38/1.78 => ( binintersectEL
% 3.38/1.79 => ( binintersectLsub
% 3.38/1.79 => ( binintersectSubset2
% 3.38/1.79 => ( binintersectSubset3
% 3.38/1.79 => ( binintersectER
% 3.38/1.79 => ( disjointsetsI1
% 3.38/1.79 => ( binintersectRsub
% 3.38/1.79 => ( binintersectSubset4
% 3.38/1.79 => ( binintersectSubset1
% 3.38/1.79 => ( bs114d
% 3.38/1.79 => ( setminusI
% 3.38/1.79 => ( setminusEL
% 3.38/1.79 => ( setminusER
% 3.38/1.79 => ( setminusSubset2
% 3.38/1.79 => ( setminusERneg
% 3.38/1.79 => ( setminusELneg
% 3.38/1.79 => ( setminusILneg
% 3.38/1.79 => ( setminusIRneg
% 3.38/1.79 => ( setminusLsub
% 3.38/1.79 => ( setminusSubset1
% 3.38/1.79 => ( symdiffE
% 3.38/1.79 => ( symdiffI1
% 3.38/1.79 => ( symdiffI2
% 3.38/1.79 => ( symdiffIneg1
% 3.38/1.79 => ( symdiffIneg2
% 3.38/1.79 => ( secondinupair
% 3.38/1.79 => ( setukpairIL
% 3.38/1.79 => ( setukpairIR
% 3.38/1.79 => ( kpairiskpair
% 3.38/1.79 => ( kpairp
% 3.38/1.79 => ( singletonsubset
% 3.38/1.79 => ( singletoninpowerset
% 3.38/1.79 => ( singletoninpowunion
% 3.38/1.79 => ( upairset2E
% 3.38/1.79 => ( upairsubunion
% 3.38/1.79 => ( upairinpowunion
% 3.38/1.79 => ( ubforcartprodlem1
% 3.38/1.79 => ( ubforcartprodlem2
% 3.38/1.79 => ( ubforcartprodlem3
% 3.38/1.79 => ( cartprodpairin
% 3.38/1.79 => ( cartprodmempair1
% 3.38/1.79 => ( cartprodmempair
% 3.38/1.79 => ( setunionE2
% 3.38/1.79 => ( setunionsingleton1
% 3.38/1.79 => ( setunionsingleton2
% 3.38/1.79 => ( setunionsingleton
% 3.38/1.79 => ( singletonprop
% 3.38/1.79 => ( ex1E1
% 3.38/1.79 => ( ex1I
% 3.38/1.79 => ( ex1I2
% 3.38/1.79 => ( singletonsuniq
% 3.38/1.79 => ( setukpairinjL1
% 3.38/1.79 => ( kfstsingleton
% 3.38/1.79 => ( theprop
% 3.38/1.79 => ( kfstpairEq
% 3.38/1.79 => ( cartprodfstin
% 3.38/1.79 => ( setukpairinjL2
% 3.38/1.79 => ( setukpairinjL
% 3.38/1.79 => ( setukpairinjR11
% 3.38/1.79 => ( setukpairinjR12
% 3.38/1.79 => ( setukpairinjR1
% 3.38/1.79 => ( upairequniteq
% 3.38/1.79 => ( setukpairinjR2
% 3.38/1.79 => ( setukpairinjR
% 3.38/1.79 => ( ksndsingleton
% 3.38/1.79 => ( ksndpairEq
% 3.38/1.79 => ( kpairsurjEq
% 3.38/1.79 => ( cartprodsndin
% 3.38/1.79 => ( cartprodpairmemEL
% 3.38/1.79 => ( cartprodpairmemER
% 3.38/1.79 => ( cartprodmempaircEq
% 3.38/1.79 => ( cartprodfstpairEq
% 3.38/1.79 => ( cartprodsndpairEq
% 3.38/1.79 => ( cartprodpairsurjEq
% 3.38/1.79 => ( dpsetconstrI
% 3.38/1.79 => ( dpsetconstrSub
% 3.38/1.79 => ( setOfPairsIsBReln
% 3.38/1.79 => ( dpsetconstrERa
% 3.38/1.79 => ( dpsetconstrEL1
% 3.38/1.79 => ( dpsetconstrEL2
% 3.38/1.79 => ( dpsetconstrER
% 3.38/1.79 => ( funcImageSingleton
% 3.38/1.79 => ( apProp
% 3.38/1.79 => ( app
% 3.38/1.79 => ( infuncsetfunc
% 3.38/1.79 => ( ap2p
% 3.38/1.79 => ( funcinfuncset
% 3.38/1.79 => ( lamProp
% 3.38/1.79 => ( lamp
% 3.38/1.79 => ( lam2p
% 3.38/1.79 => ( brelnall1
% 3.38/1.79 => ( brelnall2
% 3.38/1.79 => ( ex1E2
% 3.38/1.79 => ( funcGraphProp1
% 3.38/1.79 => ( funcGraphProp3
% 3.38/1.79 => ( funcGraphProp2
% 3.38/1.79 => ( funcextLem
% 3.38/1.79 => ( funcGraphProp4
% 3.38/1.79 => ( subbreln
% 3.38/1.79 => ( eqbreln
% 3.38/1.79 => ( funcext
% 3.38/1.79 => ( funcext2
% 3.38/1.79 => ( ap2apEq1
% 3.38/1.79 => ( ap2apEq2
% 3.38/1.79 => ( beta1
% 3.38/1.79 => ( eta1
% 3.38/1.79 => ( lam2lamEq
% 3.38/1.79 => ( beta2
% 3.38/1.79 => ( eta2
% 3.38/1.79 => ( iffalseProp1
% 3.38/1.79 => ( iffalseProp2
% 3.38/1.79 => ( iftrueProp1
% 3.38/1.79 => ( iftrueProp2
% 3.38/1.79 => ( ifSingleton
% 3.38/1.79 => ( ifp
% 3.38/1.79 => ( theeq
% 3.38/1.79 => ( iftrue
% 3.38/1.79 => ( iffalse
% 3.38/1.79 => ( iftrueorfalse
% 3.38/1.79 => ( binintersectT_lem
% 3.38/1.79 => ( binunionT_lem
% 3.38/1.79 => ( powersetT_lem
% 3.38/1.79 => ( setminusT_lem
% 3.38/1.79 => ( complementT_lem
% 3.38/1.79 => ( setextT
% 3.38/1.79 => ( subsetTI
% 3.38/1.79 => ( powersetTI1
% 3.38/1.79 => ( powersetTE1
% 3.38/1.79 => ( complementTI1
% 3.38/1.79 => ( complementTE1
% 3.38/1.79 => ( binintersectTELcontra
% 3.38/1.79 => ( binintersectTERcontra
% 3.38/1.79 => ( contrasubsetT
% 3.38/1.79 => ( contrasubsetT1
% 3.38/1.79 => ( contrasubsetT2
% 3.38/1.79 => ( contrasubsetT3
% 3.38/1.79 => ( doubleComplementI1
% 3.38/1.79 => ( doubleComplementE1
% 3.38/1.79 => ( doubleComplementSub1
% 3.38/1.79 => ( doubleComplementSub2
% 3.38/1.79 => ( doubleComplementEq
% 3.38/1.79 => ( complementTnotintersectT
% 3.38/1.79 => ( complementImpComplementIntersect
% 3.38/1.79 => ( complementSubsetComplementIntersect
% 3.38/1.79 => ( complementInPowersetComplementIntersect
% 3.38/1.79 => ( contraSubsetComplement
% 3.38/1.79 => ( complementTcontraSubset
% 3.38/1.79 => ( binunionTILcontra
% 3.38/1.79 => ( binunionTIRcontra
% 3.38/1.79 => ( inIntersectImpInUnion
% 3.38/1.79 => ( inIntersectImpInUnion2
% 3.38/1.79 => ( inIntersectImpInIntersectUnions
% 3.38/1.79 => ( intersectInPowersetIntersectUnions
% 3.38/1.79 => ( inComplementUnionImpNotIn1
% 3.38/1.79 => ( inComplementUnionImpInComplement1
% 3.38/1.79 => ( binunionTE
% 3.38/1.79 => ( binunionTEcontra
% 3.38/1.79 => ( demorgan2a1
% 3.38/1.79 => ( complementUnionInPowersetComplement
% 3.38/1.79 => ( demorgan2a2
% 3.38/1.79 => ( demorgan1a
% 3.38/1.79 => ( demorgan1b
% 3.38/1.79 => ( demorgan1
% 3.38/1.79 => ( demorgan2a
% 3.38/1.79 => ( demorgan2b2
% 3.38/1.79 => ( demorgan2b
% 3.38/1.79 => ( demorgan2
% 3.38/1.79 => ( woz13rule0
% 3.38/1.79 => ( woz13rule1
% 3.38/1.79 => ( woz13rule2
% 3.38/1.79 => ( woz13rule3
% 3.38/1.79 => ( woz13rule4
% 3.38/1.79 => ( woz1_1
% 3.38/1.79 => ( woz1_2
% 3.38/1.79 => ( woz1_3
% 3.38/1.79 => ( woz1_4
% 3.38/1.79 => ( woz1_5
% 3.38/1.79 => ( breln1all2
% 3.38/1.79 => ! [A: $i,R: $i] :
% 3.38/1.79 ( ( in @ R @ ( breln1Set @ A ) )
% 3.38/1.79 => ( breln1 @ A @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.38/1.79 ------------------------
% 3.47/1.83 % (16452)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/18Mi)
% 3.47/1.83 % (16450)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_16347 for (2999ds/2Mi)
% 3.47/1.83 % (16448)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_16347 for (2999ds/27Mi)
% 3.47/1.83 % (16446)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/183Mi)
% 3.47/1.83 % (16447)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_16347 for (2999ds/4Mi)
% 3.47/1.83 % (16449)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/2Mi)
% 3.47/1.83 % (16451)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_16347 for (2999ds/275Mi)
% 3.47/1.83 % (16450)Instruction limit reached!
% 3.47/1.83 % (16450)------------------------------
% 3.47/1.83 % (16450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.83 % (16450)Termination reason: Unknown
% 3.47/1.83 % (16450)Termination phase: shuffling
% 3.47/1.83
% 3.47/1.83 % (16450)Memory used [KB]: 1535
% 3.47/1.83 % (16450)Time elapsed: 0.003 s
% 3.47/1.83 % (16450)Instructions burned: 2 (million)
% 3.47/1.83 % (16450)------------------------------
% 3.47/1.83 % (16450)------------------------------
% 3.47/1.83 % (16449)Instruction limit reached!
% 3.47/1.83 % (16449)------------------------------
% 3.47/1.83 % (16449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.83 % (16449)Termination reason: Unknown
% 3.47/1.83 % (16449)Termination phase: shuffling
% 3.47/1.83
% 3.47/1.83 % (16449)Memory used [KB]: 1535
% 3.47/1.83 % (16449)Time elapsed: 0.003 s
% 3.47/1.83 % (16449)Instructions burned: 3 (million)
% 3.47/1.83 % (16449)------------------------------
% 3.47/1.83 % (16449)------------------------------
% 3.47/1.83 % (16447)Instruction limit reached!
% 3.47/1.83 % (16447)------------------------------
% 3.47/1.83 % (16447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.83 % (16447)Termination reason: Unknown
% 3.47/1.83 % (16447)Termination phase: shuffling
% 3.47/1.83
% 3.47/1.83 % (16447)Memory used [KB]: 1535
% 3.47/1.83 % (16447)Time elapsed: 0.004 s
% 3.47/1.83 % (16447)Instructions burned: 6 (million)
% 3.47/1.83 % (16447)------------------------------
% 3.47/1.83 % (16447)------------------------------
% 3.47/1.84 % (16452)Instruction limit reached!
% 3.47/1.84 % (16452)------------------------------
% 3.47/1.84 % (16452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.84 % (16452)Termination reason: Unknown
% 3.47/1.84 % (16452)Termination phase: shuffling
% 3.47/1.84
% 3.47/1.84 % (16452)Memory used [KB]: 1791
% 3.47/1.84 % (16452)Time elapsed: 0.010 s
% 3.47/1.84 % (16452)Instructions burned: 18 (million)
% 3.47/1.84 % (16452)------------------------------
% 3.47/1.84 % (16452)------------------------------
% 3.47/1.84 % (16448)Instruction limit reached!
% 3.47/1.84 % (16448)------------------------------
% 3.47/1.84 % (16448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.84 % (16448)Termination reason: Unknown
% 3.47/1.84 % (16448)Termination phase: shuffling
% 3.47/1.84
% 3.47/1.84 % (16448)Memory used [KB]: 2046
% 3.47/1.84 % (16448)Time elapsed: 0.015 s
% 3.47/1.84 % (16448)Instructions burned: 28 (million)
% 3.47/1.84 % (16448)------------------------------
% 3.47/1.84 % (16448)------------------------------
% 3.47/1.85 % (16453)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/3Mi)
% 3.47/1.85 % (16454)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_16347 for (2999ds/37Mi)
% 3.47/1.85 % (16455)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_16347 for (2999ds/15Mi)
% 3.47/1.85 % (16453)Instruction limit reached!
% 3.47/1.85 % (16453)------------------------------
% 3.47/1.85 % (16453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.85 % (16453)Termination reason: Unknown
% 3.47/1.85 % (16453)Termination phase: shuffling
% 3.47/1.85
% 3.47/1.85 % (16453)Memory used [KB]: 1535
% 3.47/1.85 % (16453)Time elapsed: 0.004 s
% 3.47/1.85 % (16453)Instructions burned: 5 (million)
% 3.47/1.85 % (16453)------------------------------
% 3.47/1.85 % (16453)------------------------------
% 3.47/1.85 % (16456)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_16347 for (2999ds/3Mi)
% 3.47/1.86 % (16456)Instruction limit reached!
% 3.47/1.86 % (16456)------------------------------
% 3.47/1.86 % (16456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.86 % (16456)Termination reason: Unknown
% 3.47/1.86 % (16456)Termination phase: shuffling
% 3.47/1.86
% 3.47/1.86 % (16456)Memory used [KB]: 1535
% 3.47/1.86 % (16456)Time elapsed: 0.004 s
% 3.47/1.86 % (16456)Instructions burned: 4 (million)
% 3.47/1.86 % (16456)------------------------------
% 3.47/1.86 % (16456)------------------------------
% 3.47/1.86 % (16455)Instruction limit reached!
% 3.47/1.86 % (16455)------------------------------
% 3.47/1.86 % (16455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.86 % (16455)Termination reason: Unknown
% 3.47/1.86 % (16455)Termination phase: shuffling
% 3.47/1.86
% 3.47/1.86 % (16455)Memory used [KB]: 1791
% 3.47/1.86 % (16455)Time elapsed: 0.009 s
% 3.47/1.86 % (16455)Instructions burned: 15 (million)
% 3.47/1.86 % (16455)------------------------------
% 3.47/1.86 % (16455)------------------------------
% 3.47/1.86 % (16457)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_16347 for (2999ds/1041Mi)
% 3.47/1.86 % (16454)Instruction limit reached!
% 3.47/1.86 % (16454)------------------------------
% 3.47/1.86 % (16454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.86 % (16454)Termination reason: Unknown
% 3.47/1.87 % (16454)Termination phase: shuffling
% 3.47/1.87
% 3.47/1.87 % (16454)Memory used [KB]: 2302
% 3.47/1.87 % (16454)Time elapsed: 0.019 s
% 3.47/1.87 % (16454)Instructions burned: 38 (million)
% 3.47/1.87 % (16454)------------------------------
% 3.47/1.87 % (16454)------------------------------
% 3.47/1.87 % (16458)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/7Mi)
% 3.47/1.87 % (16458)Instruction limit reached!
% 3.47/1.87 % (16458)------------------------------
% 3.47/1.87 % (16458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.87 % (16458)Termination reason: Unknown
% 3.47/1.87 % (16458)Termination phase: shuffling
% 3.47/1.87
% 3.47/1.87 % (16458)Memory used [KB]: 1535
% 3.47/1.87 % (16458)Time elapsed: 0.005 s
% 3.47/1.87 % (16458)Instructions burned: 7 (million)
% 3.47/1.87 % (16458)------------------------------
% 3.47/1.87 % (16458)------------------------------
% 3.47/1.87 % (16459)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/16Mi)
% 3.47/1.87 % (16460)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_16347 for (2999ds/3Mi)
% 3.47/1.87 % (16460)Instruction limit reached!
% 3.47/1.87 % (16460)------------------------------
% 3.47/1.87 % (16460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.87 % (16460)Termination reason: Unknown
% 3.47/1.87 % (16460)Termination phase: shuffling
% 3.47/1.87
% 3.47/1.87 % (16460)Memory used [KB]: 1535
% 3.47/1.87 % (16460)Time elapsed: 0.003 s
% 3.47/1.87 % (16460)Instructions burned: 3 (million)
% 3.47/1.87 % (16460)------------------------------
% 3.47/1.87 % (16460)------------------------------
% 3.47/1.88 % (16459)Instruction limit reached!
% 3.47/1.88 % (16459)------------------------------
% 3.47/1.88 % (16459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.88 % (16459)Termination reason: Unknown
% 3.47/1.88 % (16459)Termination phase: shuffling
% 3.47/1.88
% 3.47/1.88 % (16459)Memory used [KB]: 1791
% 3.47/1.88 % (16459)Time elapsed: 0.010 s
% 3.47/1.88 % (16459)Instructions burned: 18 (million)
% 3.47/1.88 % (16459)------------------------------
% 3.47/1.88 % (16459)------------------------------
% 3.47/1.88 % (16461)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_16347 for (2999ds/3Mi)
% 3.47/1.88 % (16461)Instruction limit reached!
% 3.47/1.88 % (16461)------------------------------
% 3.47/1.88 % (16461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.88 % (16461)Termination reason: Unknown
% 3.47/1.88 % (16461)Termination phase: shuffling
% 3.47/1.88
% 3.47/1.88 % (16461)Memory used [KB]: 1535
% 3.47/1.88 % (16461)Time elapsed: 0.003 s
% 3.47/1.88 % (16461)Instructions burned: 3 (million)
% 3.47/1.88 % (16461)------------------------------
% 3.47/1.88 % (16461)------------------------------
% 3.47/1.89 % (16462)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/7Mi)
% 3.47/1.89 % (16462)Instruction limit reached!
% 3.47/1.89 % (16462)------------------------------
% 3.47/1.89 % (16462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.89 % (16462)Termination reason: Unknown
% 3.47/1.89 % (16462)Termination phase: shuffling
% 3.47/1.89
% 3.47/1.89 % (16462)Memory used [KB]: 1663
% 3.47/1.89 % (16462)Time elapsed: 0.005 s
% 3.47/1.89 % (16462)Instructions burned: 7 (million)
% 3.47/1.89 % (16462)------------------------------
% 3.47/1.89 % (16462)------------------------------
% 3.47/1.89 % (16463)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/3Mi)
% 3.47/1.89 % (16463)Instruction limit reached!
% 3.47/1.89 % (16463)------------------------------
% 3.47/1.89 % (16463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.89 % (16463)Termination reason: Unknown
% 3.47/1.89 % (16463)Termination phase: shuffling
% 3.47/1.89
% 3.47/1.89 % (16463)Memory used [KB]: 1535
% 3.47/1.89 % (16463)Time elapsed: 0.004 s
% 3.47/1.89 % (16463)Instructions burned: 4 (million)
% 3.47/1.89 % (16463)------------------------------
% 3.47/1.89 % (16463)------------------------------
% 3.47/1.89 % (16464)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/4Mi)
% 3.47/1.90 % (16464)Instruction limit reached!
% 3.47/1.90 % (16464)------------------------------
% 3.47/1.90 % (16464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.90 % (16464)Termination reason: Unknown
% 3.47/1.90 % (16464)Termination phase: shuffling
% 3.47/1.90
% 3.47/1.90 % (16464)Memory used [KB]: 1535
% 3.47/1.90 % (16464)Time elapsed: 0.004 s
% 3.47/1.90 % (16464)Instructions burned: 4 (million)
% 3.47/1.90 % (16464)------------------------------
% 3.47/1.90 % (16464)------------------------------
% 3.47/1.90 % (16465)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/18Mi)
% 3.47/1.91 % (16466)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/710Mi)
% 3.47/1.91 % (16465)Instruction limit reached!
% 3.47/1.91 % (16465)------------------------------
% 3.47/1.91 % (16465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.91 % (16465)Termination reason: Unknown
% 3.47/1.91 % (16465)Termination phase: shuffling
% 3.47/1.91
% 3.47/1.91 % (16465)Memory used [KB]: 1791
% 3.47/1.91 % (16465)Time elapsed: 0.010 s
% 3.47/1.91 % (16465)Instructions burned: 18 (million)
% 3.47/1.91 % (16465)------------------------------
% 3.47/1.91 % (16465)------------------------------
% 3.47/1.91 % (16467)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16347 for (2999ds/6Mi)
% 3.47/1.91 % (16446)Instruction limit reached!
% 3.47/1.91 % (16446)------------------------------
% 3.47/1.91 % (16446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.91 % (16446)Termination reason: Unknown
% 3.47/1.91 % (16446)Termination phase: Saturation
% 3.47/1.91
% 3.47/1.91 % (16446)Memory used [KB]: 7419
% 3.47/1.91 % (16446)Time elapsed: 0.082 s
% 3.47/1.91 % (16446)Instructions burned: 183 (million)
% 3.47/1.91 % (16446)------------------------------
% 3.47/1.91 % (16446)------------------------------
% 3.47/1.91 % (16467)Instruction limit reached!
% 3.47/1.91 % (16467)------------------------------
% 3.47/1.91 % (16467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.91 % (16467)Termination reason: Unknown
% 3.47/1.91 % (16467)Termination phase: shuffling
% 3.47/1.91
% 3.47/1.91 % (16467)Memory used [KB]: 1535
% 3.47/1.91 % (16467)Time elapsed: 0.005 s
% 3.47/1.91 % (16467)Instructions burned: 7 (million)
% 3.47/1.91 % (16467)------------------------------
% 3.47/1.91 % (16467)------------------------------
% 3.47/1.91 % (16468)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_16347 for (2999ds/902Mi)
% 3.47/1.92 % (16469)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_16347 for (2998ds/21Mi)
% 3.47/1.93 % (16470)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_16347 for (2998ds/5Mi)
% 3.47/1.93 % (16471)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_16347 for (2998ds/6Mi)
% 3.47/1.93 % (16470)Instruction limit reached!
% 3.47/1.93 % (16470)------------------------------
% 3.47/1.93 % (16470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.93 % (16470)Termination reason: Unknown
% 3.47/1.93 % (16470)Termination phase: shuffling
% 3.47/1.93
% 3.47/1.93 % (16470)Memory used [KB]: 1535
% 3.47/1.93 % (16470)Time elapsed: 0.005 s
% 3.47/1.93 % (16470)Instructions burned: 6 (million)
% 3.47/1.93 % (16470)------------------------------
% 3.47/1.93 % (16470)------------------------------
% 3.47/1.93 % (16471)Instruction limit reached!
% 3.47/1.93 % (16471)------------------------------
% 3.47/1.93 % (16471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.93 % (16471)Termination reason: Unknown
% 3.47/1.93 % (16471)Termination phase: shuffling
% 3.47/1.93
% 3.47/1.93 % (16471)Memory used [KB]: 1535
% 3.47/1.93 % (16471)Time elapsed: 0.005 s
% 3.47/1.93 % (16471)Instructions burned: 7 (million)
% 3.47/1.93 % (16471)------------------------------
% 3.47/1.93 % (16471)------------------------------
% 3.47/1.93 % (16469)Instruction limit reached!
% 3.47/1.93 % (16469)------------------------------
% 3.47/1.93 % (16469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.93 % (16469)Termination reason: Unknown
% 3.47/1.93 % (16469)Termination phase: shuffling
% 3.47/1.93
% 3.47/1.93 % (16469)Memory used [KB]: 1918
% 3.47/1.93 % (16469)Time elapsed: 0.012 s
% 3.47/1.93 % (16469)Instructions burned: 22 (million)
% 3.47/1.93 % (16469)------------------------------
% 3.47/1.93 % (16469)------------------------------
% 3.47/1.95 % (16472)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_16347 for (2998ds/377Mi)
% 3.47/1.95 % (16473)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_16347 for (2998ds/779Mi)
% 3.47/1.95 % (16474)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_16347 for (2998ds/19Mi)
% 3.47/1.96 % (16474)Instruction limit reached!
% 3.47/1.96 % (16474)------------------------------
% 3.47/1.96 % (16474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.96 % (16474)Termination reason: Unknown
% 3.47/1.96 % (16474)Termination phase: shuffling
% 3.47/1.96
% 3.47/1.96 % (16474)Memory used [KB]: 1918
% 3.47/1.96 % (16474)Time elapsed: 0.011 s
% 3.47/1.96 % (16474)Instructions burned: 20 (million)
% 3.47/1.96 % (16474)------------------------------
% 3.47/1.96 % (16474)------------------------------
% 3.47/1.96 % (16451)Instruction limit reached!
% 3.47/1.96 % (16451)------------------------------
% 3.47/1.96 % (16451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.96 % (16451)Termination reason: Unknown
% 3.47/1.96 % (16451)Termination phase: Saturation
% 3.47/1.96
% 3.47/1.96 % (16451)Memory used [KB]: 10106
% 3.47/1.96 % (16451)Time elapsed: 0.133 s
% 3.47/1.96 % (16451)Instructions burned: 276 (million)
% 3.47/1.96 % (16451)------------------------------
% 3.47/1.96 % (16451)------------------------------
% 3.47/1.98 % (16475)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_16347 for (2998ds/879Mi)
% 3.47/1.98 % (16476)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_16347 for (2998ds/17Mi)
% 3.47/1.99 % (16476)Instruction limit reached!
% 3.47/1.99 % (16476)------------------------------
% 3.47/1.99 % (16476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.99 % (16476)Termination reason: Unknown
% 3.47/1.99 % (16476)Termination phase: shuffling
% 3.47/1.99
% 3.47/1.99 % (16476)Memory used [KB]: 1791
% 3.47/1.99 % (16476)Time elapsed: 0.010 s
% 3.47/1.99 % (16476)Instructions burned: 17 (million)
% 3.47/1.99 % (16476)------------------------------
% 3.47/1.99 % (16476)------------------------------
% 3.47/2.00 % (16477)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_16347 for (2998ds/3Mi)
% 3.47/2.01 % (16477)Instruction limit reached!
% 3.47/2.01 % (16477)------------------------------
% 3.47/2.01 % (16477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/2.01 % (16477)Termination reason: Unknown
% 3.47/2.01 % (16477)Termination phase: shuffling
% 3.47/2.01
% 3.47/2.01 % (16477)Memory used [KB]: 1535
% 3.47/2.01 % (16477)Time elapsed: 0.004 s
% 3.47/2.01 % (16477)Instructions burned: 3 (million)
% 3.47/2.01 % (16477)------------------------------
% 3.47/2.01 % (16477)------------------------------
% 4.85/2.04 % (16478)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_16347 for (2997ds/30Mi)
% 4.85/2.05 % (16478)Instruction limit reached!
% 4.85/2.05 % (16478)------------------------------
% 4.85/2.05 % (16478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.85/2.05 % (16478)Termination reason: Unknown
% 4.85/2.05 % (16478)Termination phase: shuffling
% 4.85/2.05
% 4.85/2.05 % (16478)Memory used [KB]: 2046
% 4.85/2.05 % (16478)Time elapsed: 0.016 s
% 4.85/2.05 % (16478)Instructions burned: 30 (million)
% 4.85/2.05 % (16478)------------------------------
% 4.85/2.05 % (16478)------------------------------
% 4.85/2.07 % (16479)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_16347 for (2997ds/127Mi)
% 4.85/2.11 % (16468)Refutation not found, incomplete strategy
% 4.85/2.11 % (16468)------------------------------
% 4.85/2.11 % (16468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.85/2.11 % (16468)Termination reason: Refutation not found, incomplete strategy
% 4.85/2.11
% 4.85/2.11
% 4.85/2.11 % (16468)Memory used [KB]: 10618
% 4.85/2.11 % (16468)Time elapsed: 0.194 s
% 4.85/2.11 % (16468)Instructions burned: 401 (million)
% 4.85/2.11 % (16468)------------------------------
% 4.85/2.11 % (16468)------------------------------
% 4.85/2.12 % (16472)Instruction limit reached!
% 4.85/2.12 % (16472)------------------------------
% 4.85/2.12 % (16472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.85/2.12 % (16472)Termination reason: Unknown
% 4.85/2.12 % (16472)Termination phase: Saturation
% 4.85/2.12
% 4.85/2.12 % (16472)Memory used [KB]: 9466
% 4.85/2.12 % (16472)Time elapsed: 0.173 s
% 4.85/2.12 % (16472)Instructions burned: 377 (million)
% 4.85/2.12 % (16472)------------------------------
% 4.85/2.12 % (16472)------------------------------
% 5.50/2.12 % (16480)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_16347 for (2996ds/100Mi)
% 5.50/2.13 % (16479)Instruction limit reached!
% 5.50/2.13 % (16479)------------------------------
% 5.50/2.13 % (16479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.50/2.13 % (16479)Termination reason: Unknown
% 5.50/2.13 % (16479)Termination phase: Property scanning
% 5.50/2.13
% 5.50/2.13 % (16479)Memory used [KB]: 2686
% 5.50/2.13 % (16479)Time elapsed: 0.058 s
% 5.50/2.13 % (16479)Instructions burned: 127 (million)
% 5.50/2.13 % (16479)------------------------------
% 5.50/2.13 % (16479)------------------------------
% 5.50/2.13 % (16481)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_16347 for (2996ds/3Mi)
% 5.50/2.13 % (16481)Instruction limit reached!
% 5.50/2.13 % (16481)------------------------------
% 5.50/2.13 % (16481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.50/2.13 % (16481)Termination reason: Unknown
% 5.50/2.13 % (16481)Termination phase: shuffling
% 5.50/2.13
% 5.50/2.13 % (16481)Memory used [KB]: 1535
% 5.50/2.13 % (16481)Time elapsed: 0.003 s
% 5.50/2.13 % (16481)Instructions burned: 3 (million)
% 5.50/2.13 % (16481)------------------------------
% 5.50/2.13 % (16481)------------------------------
% 5.50/2.14 % (16482)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_16347 for (2996ds/20Mi)
% 5.50/2.15 % (16483)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_16347 for (2996ds/86Mi)
% 5.50/2.15 % (16482)Instruction limit reached!
% 5.50/2.15 % (16482)------------------------------
% 5.50/2.15 % (16482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.50/2.15 % (16482)Termination reason: Unknown
% 5.50/2.15 % (16482)Termination phase: shuffling
% 5.50/2.15
% 5.50/2.15 % (16482)Memory used [KB]: 1791
% 5.50/2.15 % (16482)Time elapsed: 0.012 s
% 5.50/2.15 % (16482)Instructions burned: 20 (million)
% 5.50/2.15 % (16482)------------------------------
% 5.50/2.15 % (16482)------------------------------
% 5.50/2.17 % (16480)Instruction limit reached!
% 5.50/2.17 % (16480)------------------------------
% 5.50/2.17 % (16480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.50/2.17 % (16480)Termination reason: Unknown
% 5.50/2.17 % (16480)Termination phase: Property scanning
% 5.50/2.17
% 5.50/2.17 % (16480)Memory used [KB]: 2686
% 5.50/2.17 % (16480)Time elapsed: 0.046 s
% 5.50/2.17 % (16480)Instructions burned: 100 (million)
% 5.50/2.17 % (16480)------------------------------
% 5.50/2.17 % (16480)------------------------------
% 5.50/2.17 % (16484)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_16347 for (2996ds/107Mi)
% 5.50/2.18 % (16485)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_16347 for (2996ds/39Mi)
% 5.50/2.19 % (16483)Instruction limit reached!
% 5.50/2.19 % (16483)------------------------------
% 5.50/2.19 % (16483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.50/2.19 % (16483)Termination reason: Unknown
% 5.50/2.19 % (16483)Termination phase: Property scanning
% 5.50/2.19
% 5.50/2.19 % (16483)Memory used [KB]: 2558
% 5.50/2.19 % (16483)Time elapsed: 0.041 s
% 5.50/2.19 % (16483)Instructions burned: 88 (million)
% 5.50/2.19 % (16483)------------------------------
% 5.50/2.19 % (16483)------------------------------
% 6.03/2.20 % (16485)Instruction limit reached!
% 6.03/2.20 % (16485)------------------------------
% 6.03/2.20 % (16485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.20 % (16485)Termination reason: Unknown
% 6.03/2.20 % (16485)Termination phase: shuffling
% 6.03/2.20
% 6.03/2.20 % (16485)Memory used [KB]: 2302
% 6.03/2.20 % (16485)Time elapsed: 0.020 s
% 6.03/2.20 % (16485)Instructions burned: 39 (million)
% 6.03/2.20 % (16485)------------------------------
% 6.03/2.20 % (16485)------------------------------
% 6.03/2.21 % (16486)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_16347 for (2996ds/448Mi)
% 6.03/2.22 % (16484)Instruction limit reached!
% 6.03/2.22 % (16484)------------------------------
% 6.03/2.22 % (16484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.22 % (16484)Termination reason: Unknown
% 6.03/2.22 % (16484)Termination phase: SInE selection
% 6.03/2.22
% 6.03/2.22 % (16484)Memory used [KB]: 2558
% 6.03/2.22 % (16484)Time elapsed: 0.049 s
% 6.03/2.22 % (16484)Instructions burned: 109 (million)
% 6.03/2.22 % (16484)------------------------------
% 6.03/2.22 % (16484)------------------------------
% 6.03/2.22 % (16487)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_16347 for (2996ds/46Mi)
% 6.03/2.23 % (16488)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_16347 for (2995ds/98Mi)
% 6.03/2.24 % (16487)Instruction limit reached!
% 6.03/2.24 % (16487)------------------------------
% 6.03/2.24 % (16487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.24 % (16487)Termination reason: Unknown
% 6.03/2.24 % (16487)Termination phase: shuffling
% 6.03/2.24
% 6.03/2.24 % (16487)Memory used [KB]: 2430
% 6.03/2.24 % (16487)Time elapsed: 0.024 s
% 6.03/2.24 % (16487)Instructions burned: 47 (million)
% 6.03/2.24 % (16487)------------------------------
% 6.03/2.24 % (16487)------------------------------
% 6.03/2.25 % (16473)Instruction limit reached!
% 6.03/2.25 % (16473)------------------------------
% 6.03/2.25 % (16473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.25 % (16473)Termination reason: Unknown
% 6.03/2.25 % (16473)Termination phase: Saturation
% 6.03/2.25
% 6.03/2.25 % (16473)Memory used [KB]: 7675
% 6.03/2.25 % (16473)Time elapsed: 0.304 s
% 6.03/2.25 % (16473)Instructions burned: 779 (million)
% 6.03/2.25 % (16473)------------------------------
% 6.03/2.25 % (16473)------------------------------
% 6.03/2.26 % (16457)Instruction limit reached!
% 6.03/2.26 % (16457)------------------------------
% 6.03/2.26 % (16457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.26 % (16457)Termination reason: Unknown
% 6.03/2.26 % (16457)Termination phase: Saturation
% 6.03/2.26
% 6.03/2.26 % (16457)Memory used [KB]: 16119
% 6.03/2.26 % (16457)Time elapsed: 0.401 s
% 6.03/2.26 % (16457)Instructions burned: 1043 (million)
% 6.03/2.26 % (16457)------------------------------
% 6.03/2.26 % (16457)------------------------------
% 6.03/2.26 % (16466)Instruction limit reached!
% 6.03/2.26 % (16466)------------------------------
% 6.03/2.26 % (16466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.26 % (16466)Termination reason: Unknown
% 6.03/2.26 % (16466)Termination phase: Saturation
% 6.03/2.26
% 6.03/2.26 % (16466)Memory used [KB]: 10874
% 6.03/2.26 % (16466)Time elapsed: 0.355 s
% 6.03/2.26 % (16466)Instructions burned: 711 (million)
% 6.03/2.26 % (16466)------------------------------
% 6.03/2.26 % (16466)------------------------------
% 6.03/2.26 % (16489)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_16347 for (2995ds/507Mi)
% 6.03/2.27 % (16490)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_16347 for (2995ds/149Mi)
% 6.03/2.27 % (16491)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_16347 for (2995ds/39Mi)
% 6.03/2.28 % (16492)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_16347 for (2995ds/6Mi)
% 6.03/2.28 % (16488)Instruction limit reached!
% 6.03/2.28 % (16488)------------------------------
% 6.03/2.28 % (16488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.28 % (16488)Termination reason: Unknown
% 6.03/2.28 % (16488)Termination phase: Property scanning
% 6.03/2.28
% 6.03/2.28 % (16488)Memory used [KB]: 2558
% 6.03/2.28 % (16488)Time elapsed: 0.045 s
% 6.03/2.28 % (16488)Instructions burned: 98 (million)
% 6.03/2.28 % (16488)------------------------------
% 6.03/2.28 % (16488)------------------------------
% 6.03/2.28 % (16492)Instruction limit reached!
% 6.03/2.28 % (16492)------------------------------
% 6.03/2.28 % (16492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.28 % (16492)Termination reason: Unknown
% 6.03/2.28 % (16492)Termination phase: shuffling
% 6.03/2.28
% 6.03/2.28 % (16492)Memory used [KB]: 1535
% 6.03/2.28 % (16492)Time elapsed: 0.005 s
% 6.03/2.28 % (16492)Instructions burned: 7 (million)
% 6.03/2.28 % (16492)------------------------------
% 6.03/2.28 % (16492)------------------------------
% 6.03/2.29 % (16491)Instruction limit reached!
% 6.03/2.29 % (16491)------------------------------
% 6.03/2.29 % (16491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.03/2.29 % (16491)Termination reason: Unknown
% 6.03/2.29 % (16491)Termination phase: shuffling
% 6.03/2.29
% 6.03/2.29 % (16491)Memory used [KB]: 2302
% 6.03/2.29 % (16491)Time elapsed: 0.020 s
% 6.03/2.29 % (16491)Instructions burned: 40 (million)
% 6.03/2.29 % (16491)------------------------------
% 6.03/2.29 % (16491)------------------------------
% 6.03/2.29 % (16493)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_16347 for (2995ds/1087Mi)
% 6.03/2.29 % (16494)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_16347 for (2995ds/47Mi)
% 6.60/2.31 % (16495)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_16347 for (2994ds/44Mi)
% 6.60/2.32 % (16494)Instruction limit reached!
% 6.60/2.32 % (16494)------------------------------
% 6.60/2.32 % (16494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.60/2.32 % (16494)Termination reason: Unknown
% 6.60/2.32 % (16494)Termination phase: shuffling
% 6.60/2.32
% 6.60/2.32 % (16494)Memory used [KB]: 2430
% 6.60/2.32 % (16494)Time elapsed: 0.023 s
% 6.60/2.32 % (16494)Instructions burned: 47 (million)
% 6.60/2.32 % (16494)------------------------------
% 6.60/2.32 % (16494)------------------------------
% 6.60/2.33 % (16495)Instruction limit reached!
% 6.60/2.33 % (16495)------------------------------
% 6.60/2.33 % (16495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.60/2.33 % (16495)Termination reason: Unknown
% 6.60/2.33 % (16495)Termination phase: shuffling
% 6.60/2.33
% 6.60/2.33 % (16495)Memory used [KB]: 2302
% 6.60/2.33 % (16495)Time elapsed: 0.022 s
% 6.60/2.33 % (16495)Instructions burned: 44 (million)
% 6.60/2.33 % (16495)------------------------------
% 6.60/2.33 % (16495)------------------------------
% 6.60/2.33 % (16496)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_16347 for (2994ds/432Mi)
% 6.60/2.33 % (16490)Instruction limit reached!
% 6.60/2.33 % (16490)------------------------------
% 6.60/2.33 % (16490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.60/2.33 % (16490)Termination reason: Unknown
% 6.60/2.33 % (16490)Termination phase: Twee Goal Transformation
% 6.60/2.33
% 6.60/2.33 % (16490)Memory used [KB]: 3326
% 6.60/2.33 % (16490)Time elapsed: 0.069 s
% 6.60/2.33 % (16490)Instructions burned: 149 (million)
% 6.60/2.33 % (16490)------------------------------
% 6.60/2.33 % (16490)------------------------------
% 6.60/2.35 % (16497)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_16347 for (2994ds/62Mi)
% 6.60/2.35 % (16498)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_16347 for (2994ds/31Mi)
% 6.60/2.36 % (16498)Instruction limit reached!
% 6.60/2.36 % (16498)------------------------------
% 6.60/2.36 % (16498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.60/2.36 % (16498)Termination reason: Unknown
% 6.60/2.36 % (16498)Termination phase: shuffling
% 6.60/2.36
% 6.60/2.36 % (16498)Memory used [KB]: 2174
% 6.60/2.36 % (16498)Time elapsed: 0.016 s
% 6.60/2.36 % (16498)Instructions burned: 32 (million)
% 6.60/2.36 % (16498)------------------------------
% 6.60/2.36 % (16498)------------------------------
% 6.60/2.37 % (16497)Instruction limit reached!
% 6.60/2.37 % (16497)------------------------------
% 6.60/2.37 % (16497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.60/2.37 % (16497)Termination reason: Unknown
% 6.60/2.37 % (16497)Termination phase: Property scanning
% 6.60/2.37
% 6.60/2.37 % (16497)Memory used [KB]: 2558
% 6.60/2.37 % (16497)Time elapsed: 0.030 s
% 6.60/2.37 % (16497)Instructions burned: 63 (million)
% 6.60/2.37 % (16497)------------------------------
% 6.60/2.37 % (16497)------------------------------
% 6.60/2.38 % (16499)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_16347 for (2994ds/96Mi)
% 6.60/2.39 % (16486)Instruction limit reached!
% 6.60/2.39 % (16486)------------------------------
% 6.60/2.39 % (16486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.60/2.39 % (16486)Termination reason: Unknown
% 6.60/2.39 % (16486)Termination phase: Saturation
% 6.60/2.39
% 6.60/2.39 % (16486)Memory used [KB]: 7547
% 6.60/2.39 % (16486)Time elapsed: 0.185 s
% 6.60/2.39 % (16486)Instructions burned: 450 (million)
% 6.60/2.39 % (16486)------------------------------
% 6.60/2.39 % (16486)------------------------------
% 6.60/2.39 % (16475)Refutation not found, non-redundant clauses discarded% (16475)------------------------------
% 6.60/2.39 % (16475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.60/2.39 % (16475)Termination reason: Refutation not found, non-redundant clauses discarded
% 6.60/2.39
% 6.60/2.39 % (16475)Memory used [KB]: 10106
% 6.60/2.39 % (16475)Time elapsed: 0.418 s
% 6.60/2.39 % (16475)Instructions burned: 867 (million)
% 6.60/2.39 % (16475)------------------------------
% 6.60/2.39 % (16475)------------------------------
% 6.60/2.39 % (16500)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_16347 for (2994ds/453Mi)
% 7.13/2.41 % (16501)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_16347 for (2994ds/71Mi)
% 7.13/2.41 % (16502)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_16347 for (2994ds/83Mi)
% 7.13/2.42 % (16499)Instruction limit reached!
% 7.13/2.42 % (16499)------------------------------
% 7.13/2.42 % (16499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.42 % (16499)Termination reason: Unknown
% 7.13/2.42 % (16499)Termination phase: Property scanning
% 7.13/2.42
% 7.13/2.42 % (16499)Memory used [KB]: 2686
% 7.13/2.43 % (16499)Time elapsed: 0.046 s
% 7.13/2.43 % (16499)Instructions burned: 98 (million)
% 7.13/2.43 % (16499)------------------------------
% 7.13/2.43 % (16499)------------------------------
% 7.13/2.44 % (16501)Instruction limit reached!
% 7.13/2.44 % (16501)------------------------------
% 7.13/2.44 % (16501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.44 % (16501)Termination reason: Unknown
% 7.13/2.44 % (16501)Termination phase: Property scanning
% 7.13/2.44
% 7.13/2.44 % (16501)Memory used [KB]: 2558
% 7.13/2.44 % (16501)Time elapsed: 0.033 s
% 7.13/2.44 % (16501)Instructions burned: 72 (million)
% 7.13/2.44 % (16501)------------------------------
% 7.13/2.44 % (16501)------------------------------
% 7.13/2.44 % (16503)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_16347 for (2993ds/11Mi)
% 7.13/2.45 % (16503)Instruction limit reached!
% 7.13/2.45 % (16503)------------------------------
% 7.13/2.45 % (16503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.45 % (16503)Termination reason: Unknown
% 7.13/2.45 % (16503)Termination phase: shuffling
% 7.13/2.45
% 7.13/2.45 % (16503)Memory used [KB]: 1791
% 7.13/2.45 % (16503)Time elapsed: 0.007 s
% 7.13/2.45 % (16503)Instructions burned: 11 (million)
% 7.13/2.45 % (16503)------------------------------
% 7.13/2.45 % (16503)------------------------------
% 7.13/2.45 % (16502)Instruction limit reached!
% 7.13/2.45 % (16502)------------------------------
% 7.13/2.45 % (16502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.45 % (16502)Termination reason: Unknown
% 7.13/2.45 % (16502)Termination phase: Property scanning
% 7.13/2.45
% 7.13/2.45 % (16502)Memory used [KB]: 2558
% 7.13/2.45 % (16502)Time elapsed: 0.040 s
% 7.13/2.45 % (16502)Instructions burned: 83 (million)
% 7.13/2.45 % (16502)------------------------------
% 7.13/2.45 % (16502)------------------------------
% 7.13/2.45 % (16504)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_16347 for (2993ds/41Mi)
% 7.13/2.46 % (16506)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_16347 for (2993ds/150Mi)
% 7.13/2.46 % (16505)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16347 for (2993ds/6Mi)
% 7.13/2.47 % (16505)Instruction limit reached!
% 7.13/2.47 % (16505)------------------------------
% 7.13/2.47 % (16505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.47 % (16505)Termination reason: Unknown
% 7.13/2.47 % (16505)Termination phase: shuffling
% 7.13/2.47
% 7.13/2.47 % (16505)Memory used [KB]: 1535
% 7.13/2.47 % (16505)Time elapsed: 0.005 s
% 7.13/2.47 % (16505)Instructions burned: 7 (million)
% 7.13/2.47 % (16505)------------------------------
% 7.13/2.47 % (16505)------------------------------
% 7.13/2.47 % (16504)Instruction limit reached!
% 7.13/2.47 % (16504)------------------------------
% 7.13/2.47 % (16504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.47 % (16504)Termination reason: Unknown
% 7.13/2.47 % (16504)Termination phase: shuffling
% 7.13/2.47
% 7.13/2.47 % (16504)Memory used [KB]: 2302
% 7.13/2.47 % (16504)Time elapsed: 0.020 s
% 7.13/2.47 % (16504)Instructions burned: 41 (million)
% 7.13/2.47 % (16504)------------------------------
% 7.13/2.47 % (16504)------------------------------
% 7.13/2.48 % (16507)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_16347 for (2993ds/343Mi)
% 7.13/2.49 % (16508)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_16347 for (2993ds/109Mi)
% 7.38/2.50 % (16489)Instruction limit reached!
% 7.38/2.50 % (16489)------------------------------
% 7.38/2.50 % (16489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.50 % (16489)Termination reason: Unknown
% 7.38/2.50 % (16489)Termination phase: Saturation
% 7.38/2.50
% 7.38/2.50 % (16489)Memory used [KB]: 8187
% 7.38/2.50 % (16489)Time elapsed: 0.242 s
% 7.38/2.50 % (16489)Instructions burned: 507 (million)
% 7.38/2.50 % (16489)------------------------------
% 7.38/2.50 % (16489)------------------------------
% 7.38/2.51 % (16496)Instruction limit reached!
% 7.38/2.51 % (16496)------------------------------
% 7.38/2.51 % (16496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.51 % (16496)Termination reason: Unknown
% 7.38/2.51 % (16496)Termination phase: Saturation
% 7.38/2.51
% 7.38/2.51 % (16496)Memory used [KB]: 3070
% 7.38/2.51 % (16496)Time elapsed: 0.179 s
% 7.38/2.51 % (16496)Instructions burned: 432 (million)
% 7.38/2.51 % (16496)------------------------------
% 7.38/2.51 % (16496)------------------------------
% 7.38/2.52 % (16509)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_16347 for (2993ds/253Mi)
% 7.38/2.53 % (16510)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_16347 for (2992ds/140Mi)
% 7.38/2.53 % (16506)Instruction limit reached!
% 7.38/2.53 % (16506)------------------------------
% 7.38/2.53 % (16506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.53 % (16506)Termination reason: Unknown
% 7.38/2.53 % (16506)Termination phase: Property scanning
% 7.38/2.53
% 7.38/2.53 % (16506)Memory used [KB]: 4989
% 7.38/2.53 % (16506)Time elapsed: 0.073 s
% 7.38/2.53 % (16506)Instructions burned: 150 (million)
% 7.38/2.53 % (16506)------------------------------
% 7.38/2.53 % (16506)------------------------------
% 7.38/2.54 % (16508)Instruction limit reached!
% 7.38/2.54 % (16508)------------------------------
% 7.38/2.54 % (16508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.54 % (16508)Termination reason: Unknown
% 7.38/2.54 % (16508)Termination phase: Naming
% 7.38/2.54
% 7.38/2.54 % (16508)Memory used [KB]: 2686
% 7.38/2.54 % (16508)Time elapsed: 0.050 s
% 7.38/2.54 % (16508)Instructions burned: 111 (million)
% 7.38/2.54 % (16508)------------------------------
% 7.38/2.54 % (16508)------------------------------
% 7.38/2.55 % (16511)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_16347 for (2992ds/614Mi)
% 7.38/2.55 % (16512)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_16347 for (2992ds/202Mi)
% 7.38/2.58 % (16500)Instruction limit reached!
% 7.38/2.58 % (16500)------------------------------
% 7.38/2.58 % (16500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.58 % (16500)Termination reason: Unknown
% 7.38/2.58 % (16500)Termination phase: Saturation
% 7.38/2.58
% 7.38/2.58 % (16500)Memory used [KB]: 7547
% 7.38/2.58 % (16500)Time elapsed: 0.188 s
% 7.38/2.58 % (16500)Instructions burned: 454 (million)
% 7.38/2.58 % (16500)------------------------------
% 7.38/2.58 % (16500)------------------------------
% 7.38/2.59 % (16510)Instruction limit reached!
% 7.38/2.59 % (16510)------------------------------
% 7.38/2.59 % (16510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.38/2.59 % (16510)Termination reason: Unknown
% 7.38/2.59 % (16510)Termination phase: Function definition elimination
% 7.38/2.59
% 7.38/2.59 % (16510)Memory used [KB]: 2814
% 7.38/2.59 % (16510)Time elapsed: 0.063 s
% 7.38/2.59 % (16510)Instructions burned: 140 (million)
% 7.38/2.59 % (16510)------------------------------
% 7.38/2.59 % (16510)------------------------------
% 7.38/2.60 % (16513)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_16347 for (2992ds/339Mi)
% 8.26/2.60 % (16514)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_16347 for (2992ds/283Mi)
% 8.26/2.63 % (16507)Instruction limit reached!
% 8.26/2.63 % (16507)------------------------------
% 8.26/2.63 % (16507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.63 % (16507)Termination reason: Unknown
% 8.26/2.63 % (16507)Termination phase: Saturation
% 8.26/2.63
% 8.26/2.63 % (16507)Memory used [KB]: 8443
% 8.26/2.63 % (16507)Time elapsed: 0.150 s
% 8.26/2.63 % (16507)Instructions burned: 344 (million)
% 8.26/2.63 % (16507)------------------------------
% 8.26/2.63 % (16507)------------------------------
% 8.26/2.64 % (16509)Instruction limit reached!
% 8.26/2.64 % (16509)------------------------------
% 8.26/2.64 % (16509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.64 % (16509)Termination reason: Unknown
% 8.26/2.64 % (16509)Termination phase: Saturation
% 8.26/2.64
% 8.26/2.64 % (16509)Memory used [KB]: 10106
% 8.26/2.64 % (16509)Time elapsed: 0.121 s
% 8.26/2.64 % (16509)Instructions burned: 254 (million)
% 8.26/2.64 % (16509)------------------------------
% 8.26/2.64 % (16509)------------------------------
% 8.26/2.65 % (16512)Instruction limit reached!
% 8.26/2.65 % (16512)------------------------------
% 8.26/2.65 % (16512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.26/2.65 % (16512)Termination reason: Unknown
% 8.26/2.65 % (16512)Termination phase: Property scanning
% 8.26/2.65
% 8.26/2.65 % (16512)Memory used [KB]: 5117
% 8.26/2.65 % (16512)Time elapsed: 0.095 s
% 8.26/2.65 % (16512)Instructions burned: 202 (million)
% 8.26/2.65 % (16512)------------------------------
% 8.26/2.65 % (16512)------------------------------
% 8.26/2.65 % (16515)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_16347 for (2991ds/134Mi)
% 8.26/2.65 % (16516)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_16347 for (2991ds/298Mi)
% 8.26/2.66 % (16517)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_16347 for (2991ds/192Mi)
% 8.67/2.71 % (16515)Instruction limit reached!
% 8.67/2.71 % (16515)------------------------------
% 8.67/2.71 % (16515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.67/2.71 % (16515)Termination reason: Unknown
% 8.67/2.71 % (16515)Termination phase: Function definition elimination
% 8.67/2.71
% 8.67/2.71 % (16515)Memory used [KB]: 2814
% 8.67/2.71 % (16515)Time elapsed: 0.060 s
% 8.67/2.71 % (16515)Instructions burned: 134 (million)
% 8.67/2.71 % (16515)------------------------------
% 8.67/2.71 % (16515)------------------------------
% 8.67/2.72 % (16518)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_16347 for (2990ds/258Mi)
% 8.67/2.72 % (16514)Instruction limit reached!
% 8.67/2.72 % (16514)------------------------------
% 8.67/2.72 % (16514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.67/2.72 % (16514)Termination reason: Unknown
% 8.67/2.72 % (16514)Termination phase: Saturation
% 8.67/2.72
% 8.67/2.72 % (16514)Memory used [KB]: 7547
% 8.67/2.72 % (16514)Time elapsed: 0.120 s
% 8.67/2.72 % (16514)Instructions burned: 285 (million)
% 8.67/2.72 % (16514)------------------------------
% 8.67/2.72 % (16514)------------------------------
% 8.67/2.74 % (16519)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_16347 for (2990ds/90Mi)
% 8.67/2.75 % (16513)Instruction limit reached!
% 8.67/2.75 % (16513)------------------------------
% 8.67/2.75 % (16513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.67/2.75 % (16513)Termination reason: Unknown
% 8.67/2.75 % (16513)Termination phase: Saturation
% 8.67/2.75
% 8.67/2.75 % (16513)Memory used [KB]: 7803
% 8.67/2.75 % (16513)Time elapsed: 0.152 s
% 8.67/2.75 % (16513)Instructions burned: 339 (million)
% 8.67/2.75 % (16513)------------------------------
% 8.67/2.75 % (16513)------------------------------
% 8.67/2.75 % (16517)Instruction limit reached!
% 8.67/2.75 % (16517)------------------------------
% 8.67/2.75 % (16517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.67/2.75 % (16517)Termination reason: Unknown
% 8.67/2.75 % (16517)Termination phase: Saturation
% 8.67/2.75
% 8.67/2.75 % (16517)Memory used [KB]: 7547
% 8.67/2.75 % (16517)Time elapsed: 0.086 s
% 8.67/2.75 % (16517)Instructions burned: 193 (million)
% 8.67/2.75 % (16517)------------------------------
% 8.67/2.75 % (16517)------------------------------
% 8.67/2.76 % (16520)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_16347 for (2990ds/231Mi)
% 8.67/2.76 % (16521)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_16347 for (2990ds/458Mi)
% 8.67/2.78 % (16516)Instruction limit reached!
% 8.67/2.78 % (16516)------------------------------
% 8.67/2.78 % (16516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.67/2.78 % (16516)Termination reason: Unknown
% 8.67/2.78 % (16516)Termination phase: Saturation
% 8.67/2.78
% 8.67/2.78 % (16516)Memory used [KB]: 7547
% 8.67/2.78 % (16516)Time elapsed: 0.127 s
% 8.67/2.78 % (16516)Instructions burned: 300 (million)
% 8.67/2.78 % (16516)------------------------------
% 8.67/2.78 % (16516)------------------------------
% 8.67/2.78 % (16519)Instruction limit reached!
% 8.67/2.78 % (16519)------------------------------
% 8.67/2.78 % (16519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.67/2.78 % (16519)Termination reason: Unknown
% 8.67/2.78 % (16519)Termination phase: Property scanning
% 8.67/2.78
% 8.67/2.78 % (16519)Memory used [KB]: 2686
% 8.67/2.78 % (16519)Time elapsed: 0.043 s
% 8.67/2.78 % (16519)Instructions burned: 90 (million)
% 8.67/2.78 % (16519)------------------------------
% 8.67/2.78 % (16519)------------------------------
% 8.67/2.80 % (16522)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_16347 for (2990ds/31Mi)
% 9.35/2.80 % (16523)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_16347 for (2990ds/297Mi)
% 9.35/2.81 % (16522)Instruction limit reached!
% 9.35/2.81 % (16522)------------------------------
% 9.35/2.81 % (16522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.35/2.81 % (16522)Termination reason: Unknown
% 9.35/2.81 % (16522)Termination phase: shuffling
% 9.35/2.81
% 9.35/2.81 % (16522)Memory used [KB]: 2174
% 9.35/2.81 % (16522)Time elapsed: 0.017 s
% 9.35/2.81 % (16522)Instructions burned: 32 (million)
% 9.35/2.81 % (16522)------------------------------
% 9.35/2.81 % (16522)------------------------------
% 9.35/2.82 % (16493)First to succeed.
% 9.35/2.83 % (16524)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_16347 for (2989ds/987Mi)
% 9.35/2.83 % (16493)Refutation found. Thanks to Tanya!
% 9.35/2.83 % SZS status Theorem for DTF2THF_16347
% 9.35/2.83 % SZS output start Proof for DTF2THF_16347
% 9.35/2.83 thf(func_def_0, type, in: $i > $i > $o).
% 9.35/2.83 thf(func_def_1, type, exu: ($i > $o) > $o).
% 9.35/2.83 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 9.35/2.83 thf(func_def_8, type, powerset: $i > $i).
% 9.35/2.83 thf(func_def_10, type, setunion: $i > $i).
% 9.35/2.83 thf(func_def_19, type, descr: ($i > $o) > $i).
% 9.35/2.83 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_26, type, prop2set: $o > $i).
% 9.35/2.83 thf(func_def_36, type, nonempty: $i > $o).
% 9.35/2.83 thf(func_def_69, type, set2prop: $i > $o).
% 9.35/2.83 thf(func_def_88, type, subset: $i > $i > $o).
% 9.35/2.83 thf(func_def_89, type, disjoint: $i > $i > $o).
% 9.35/2.83 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 9.35/2.83 thf(func_def_114, type, binunion: $i > $i > $i).
% 9.35/2.83 thf(func_def_122, type, binintersect: $i > $i > $i).
% 9.35/2.83 thf(func_def_135, type, regular: $i > $o).
% 9.35/2.83 thf(func_def_136, type, setminus: $i > $i > $i).
% 9.35/2.83 thf(func_def_147, type, symdiff: $i > $i > $i).
% 9.35/2.83 thf(func_def_153, type, iskpair: $i > $o).
% 9.35/2.83 thf(func_def_158, type, kpair: $i > $i > $i).
% 9.35/2.83 thf(func_def_160, type, cartprod: $i > $i > $i).
% 9.35/2.83 thf(func_def_177, type, singleton: $i > $o).
% 9.35/2.83 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_184, type, atmost1p: $i > $o).
% 9.35/2.83 thf(func_def_185, type, atleast2p: $i > $o).
% 9.35/2.83 thf(func_def_186, type, atmost2p: $i > $o).
% 9.35/2.83 thf(func_def_187, type, upairsetp: $i > $o).
% 9.35/2.83 thf(func_def_191, type, kfst: $i > $i).
% 9.35/2.83 thf(func_def_203, type, ksnd: $i > $i).
% 9.35/2.83 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 9.35/2.83 thf(func_def_222, type, func: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_223, type, funcSet: $i > $i > $i).
% 9.35/2.83 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 9.35/2.83 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 9.35/2.83 thf(func_def_323, type, breln1: $i > $i > $o).
% 9.35/2.83 thf(func_def_325, type, breln1Set: $i > $i).
% 9.35/2.83 thf(func_def_339, type, sP0: $i > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_340, type, sP1: ($i > $o) > $i > $o).
% 9.35/2.83 thf(func_def_342, type, sP3: ($i > $o) > $i > $o).
% 9.35/2.83 thf(func_def_344, type, sP5: ($i > $o) > $o).
% 9.35/2.83 thf(func_def_345, type, sP6: $i > $o).
% 9.35/2.83 thf(func_def_346, type, sP7: $i > $o).
% 9.35/2.83 thf(func_def_347, type, sP8: $i > $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_348, type, sP9: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_349, type, sP10: $i > $o).
% 9.35/2.83 thf(func_def_350, type, sP11: $i > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_351, type, sP12: $i > ($i > $o) > $i > $i > $o).
% 9.35/2.83 thf(func_def_352, type, sP13: ($i > $o) > $o).
% 9.35/2.83 thf(func_def_353, type, sP14: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_354, type, sP15: $i > $i > $o).
% 9.35/2.83 thf(func_def_355, type, sP16: $i > $o).
% 9.35/2.83 thf(func_def_356, type, sP17: $i > $i > $o).
% 9.35/2.83 thf(func_def_357, type, sP18: $i > $i > $o).
% 9.35/2.83 thf(func_def_358, type, sP19: $i > $i > $o).
% 9.35/2.83 thf(func_def_359, type, sP20: ($i > $o) > $i > $o).
% 9.35/2.83 thf(func_def_361, type, sP22: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_362, type, sP23: $i > $o > $i > $i > $o).
% 9.35/2.83 thf(func_def_363, type, sP24: $i > $o > $i > $i > $o).
% 9.35/2.83 thf(func_def_364, type, sP25: $i > ($i > $i > $o) > $i > $o).
% 9.35/2.83 thf(func_def_365, type, sP26: $i > ($i > $i > $o) > $o).
% 9.35/2.83 thf(func_def_367, type, sP28: $i > $i > $o).
% 9.35/2.83 thf(func_def_369, type, sP30: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_370, type, sP31: $i > $i > $o).
% 9.35/2.83 thf(func_def_372, type, sP33: ($i > $o) > $o).
% 9.35/2.83 thf(func_def_374, type, sP35: $i > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_375, type, sP36: $i > $i > $i > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_376, type, sP37: $i > ($i > $i) > $i > $o).
% 9.35/2.83 thf(func_def_378, type, sP39: ($i > $o) > $i > $o).
% 9.35/2.83 thf(func_def_380, type, sP41: $i > $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_381, type, sP42: $i > $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_383, type, sP44: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_384, type, sP45: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_385, type, sP46: $i > ($i > $i) > $i > $o).
% 9.35/2.83 thf(func_def_387, type, sP48: $i > $i > $o > $o).
% 9.35/2.83 thf(func_def_388, type, sP49: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_391, type, sP52: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_392, type, sP53: $i > $i > ($i > $o) > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_393, type, sP54: $i > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_394, type, sP55: ($i > $o) > $i > $i > $o).
% 9.35/2.83 thf(func_def_395, type, sP56: ($i > $o) > $o).
% 9.35/2.83 thf(func_def_396, type, sP57: $i > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_399, type, sP60: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_401, type, sP62: $i > ($i > $o) > $o).
% 9.35/2.83 thf(func_def_403, type, sP64: ($i > $o) > $i > $o).
% 9.35/2.83 thf(func_def_405, type, sP66: $i > $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_407, type, sP68: $o > $i > $i > $o).
% 9.35/2.83 thf(func_def_408, type, sP69: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_412, type, sP73: ($i > $o) > $i > $o).
% 9.35/2.83 thf(func_def_413, type, sP74: ($i > $o) > $i > $o).
% 9.35/2.83 thf(func_def_414, type, sP75: ($i > $o) > $o).
% 9.35/2.83 thf(func_def_415, type, sP76: $o > $i > $i > $o).
% 9.35/2.83 thf(func_def_417, type, sP78: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_418, type, sP79: ($i > $o) > $i > $i > $o).
% 9.35/2.83 thf(func_def_419, type, sP80: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_420, type, sP81: $i > $i > $o).
% 9.35/2.83 thf(func_def_423, type, sP84: $i > $i > $o > $o).
% 9.35/2.83 thf(func_def_451, type, sK112: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_452, type, sK113: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_453, type, sK114: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_455, type, sK116: $i > $o).
% 9.35/2.83 thf(func_def_464, type, sK125: $i > $i > $i).
% 9.35/2.83 thf(func_def_468, type, sK129: $i > $o).
% 9.35/2.83 thf(func_def_469, type, sK130: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_470, type, sK131: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_472, type, sK133: ($i > $o) > $i).
% 9.35/2.83 thf(func_def_473, type, sK134: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_474, type, sK135: $i > $o).
% 9.35/2.83 thf(func_def_486, type, sK147: $i > $i).
% 9.35/2.83 thf(func_def_487, type, sK148: $i > $i).
% 9.35/2.83 thf(func_def_504, type, sK165: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_505, type, sK166: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_506, type, sK167: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_507, type, sK168: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_512, type, sK173: $i > $o).
% 9.35/2.83 thf(func_def_530, type, sK191: $i > $i > $i).
% 9.35/2.83 thf(func_def_531, type, sK192: $i > $i).
% 9.35/2.83 thf(func_def_540, type, sK201: $i > ($i > $o) > $i > $i > $i).
% 9.35/2.83 thf(func_def_541, type, sK202: $i > ($i > $o) > $i > $i > $i).
% 9.35/2.83 thf(func_def_542, type, sK203: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_546, type, sK207: $i > $o).
% 9.35/2.83 thf(func_def_562, type, sK223: ($i > $o) > $i).
% 9.35/2.83 thf(func_def_563, type, sK224: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_564, type, sK225: $i > $o).
% 9.35/2.83 thf(func_def_582, type, sK243: $i > $i > $i).
% 9.35/2.83 thf(func_def_583, type, sK244: $i > $i > $i).
% 9.35/2.83 thf(func_def_584, type, sK245: $i > $i > $i).
% 9.35/2.83 thf(func_def_585, type, sK246: $i > $i > $i).
% 9.35/2.83 thf(func_def_586, type, sK247: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_587, type, sK248: $i > $i > $i).
% 9.35/2.83 thf(func_def_588, type, sK249: $i > $i > $i).
% 9.35/2.83 thf(func_def_589, type, sK250: $i > $i > $i).
% 9.35/2.83 thf(func_def_590, type, sK251: $i > $i).
% 9.35/2.83 thf(func_def_591, type, sK252: $i > $i).
% 9.35/2.83 thf(func_def_592, type, sK253: $i > $i).
% 9.35/2.83 thf(func_def_593, type, sK254: $i > $i).
% 9.35/2.83 thf(func_def_594, type, sK255: $i > $i > $i).
% 9.35/2.83 thf(func_def_595, type, sK256: $i > $i > $i).
% 9.35/2.83 thf(func_def_596, type, sK257: $i > $i > $i).
% 9.35/2.83 thf(func_def_597, type, sK258: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_598, type, sK259: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_599, type, sK260: $i > $i).
% 9.35/2.83 thf(func_def_606, type, sK267: $i > $o).
% 9.35/2.83 thf(func_def_608, type, sK269: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_609, type, sK270: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_610, type, sK271: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_611, type, sK272: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_620, type, sK281: $i > $i > $o).
% 9.35/2.83 thf(func_def_621, type, sK282: ($i > $i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_622, type, sK283: $i > $i > ($i > $i > $o) > $i).
% 9.35/2.83 thf(func_def_623, type, sK284: $i > ($i > $i > $o) > $i).
% 9.35/2.83 thf(func_def_624, type, sK285: $i > ($i > $i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_646, type, sK307: $i > $i).
% 9.35/2.83 thf(func_def_654, type, sK315: $i > $i > $i).
% 9.35/2.83 thf(func_def_672, type, sK333: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_676, type, sK337: $i > $o).
% 9.35/2.83 thf(func_def_687, type, sK348: $i > $i > $i).
% 9.35/2.83 thf(func_def_688, type, sK349: $i > $i > $i).
% 9.35/2.83 thf(func_def_698, type, sK359: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_699, type, sK360: ($i > $o) > $i).
% 9.35/2.83 thf(func_def_700, type, sK361: $i > $o).
% 9.35/2.83 thf(func_def_714, type, sK375: $i > $i > $i).
% 9.35/2.83 thf(func_def_717, type, sK378: $i > $i > $o).
% 9.35/2.83 thf(func_def_730, type, sK391: $i > $i > $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_731, type, sK392: $i > $i > $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_732, type, sK393: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_736, type, sK397: $i > $o).
% 9.35/2.83 thf(func_def_758, type, sK419: $i > $o).
% 9.35/2.83 thf(func_def_765, type, sK426: $i > $i).
% 9.35/2.83 thf(func_def_767, type, sK428: $i > ($i > $i) > $i > $i).
% 9.35/2.83 thf(func_def_768, type, sK429: $i > ($i > $i) > $i > $i).
% 9.35/2.83 thf(func_def_779, type, sK440: $i > $o).
% 9.35/2.83 thf(func_def_781, type, sK442: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_782, type, sK443: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_793, type, sK454: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_794, type, sK455: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_795, type, sK456: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_796, type, sK457: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_801, type, sK462: $i > $o).
% 9.35/2.83 thf(func_def_802, type, sK463: $i > $o).
% 9.35/2.83 thf(func_def_803, type, sK464: ($i > $o) > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_804, type, sK465: ($i > $o) > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_828, type, sK489: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_829, type, sK490: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_840, type, sK501: $i > $i).
% 9.35/2.83 thf(func_def_841, type, sK502: ($i > $i) > $i > $i > $i).
% 9.35/2.83 thf(func_def_842, type, sK503: $i > ($i > $i) > $i > $i).
% 9.35/2.83 thf(func_def_847, type, sK508: $i > $o).
% 9.35/2.83 thf(func_def_848, type, sK509: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_856, type, sK517: $i > $i).
% 9.35/2.83 thf(func_def_859, type, sK520: $i > $i > ($i > $i) > $i).
% 9.35/2.83 thf(func_def_868, type, sK529: $i > $o).
% 9.35/2.83 thf(func_def_869, type, sK530: ($i > $o) > $i).
% 9.35/2.83 thf(func_def_876, type, sK537: $i > $o).
% 9.35/2.83 thf(func_def_879, type, sK540: $i > $i > $o).
% 9.35/2.83 thf(func_def_914, type, sK575: $i > $o).
% 9.35/2.83 thf(func_def_929, type, sK590: $i > $i > $i).
% 9.35/2.83 thf(func_def_930, type, sK591: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_931, type, sK592: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_936, type, sK597: $i > $o).
% 9.35/2.83 thf(func_def_946, type, sK607: $i > $i > ($i > $o) > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_947, type, sK608: $i > $i > ($i > $o) > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_950, type, sK611: $i > $o).
% 9.35/2.83 thf(func_def_951, type, sK612: $i > $o).
% 9.35/2.83 thf(func_def_952, type, sK613: $i > $o).
% 9.35/2.83 thf(func_def_974, type, sK635: ($i > $o) > $i > $i > $i).
% 9.35/2.83 thf(func_def_975, type, sK636: ($i > $o) > $i > $i > $i).
% 9.35/2.83 thf(func_def_976, type, sK637: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_979, type, sK640: $i > $o).
% 9.35/2.83 thf(func_def_980, type, sK641: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_981, type, sK642: ($i > $o) > $i).
% 9.35/2.83 thf(func_def_982, type, sK643: $i > $o).
% 9.35/2.83 thf(func_def_983, type, sK644: $i > $o).
% 9.35/2.83 thf(func_def_985, type, sK646: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_986, type, sK647: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1006, type, sK667: $i > $i > $o).
% 9.35/2.83 thf(func_def_1015, type, sK676: $i > $o).
% 9.35/2.83 thf(func_def_1016, type, sK677: $i > $i > $o).
% 9.35/2.83 thf(func_def_1028, type, sK689: $i > $i).
% 9.35/2.83 thf(func_def_1046, type, sK707: $i > $i).
% 9.35/2.83 thf(func_def_1048, type, sK709: $i > ($i > $i) > $i > $i).
% 9.35/2.83 thf(func_def_1055, type, sK716: $i > $o).
% 9.35/2.83 thf(func_def_1064, type, sK725: $i > $o).
% 9.35/2.83 thf(func_def_1066, type, sK727: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1067, type, sK728: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1072, type, sK733: $i > $o).
% 9.35/2.83 thf(func_def_1074, type, sK735: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1075, type, sK736: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_1076, type, sK737: $i > $i > $i).
% 9.35/2.83 thf(func_def_1088, type, sK749: $i > $i).
% 9.35/2.83 thf(func_def_1095, type, sK756: $i > $o).
% 9.35/2.83 thf(func_def_1117, type, sK778: $i > $i > $o).
% 9.35/2.83 thf(func_def_1120, type, sK781: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_1121, type, sK782: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_1126, type, sK787: $i > $o).
% 9.35/2.83 thf(func_def_1127, type, sK788: $i > $o).
% 9.35/2.83 thf(func_def_1128, type, sK789: ($i > $o) > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1129, type, sK790: ($i > $o) > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1143, type, sK804: $i > $i > $i > $o).
% 9.35/2.83 thf(func_def_1144, type, sK805: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_1158, type, sK819: $i > $i).
% 9.35/2.83 thf(func_def_1159, type, sK820: ($i > $i) > $i > $i > $i).
% 9.35/2.83 thf(func_def_1197, type, sK858: $i > $i > $o).
% 9.35/2.83 thf(func_def_1202, type, sK863: $i > $i).
% 9.35/2.83 thf(func_def_1205, type, sK866: $i > $i > ($i > $i) > $i).
% 9.35/2.83 thf(func_def_1215, type, sK876: $i > $i).
% 9.35/2.83 thf(func_def_1218, type, sK879: $i > $i > $i).
% 9.35/2.83 thf(func_def_1219, type, sK880: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_1220, type, sK881: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_1221, type, sK882: ($i > $o) > $i > $i).
% 9.35/2.83 thf(func_def_1222, type, sK883: $i > $o).
% 9.35/2.83 thf(func_def_1224, type, sK885: ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1225, type, sK886: ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1226, type, sK887: $i > $o).
% 9.35/2.83 thf(func_def_1232, type, sK893: $i > $o).
% 9.35/2.83 thf(func_def_1260, type, sK921: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_1268, type, sK929: ($i > $o) > $i > $i > $i).
% 9.35/2.83 thf(func_def_1269, type, sK930: $i > $o).
% 9.35/2.83 thf(func_def_1277, type, sK938: $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_1278, type, sK939: $i > $i > $i > $i > $i).
% 9.35/2.83 thf(func_def_1286, type, sK947: $i > $i > $i).
% 9.35/2.83 thf(func_def_1313, type, sK974: $i > $o).
% 9.35/2.83 thf(func_def_1327, type, sK988: $i > $i > $i).
% 9.35/2.83 thf(func_def_1328, type, sK989: $i > $i > $o > $i).
% 9.35/2.83 thf(func_def_1334, type, sK995: $i > $i > $o).
% 9.35/2.83 thf(func_def_1337, type, sK998: $i > $o).
% 9.35/2.83 thf(func_def_1339, type, sK1000: $i > ($i > $o) > $i).
% 9.35/2.83 thf(func_def_1341, type, ph1002: !>[X0: $tType]:(X0)).
% 9.35/2.83 thf(f9615,plain,(
% 9.35/2.83 $false),
% 9.35/2.83 inference(subsumption_resolution,[status(thm)],[f9560,f5904])).
% 9.35/2.83 thf(f5904,plain,(
% 9.35/2.83 ((breln @ sK295 @ sK295 @ sK294) != $true)),
% 9.35/2.83 inference(beta_eta_normalization,[status(thm)],[f4890])).
% 9.35/2.83 thf(f4890,plain,(
% 9.35/2.83 (((^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))) @ sK295 @ sK294) != $true)),
% 9.35/2.83 inference(definition_unfolding,[status(thm)],[f3346,f3736])).
% 9.35/2.83 thf(f3736,plain,(
% 9.35/2.83 (breln1 = (^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))))),
% 9.35/2.83 inference(cnf_transformation,[status(thm)],[f812])).
% 9.35/2.83 thf(f812,plain,(
% 9.35/2.83 (breln1 = (^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))))),
% 9.35/2.83 inference(fool_elimination,[status(thm)],[f811])).
% 9.35/2.83 thf(f811,plain,(
% 9.35/2.83 (breln1 = (^[X0 : $i, X1 : $i] : (breln @ X0 @ X0 @ X1)))),
% 9.35/2.83 inference(rectify,[status(thm)],[f289])).
% 9.35/2.83 thf(f289,axiom,(
% 9.35/2.83 (breln1 = (^[X3 : $i, X13 : $i] : (breln @ X3 @ X3 @ X13)))),
% 9.35/2.83 file('/export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2THF_16347.p',breln1)).
% 9.35/2.83 thf(f3346,plain,(
% 9.35/2.83 ((breln1 @ sK295 @ sK294) != $true)),
% 9.35/2.83 inference(cnf_transformation,[status(thm)],[f1794])).
% 9.35/2.83 thf(f1794,plain,(
% 9.35/2.83 (binunionLsub = $true) & (setunionsingleton1 = $true) & (replAx = $true) & (woz13rule0 = $true) & (eta2 = $true) & (setextT = $true) & (secondinupair = $true) & (setukpairinjR = $true) & (ap2p = $true) & (exuE1 = $true) & (quantDeMorgan1 = $true) & (symdiffI2 = $true) & (exuI3 = $true) & (setunionE = $true) & (subsetRefl = $true) & (doubleComplementE1 = $true) & (vacuousDall = $true) & (disjointsetsI1 = $true) & (iffalse = $true) & (binintersectRsub = $true) & (setukpairinjR11 = $true) & (setminusSubset1 = $true) & (setukpairinjR2 = $true) & (exu__Cong = $true) & (funcGraphProp1 = $true) & (binintersectEL = $true) & (setminusILneg = $true) & (doubleComplementSub2 = $true) & (subsetTrans = $true) & (complementUnionInPowersetComplement = $true) & (subbreln = $true) & (symdiffIneg1 = $true) & (complementTE1 = $true) & (setunionsingleton = $true) & (eqimpsubset1 = $true) & (kpairp = $true) & (complementT_lem = $true) & (prop2setI = $true) & (binintersectTERcontra = $true) & (emptyinPowerset = $true) & (setadjoinSub = $true) & (contraSubsetComplement = $true) & (woz13rule1 = $true) & (powersetI1 = $true) & (ap2apEq1 = $true) & (setminusT_lem = $true) & (brelnall1 = $true) & (breln1all2 = $true) & (inIntersectImpInIntersectUnions = $true) & (cartprodsndpairEq = $true) & (theeq = $true) & (upairset2E = $true) & (ifSingleton = $true) & (demorgan2b2 = $true) & (demorgan2a1 = $true) & (dpsetconstrER = $true) & (inIntersectImpInUnion2 = $true) & (singletonsuniq = $true) & (woz1_1 = $true) & (sepInPowerset = $true) & (demorgan1 = $true) & (upairsetIR = $true) & (setukpairinjR1 = $true) & (contrasubsetT3 = $true) & (setadjoinIR = $true) & (doubleComplementSub1 = $true) & (emptyInPowerset = $true) & (powersetI = $true) & (upairset2IR = $true) & (emptyinunitempty = $true) & (complementTnotintersectT = $true) & (quantDeMorgan2 = $true) & (nonemptyI1 = $true) & (upairequniteq = $true) & (exuE3e = $true) & (notinsingleton = $true) & (ex1E1 = $true) & (setunionI = $true) & (ubforcartprodlem1 = $true) & (dpsetconstrSub = $true) & (exuEu = $true) & (emptysetE = $true) & (singletonsubset = $true) & (complementTcontraSubset = $true) & (cartprodmempaircEq = $true) & (setadjoin__Cong = $true) & (binunionEcases = $true) & (doubleComplementEq = $true) & (lam2p = $true) & (foundationAx = $true) & (funcImageSingleton = $true) & (demorgan2a = $true) & (binunionIR = $true) & (notdexE = $true) & (binunionE = $true) & (upairinpowunion = $true) & (eqbreln = $true) & (ksndsingleton = $true) & (binintersectSubset5 = $true) & (lamProp = $true) & (ifp = $true) & (symdiffIneg2 = $true) & (theprop = $true) & (lamp = $true) & (binintersectSubset3 = $true) & (in__Cong = $true) & (quantDeMorgan4 = $true) & (descr__Cong = $true) & (iftrueorfalse = $true) & (subsetE2 = $true) & (powersetTI1 = $true) & (complementImpComplementIntersect = $true) & (omega0Ax = $true) & (binintersectTELcontra = $true) & (wellorderingAx = $true) & (subset2powerset = $true) & (dsetconstr__Cong = $true) & (cartprodpairin = $true) & (binintersectER = $true) & (binintersectLsub = $true) & (binintersectSubset4 = $true) & (binunionTEcontra = $true) & (dpsetconstrI = $true) & (subsetI2 = $true) & (setunionAx = $true) & (beta2 = $true) & (contrasubsetT2 = $true) & (binunionTIRcontra = $true) & (ex1I = $true) & (eta1 = $true) & (setukpairIL = $true) & (nonemptyI = $true) & (subsetI1 = $true) & (emptyE1 = $true) & (cartprodmempair = $true) & (nonemptyImpWitness = $true) & (funcGraphProp2 = $true) & (symdiffI1 = $true) & (notinemptyset = $true) & (sepSubset = $true) & (eqimpsubset2 = $true) & (kfstpairEq = $true) & (funcGraphProp3 = $true) & (inIntersectImpInUnion = $true) & (notequalI1 = $true) & (setunion__Cong = $true) & (demorgan2 = $true) & (setminusER = $true) & (setoftrueEq = $true) & (binintersectSubset1 = $true) & (kpairiskpair = $true) & (omegaSAx = $true) & (setext = $true) & (powersetE1 = $true) & (setminusIRneg = $true) & (ubforcartprodlem2 = $true) & (intersectInPowersetIntersectUnions = $true) & (funcext2 = $true) & (bs114d = $true) & (cartprodmempair1 = $true) & (uniqinunit = $true) & (dpsetconstrEL2 = $true) & (dsetconstrEL = $true) & (cartprodpairmemER = $true) & (eqinunit = $true) & (woz13rule2 = $true) & (setminusI = $true) & (powerset__Cong = $true) & (powersetE = $true) & (emptysetAx = $true) & (dpsetconstrERa = $true) & (dsetconstrER = $true) & (setukpairinjL1 = $true) & (ex1E2 = $true) & (brelnall2 = $true) & (setbeta = $true) & (subsetE = $true) & (setOfPairsIsBReln = $true) & (omega__Cong = $true) & (setminusEL = $true) & (iftrueProp1 = $true) & (omegaIndAx = $true) & (doubleComplementI1 = $true) & (setextsub = $true) & (subsetTI = $true) & (complementTI1 = $true) & (cartprodfstin = $true) & (funcGraphProp4 = $true) & (singletonsswitch = $true) & (setminusSubset2 = $true) & (symdiffE = $true) & (upairsubunion = $true) & (setadjoinIL = $true) & (setukpairinjL = $true) & (inPowerset = $true) & (cartprodfstpairEq = $true) & (singletoninpowunion = $true) & (woz1_3 = $true) & (demorgan2a2 = $true) & (funcextLem = $true) & (upairsetIL = $true) & (demorgan1b = $true) & (woz1_4 = $true) & (complementSubsetComplementIntersect = $true) & (funcext = $true) & (contrasubsetT1 = $true) & (iftrueProp2 = $true) & (setukpairinjR12 = $true) & (subPowSU = $true) & (setminusERneg = $true) & (binunionTE = $true) & (notsubsetI = $true) & (infuncsetfunc = $true) & (setadjoinE = $true) & (setunionE2 = $true) & (binunionT_lem = $true) & (demorgan2b = $true) & (exuE3u = $true) & (complementInPowersetComplementIntersect = $true) & (notequalI2 = $true) & (woz13rule3 = $true) & (subsetemptysetimpeq = $true) & (app = $true) & (lam2lamEq = $true) & (emptyI = $true) & (exuI2 = $true) & (powersetTE1 = $true) & (emptysetsubset = $true) & (notdallE = $true) & (kfstsingleton = $true) & (powersetT_lem = $true) & (setadjoinAx = $true) & (woz13rule4 = $true) & (woz1_5 = $true) & (contrasubsetT = $true) & (setminusLsub = $true) & (dpsetconstrEL1 = $true) & (noeltsimpempty = $true) & (setukpairIR = $true) & (cartprodpairsurjEq = $true) & (upairsetE = $true) & (setminusELneg = $true) & (setukpairinjL2 = $true) & (singletoninpowerset = $true) & (demorgan1a = $true) & (binunionIL = $true) & (dsetconstrI = $true) & (ksndpairEq = $true) & (beta1 = $true) & (setadjoinSub2 = $true) & (inCongP = $true) & (iffalseProp1 = $true) & (emptysetimpfalse = $true) & (descrp = $true) & (ubforcartprodlem3 = $true) & (funcinfuncset = $true) & (singletonprop = $true) & (prop2set2propI = $true) & (iftrue = $true) & (kpairsurjEq = $true) & (binunionTILcontra = $true) & (ex1I2 = $true) & (cartprodsndin = $true) & (nonemptyE1 = $true) & (cartprodpairmemEL = $true) & (emptyset__Cong = $true) & (binunionRsub = $true) & (inComplementUnionImpNotIn1 = $true) & (inComplementUnionImpInComplement1 = $true) & (setextAx = $true) & (iffalseProp2 = $true) & (ap2apEq2 = $true) & (powersetsubset = $true) & (binintersectI = $true) & (binintersectSubset2 = $true) & (exuE2 = $true) & (prop2setE = $true) & (apProp = $true) & (powersetAx = $true) & (setunionsingleton2 = $true) & (((in @ sK294 @ (breln1Set @ sK295)) = $true) & ((breln1 @ sK295 @ sK294) != $true)) & (woz1_2 = $true) & (binintersectT_lem = $true) & (quantDeMorgan3 = $true) & (setadjoinOr = $true) & (exuI1 = $true)),
% 9.35/2.83 inference(skolemisation,[status(esa),new_symbols(skolem,[sK294,sK295])],[f1182,f1793])).
% 9.35/2.83 thf(f1793,plain,(
% 9.35/2.83 ? [X0 : $i,X1 : $i] : (((in @ X0 @ (breln1Set @ X1)) = $true) & ((breln1 @ X1 @ X0) != $true)) => (((in @ sK294 @ (breln1Set @ sK295)) = $true) & ((breln1 @ sK295 @ sK294) != $true))),
% 9.35/2.83 introduced(definition,[],[choice_axiom])).
% 9.35/2.83 thf(f1182,plain,(
% 9.35/2.83 (binunionLsub = $true) & (setunionsingleton1 = $true) & (replAx = $true) & (woz13rule0 = $true) & (eta2 = $true) & (setextT = $true) & (secondinupair = $true) & (setukpairinjR = $true) & (ap2p = $true) & (exuE1 = $true) & (quantDeMorgan1 = $true) & (symdiffI2 = $true) & (exuI3 = $true) & (setunionE = $true) & (subsetRefl = $true) & (doubleComplementE1 = $true) & (vacuousDall = $true) & (disjointsetsI1 = $true) & (iffalse = $true) & (binintersectRsub = $true) & (setukpairinjR11 = $true) & (setminusSubset1 = $true) & (setukpairinjR2 = $true) & (exu__Cong = $true) & (funcGraphProp1 = $true) & (binintersectEL = $true) & (setminusILneg = $true) & (doubleComplementSub2 = $true) & (subsetTrans = $true) & (complementUnionInPowersetComplement = $true) & (subbreln = $true) & (symdiffIneg1 = $true) & (complementTE1 = $true) & (setunionsingleton = $true) & (eqimpsubset1 = $true) & (kpairp = $true) & (complementT_lem = $true) & (prop2setI = $true) & (binintersectTERcontra = $true) & (emptyinPowerset = $true) & (setadjoinSub = $true) & (contraSubsetComplement = $true) & (woz13rule1 = $true) & (powersetI1 = $true) & (ap2apEq1 = $true) & (setminusT_lem = $true) & (brelnall1 = $true) & (breln1all2 = $true) & (inIntersectImpInIntersectUnions = $true) & (cartprodsndpairEq = $true) & (theeq = $true) & (upairset2E = $true) & (ifSingleton = $true) & (demorgan2b2 = $true) & (demorgan2a1 = $true) & (dpsetconstrER = $true) & (inIntersectImpInUnion2 = $true) & (singletonsuniq = $true) & (woz1_1 = $true) & (sepInPowerset = $true) & (demorgan1 = $true) & (upairsetIR = $true) & (setukpairinjR1 = $true) & (contrasubsetT3 = $true) & (setadjoinIR = $true) & (doubleComplementSub1 = $true) & (emptyInPowerset = $true) & (powersetI = $true) & (upairset2IR = $true) & (emptyinunitempty = $true) & (complementTnotintersectT = $true) & (quantDeMorgan2 = $true) & (nonemptyI1 = $true) & (upairequniteq = $true) & (exuE3e = $true) & (notinsingleton = $true) & (ex1E1 = $true) & (setunionI = $true) & (ubforcartprodlem1 = $true) & (dpsetconstrSub = $true) & (exuEu = $true) & (emptysetE = $true) & (singletonsubset = $true) & (complementTcontraSubset = $true) & (cartprodmempaircEq = $true) & (setadjoin__Cong = $true) & (binunionEcases = $true) & (doubleComplementEq = $true) & (lam2p = $true) & (foundationAx = $true) & (funcImageSingleton = $true) & (demorgan2a = $true) & (binunionIR = $true) & (notdexE = $true) & (binunionE = $true) & (upairinpowunion = $true) & (eqbreln = $true) & (ksndsingleton = $true) & (binintersectSubset5 = $true) & (lamProp = $true) & (ifp = $true) & (symdiffIneg2 = $true) & (theprop = $true) & (lamp = $true) & (binintersectSubset3 = $true) & (in__Cong = $true) & (quantDeMorgan4 = $true) & (descr__Cong = $true) & (iftrueorfalse = $true) & (subsetE2 = $true) & (powersetTI1 = $true) & (complementImpComplementIntersect = $true) & (omega0Ax = $true) & (binintersectTELcontra = $true) & (wellorderingAx = $true) & (subset2powerset = $true) & (dsetconstr__Cong = $true) & (cartprodpairin = $true) & (binintersectER = $true) & (binintersectLsub = $true) & (binintersectSubset4 = $true) & (binunionTEcontra = $true) & (dpsetconstrI = $true) & (subsetI2 = $true) & (setunionAx = $true) & (beta2 = $true) & (contrasubsetT2 = $true) & (binunionTIRcontra = $true) & (ex1I = $true) & (eta1 = $true) & (setukpairIL = $true) & (nonemptyI = $true) & (subsetI1 = $true) & (emptyE1 = $true) & (cartprodmempair = $true) & (nonemptyImpWitness = $true) & (funcGraphProp2 = $true) & (symdiffI1 = $true) & (notinemptyset = $true) & (sepSubset = $true) & (eqimpsubset2 = $true) & (kfstpairEq = $true) & (funcGraphProp3 = $true) & (inIntersectImpInUnion = $true) & (notequalI1 = $true) & (setunion__Cong = $true) & (demorgan2 = $true) & (setminusER = $true) & (setoftrueEq = $true) & (binintersectSubset1 = $true) & (kpairiskpair = $true) & (omegaSAx = $true) & (setext = $true) & (powersetE1 = $true) & (setminusIRneg = $true) & (ubforcartprodlem2 = $true) & (intersectInPowersetIntersectUnions = $true) & (funcext2 = $true) & (bs114d = $true) & (cartprodmempair1 = $true) & (uniqinunit = $true) & (dpsetconstrEL2 = $true) & (dsetconstrEL = $true) & (cartprodpairmemER = $true) & (eqinunit = $true) & (woz13rule2 = $true) & (setminusI = $true) & (powerset__Cong = $true) & (powersetE = $true) & (emptysetAx = $true) & (dpsetconstrERa = $true) & (dsetconstrER = $true) & (setukpairinjL1 = $true) & (ex1E2 = $true) & (brelnall2 = $true) & (setbeta = $true) & (subsetE = $true) & (setOfPairsIsBReln = $true) & (omega__Cong = $true) & (setminusEL = $true) & (iftrueProp1 = $true) & (omegaIndAx = $true) & (doubleComplementI1 = $true) & (setextsub = $true) & (subsetTI = $true) & (complementTI1 = $true) & (cartprodfstin = $true) & (funcGraphProp4 = $true) & (singletonsswitch = $true) & (setminusSubset2 = $true) & (symdiffE = $true) & (upairsubunion = $true) & (setadjoinIL = $true) & (setukpairinjL = $true) & (inPowerset = $true) & (cartprodfstpairEq = $true) & (singletoninpowunion = $true) & (woz1_3 = $true) & (demorgan2a2 = $true) & (funcextLem = $true) & (upairsetIL = $true) & (demorgan1b = $true) & (woz1_4 = $true) & (complementSubsetComplementIntersect = $true) & (funcext = $true) & (contrasubsetT1 = $true) & (iftrueProp2 = $true) & (setukpairinjR12 = $true) & (subPowSU = $true) & (setminusERneg = $true) & (binunionTE = $true) & (notsubsetI = $true) & (infuncsetfunc = $true) & (setadjoinE = $true) & (setunionE2 = $true) & (binunionT_lem = $true) & (demorgan2b = $true) & (exuE3u = $true) & (complementInPowersetComplementIntersect = $true) & (notequalI2 = $true) & (woz13rule3 = $true) & (subsetemptysetimpeq = $true) & (app = $true) & (lam2lamEq = $true) & (emptyI = $true) & (exuI2 = $true) & (powersetTE1 = $true) & (emptysetsubset = $true) & (notdallE = $true) & (kfstsingleton = $true) & (powersetT_lem = $true) & (setadjoinAx = $true) & (woz13rule4 = $true) & (woz1_5 = $true) & (contrasubsetT = $true) & (setminusLsub = $true) & (dpsetconstrEL1 = $true) & (noeltsimpempty = $true) & (setukpairIR = $true) & (cartprodpairsurjEq = $true) & (upairsetE = $true) & (setminusELneg = $true) & (setukpairinjL2 = $true) & (singletoninpowerset = $true) & (demorgan1a = $true) & (binunionIL = $true) & (dsetconstrI = $true) & (ksndpairEq = $true) & (beta1 = $true) & (setadjoinSub2 = $true) & (inCongP = $true) & (iffalseProp1 = $true) & (emptysetimpfalse = $true) & (descrp = $true) & (ubforcartprodlem3 = $true) & (funcinfuncset = $true) & (singletonprop = $true) & (prop2set2propI = $true) & (iftrue = $true) & (kpairsurjEq = $true) & (binunionTILcontra = $true) & (ex1I2 = $true) & (cartprodsndin = $true) & (nonemptyE1 = $true) & (cartprodpairmemEL = $true) & (emptyset__Cong = $true) & (binunionRsub = $true) & (inComplementUnionImpNotIn1 = $true) & (inComplementUnionImpInComplement1 = $true) & (setextAx = $true) & (iffalseProp2 = $true) & (ap2apEq2 = $true) & (powersetsubset = $true) & (binintersectI = $true) & (binintersectSubset2 = $true) & (exuE2 = $true) & (prop2setE = $true) & (apProp = $true) & (powersetAx = $true) & (setunionsingleton2 = $true) & ? [X0 : $i,X1 : $i] : (((in @ X0 @ (breln1Set @ X1)) = $true) & ((breln1 @ X1 @ X0) != $true)) & (woz1_2 = $true) & (binintersectT_lem = $true) & (quantDeMorgan3 = $true) & (setadjoinOr = $true) & (exuI1 = $true)),
% 9.35/2.83 inference(flattening,[status(thm)],[f1181])).
% 9.35/2.83 thf(f1181,plain,(
% 9.35/2.83 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((in @ X0 @ (breln1Set @ X1)) = $true) & ((breln1 @ X1 @ X0) != $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)),
% 9.35/2.83 inference(ennf_transformation,[status(thm)],[f459])).
% 9.35/2.83 thf(f459,plain,(
% 9.35/2.83 ~((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) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (breln1Set @ X1)) = $true) => ((breln1 @ X1 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.35/2.83 inference(fool_elimination,[status(thm)],[f458])).
% 9.35/2.83 thf(f458,plain,(
% 9.35/2.83 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (breln1Set @ X1)) => (breln1 @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.35/2.83 inference(rectify,[status(thm)],[f293])).
% 9.35/2.83 thf(f293,negated_conjecture,(
% 9.35/2.83 ~(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 => ! [X13 : $i,X3 : $i] : ((in @ X13 @ (breln1Set @ X3)) => (breln1 @ X3 @ X13)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.35/2.83 inference(negated_conjecture,[status(cth)],[f292])).
% 9.35/2.83 thf(f292,conjecture,(
% 9.35/2.83 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 => ! [X13 : $i,X3 : $i] : ((in @ X13 @ (breln1Set @ X3)) => (breln1 @ X3 @ X13))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.35/2.83 file('/export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2THF_16347.p',breln1SetBreln1)).
% 9.35/2.83 thf(f9560,plain,(
% 9.35/2.83 ((breln @ sK295 @ sK295 @ sK294) = $true)),
% 9.35/2.83 inference(trivial_inequality_removal,[status(thm)],[f9556])).
% 9.35/2.83 thf(f9556,plain,(
% 9.35/2.83 ($true != $true) | ((breln @ sK295 @ sK295 @ sK294) = $true)),
% 9.35/2.83 inference(superposition,[status(thm)],[f5915,f5951])).
% 9.35/2.83 thf(f5951,plain,(
% 9.35/2.83 ((in @ sK294 @ (dsetconstr @ (powerset @ (cartprod @ sK295 @ sK295)) @ (breln @ sK295 @ sK295))) = $true)),
% 9.35/2.83 inference(beta_eta_normalization,[status(thm)],[f4889])).
% 9.35/2.83 thf(f4889,plain,(
% 9.35/2.83 ((in @ sK294 @ ((^[Y0 : $i]: (dsetconstr @ (powerset @ (cartprod @ Y0 @ Y0)) @ (^[Y1 : $i]: ((^[Y2 : $i]: ((^[Y3 : $i]: (breln @ Y2 @ Y2 @ Y3)))) @ Y0 @ Y1)))) @ sK295)) = $true)),
% 9.35/2.83 inference(definition_unfolding,[status(thm)],[f3347,f4667])).
% 9.35/2.83 thf(f4667,plain,(
% 9.35/2.83 (breln1Set = (^[Y0 : $i]: (dsetconstr @ (powerset @ (cartprod @ Y0 @ Y0)) @ (^[Y1 : $i]: ((^[Y2 : $i]: ((^[Y3 : $i]: (breln @ Y2 @ Y2 @ Y3)))) @ Y0 @ Y1)))))),
% 9.35/2.83 inference(definition_unfolding,[status(thm)],[f3194,f3736])).
% 9.35/2.83 thf(f3194,plain,(
% 9.35/2.83 (breln1Set = (^[Y0 : $i]: (dsetconstr @ (powerset @ (cartprod @ Y0 @ Y0)) @ (^[Y1 : $i]: (breln1 @ Y0 @ Y1)))))),
% 9.35/2.83 inference(cnf_transformation,[status(thm)],[f568])).
% 9.35/2.83 thf(f568,plain,(
% 9.35/2.83 (breln1Set = (^[Y0 : $i]: (dsetconstr @ (powerset @ (cartprod @ Y0 @ Y0)) @ (^[Y1 : $i]: (breln1 @ Y0 @ Y1)))))),
% 9.35/2.83 inference(fool_elimination,[status(thm)],[f567])).
% 9.35/2.83 thf(f567,plain,(
% 9.35/2.83 (breln1Set = (^[X0 : $i] : (dsetconstr @ (powerset @ (cartprod @ X0 @ X0)) @ (^[X1 : $i] : (breln1 @ X0 @ X1)))))),
% 9.35/2.83 inference(rectify,[status(thm)],[f291])).
% 9.35/2.83 thf(f291,axiom,(
% 9.35/2.83 (breln1Set = (^[X3 : $i] : (dsetconstr @ (powerset @ (cartprod @ X3 @ X3)) @ (^[X13 : $i] : (breln1 @ X3 @ X13)))))),
% 9.35/2.83 file('/export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2THF_16347.p',breln1Set)).
% 9.35/2.83 thf(f3347,plain,(
% 9.35/2.83 ((in @ sK294 @ (breln1Set @ sK295)) = $true)),
% 9.35/2.83 inference(cnf_transformation,[status(thm)],[f1794])).
% 9.35/2.83 thf(f5915,plain,(
% 9.35/2.83 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ X0)) != $true) | ((X0 @ X1) = $true)) )),
% 9.35/2.83 inference(beta_eta_normalization,[status(thm)],[f5914])).
% 9.35/2.83 thf(f5914,plain,(
% 9.35/2.83 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (((X0 @ X1) = $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true)) )),
% 9.35/2.83 inference(trivial_inequality_removal,[status(thm)],[f5260])).
% 9.35/2.83 thf(f5260,plain,(
% 9.35/2.83 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (((X0 @ X1) = $true) | ($true != $true) | ((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true)) )),
% 9.35/2.83 inference(definition_unfolding,[status(thm)],[f4103,f3456])).
% 9.35/2.83 thf(f3456,plain,(
% 9.35/2.83 (dsetconstrER = $true)),
% 9.35/2.83 inference(cnf_transformation,[status(thm)],[f1794])).
% 9.35/2.83 thf(f4103,plain,(
% 9.35/2.83 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true) | ((X0 @ X1) = $true) | (dsetconstrER != $true)) )),
% 9.35/2.83 inference(cnf_transformation,[status(thm)],[f2345])).
% 9.35/2.83 thf(f2345,plain,(
% 9.35/2.83 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true) | ((X0 @ X1) = $true)) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | (((in @ sK614 @ (dsetconstr @ sK615 @ (^[Y0 : $i]: (sK613 @ Y0)))) = $true) & ((sK613 @ sK614) != $true)))),
% 9.35/2.83 inference(skolemisation,[status(esa),new_symbols(skolem,[sK613,sK614,sK615])],[f2343,f2344])).
% 9.35/2.83 thf(f2344,plain,(
% 9.35/2.83 ? [X3 : $i > $o,X4 : $i,X5 : $i] : (((in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0)))) = $true) & ((X3 @ X4) != $true)) => (((in @ sK614 @ (dsetconstr @ sK615 @ (^[Y0 : $i]: (sK613 @ Y0)))) = $true) & ((sK613 @ sK614) != $true))),
% 9.35/2.83 introduced(definition,[],[choice_axiom])).
% 9.35/2.83 thf(f2343,plain,(
% 9.35/2.83 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true) | ((X0 @ X1) = $true)) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | ? [X3 : $i > $o,X4 : $i,X5 : $i] : (((in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0)))) = $true) & ((X3 @ X4) != $true)))),
% 9.35/2.83 inference(rectify,[status(thm)],[f2342])).
% 9.35/2.83 thf(f2342,plain,(
% 9.35/2.83 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true) | ((X0 @ X1) = $true)) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | ? [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) = $true) & ((X0 @ X1) != $true)))),
% 9.35/2.83 inference(nnf_transformation,[status(thm)],[f1204])).
% 9.35/2.83 thf(f1204,plain,(
% 9.35/2.83 ! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true) | ((X0 @ X1) = $true)) <=> (dsetconstrER = $true)),
% 9.35/2.83 inference(ennf_transformation,[status(thm)],[f457])).
% 9.35/2.83 thf(f457,plain,(
% 9.35/2.83 ! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) = $true) => ((X0 @ X1) = $true)) <=> (dsetconstrER = $true)),
% 9.35/2.83 inference(fool_elimination,[status(thm)],[f456])).
% 9.35/2.83 thf(f456,plain,(
% 9.35/2.83 (dsetconstrER = ! [X0 : $i > $o,X1 : $i,X2 : $i] : ((in @ X1 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X0 @ X3)))) => (X0 @ X1)))),
% 9.35/2.83 inference(rectify,[status(thm)],[f16])).
% 9.35/2.83 thf(f16,axiom,(
% 9.35/2.83 (dsetconstrER = ! [X0 : $i > $o,X1 : $i,X3 : $i] : ((in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))) => (X0 @ X1)))),
% 9.35/2.83 file('/export/starexec/sandbox2/tmp/tmp.JTRsgbwAEY/DTF2THF_16347.p',dsetconstrER)).
% 9.35/2.83 % SZS output end Proof for DTF2THF_16347
% 9.35/2.83 % (16493)------------------------------
% 9.35/2.83 % (16493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.35/2.83 % (16493)Termination reason: Refutation
% 9.35/2.83
% 9.35/2.83 % (16493)Memory used [KB]: 12153
% 9.35/2.83 % (16493)Time elapsed: 0.536 s
% 9.35/2.83 % (16493)Instructions burned: 1044 (million)
% 9.35/2.83 % (16493)------------------------------
% 9.35/2.83 % (16493)------------------------------
% 9.35/2.83 % (16445)Success in time 1.022 s
%------------------------------------------------------------------------------