%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU808^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n026.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:37:22 AM UTC 2026
% Result : Theorem 6.77s 2.19s
% Output : Refutation 6.77s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 5
% Syntax : Number of formulae : 31 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 5118 (1358 equ; 0 cnn)
% Maximal formula atoms : 328 ( 165 avg)
% Number of connectives : 2500 ( 31 ~; 16 |; 985 &; 161 @)
% ( 2 <=>;1305 =>; 0 <=; 0 <~>)
% Maximal formula depth : 331 ( 76 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 663 ( 660 usr; 329 con; 0-6 aty)
% Number of variables : 25 ( 5 ^; 14 !; 6 ?; 25 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_360,type,
image1: $i > ( $i > $i ) > $i ).
thf(func_def_364,type,
injective: $i > $i > $i > $o ).
thf(func_def_365,type,
injFuncSet: $i > $i > $i ).
thf(func_def_369,type,
surjective: $i > $i > $i > $o ).
thf(func_def_370,type,
surjFuncSet: $i > $i > $i ).
thf(func_def_378,type,
omegaS: $i > $i ).
thf(func_def_393,type,
sP0: $i > $i > $o ).
thf(func_def_395,type,
sP2: $i > $i > $i > $i > $o ).
thf(func_def_396,type,
sP3: $i > $i > $i > $o ).
thf(func_def_397,type,
sP4: $i > ( $i > $o ) > $o ).
thf(func_def_398,type,
sP5: ( $i > $o ) > $i > $i > $o ).
thf(func_def_399,type,
sP6: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_400,type,
sP7: $i > ( $i > $i > $o ) > $o ).
thf(func_def_403,type,
sP10: $i > $i > $i > $i > $o ).
thf(func_def_404,type,
sP11: $i > ( $i > $o ) > $o ).
thf(func_def_406,type,
sP13: $i > $i > $i > $o ).
thf(func_def_407,type,
sP14: $i > $i > $i > $o ).
thf(func_def_408,type,
sP15: $i > $i > $o ).
thf(func_def_409,type,
sP16: $i > $o ).
thf(func_def_410,type,
sP17: $i > $i > $o ).
thf(func_def_411,type,
sP18: $i > $i > $o ).
thf(func_def_412,type,
sP19: $i > $i > $o ).
thf(func_def_413,type,
sP20: $i > $i > $i > $o ).
thf(func_def_414,type,
sP21: $i > $i > $i > $o ).
thf(func_def_415,type,
sP22: $i > ( $i > $o ) > $o ).
thf(func_def_417,type,
sP24: $i > $i > $i > $o ).
thf(func_def_419,type,
sP26: $i > ( $i > $o ) > $o ).
thf(func_def_420,type,
sP27: $i > ( $i > $o ) > $o ).
thf(func_def_422,type,
sP29: $i > ( $i > $o ) > $i > $o ).
thf(func_def_423,type,
sP30: $o > $i > $i > $i > $o ).
thf(func_def_424,type,
sP31: $i > $i > $i > $o > $o ).
thf(func_def_425,type,
sP32: $i > $i > $o > $i > $i > $i > $o ).
thf(func_def_426,type,
sP33: $i > ( $i > $o ) > $o ).
thf(func_def_428,type,
sP35: $i > $i > $i > $i > $i > $o ).
thf(func_def_431,type,
sP38: $i > ( $i > $o ) > $o ).
thf(func_def_434,type,
sP41: $i > ( $i > $o ) > $o ).
thf(func_def_436,type,
sP43: $i > $i > ( $i > $i ) > $o ).
thf(func_def_438,type,
sP45: ( $i > $o ) > $o ).
thf(func_def_439,type,
sP46: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_440,type,
sP47: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_442,type,
sP49: $i > $i > $i > $i > $o ).
thf(func_def_443,type,
sP50: $i > $i > $i > $i > $o ).
thf(func_def_444,type,
sP51: $i > $i > ( $i > $i ) > $o ).
thf(func_def_446,type,
sP53: $i > ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_448,type,
sP55: $i > $i > $o ).
thf(func_def_451,type,
sP58: $i > $i > $o > $o ).
thf(func_def_452,type,
sP59: $i > $i > $i > $o ).
thf(func_def_453,type,
sP60: $i > ( $i > $o ) > $o ).
thf(func_def_454,type,
sP61: ( $i > $o ) > $i > $i > $i > $o ).
thf(func_def_455,type,
sP62: $i > ( $i > $o ) > $o ).
thf(func_def_456,type,
sP63: $i > ( $i > $o ) > $i > $o ).
thf(func_def_457,type,
sP64: $i > $i > ( $i > $i ) > $o ).
thf(func_def_460,type,
sP67: ( $i > $o ) > $i > $o ).
thf(func_def_461,type,
sP68: $i > $i > $i > ( $i > $o ) > $o ).
thf(func_def_462,type,
sP69: $i > $i > $i > $o ).
thf(func_def_464,type,
sP71: ( $i > $o ) > $o ).
thf(func_def_465,type,
sP72: $i > $i > $o > $o ).
thf(func_def_466,type,
sP73: ( $i > $o ) > $o ).
thf(func_def_467,type,
sP74: $o > $i > $i > $o ).
thf(func_def_468,type,
sP75: $i > $i > $i > $o ).
thf(func_def_469,type,
sP76: $i > $i > $i > $o ).
thf(func_def_470,type,
sP77: $i > ( $i > $o ) > $o ).
thf(func_def_471,type,
sP78: ( $i > $o ) > $i > $o ).
thf(func_def_472,type,
sP79: $i > $i > $i > $i > $o ).
thf(func_def_473,type,
sP80: $i > $o ).
thf(func_def_475,type,
sP82: ( $i > $o ) > $o ).
thf(func_def_476,type,
sP83: $o > $i > $i > $i > $o ).
thf(func_def_477,type,
sP84: $i > $i > $i > $i > $o ).
thf(func_def_478,type,
sP85: $i > $o ).
thf(func_def_479,type,
sP86: $i > $o ).
thf(func_def_480,type,
sP87: $i > $i > $i > $o ).
thf(func_def_481,type,
sP88: $i > $i > $i > $o ).
thf(func_def_482,type,
sP89: $i > ( $i > $o ) > $o ).
thf(func_def_484,type,
sP91: $i > ( $i > $i ) > $i > $o ).
thf(func_def_486,type,
sP93: $i > $i > $i > $o ).
thf(func_def_487,type,
sP94: $i > $i > $i > $o ).
thf(func_def_488,type,
sP95: $i > $i > $o > $o ).
thf(func_def_490,type,
sP97: $i > $i > $i > $o ).
thf(func_def_494,type,
sP101: ( $i > $i ) > $i > $i > $o ).
thf(func_def_496,type,
sP103: $i > $i > $o ).
thf(func_def_498,type,
sP105: ( $i > $o ) > $o ).
thf(func_def_514,type,
sK121: $i > $i > $i ).
thf(func_def_517,type,
sK124: $i > $i > $i > $i ).
thf(func_def_518,type,
sK125: $i > $i > $i > $i > $i ).
thf(func_def_519,type,
sK126: $i > $i > $i > $i > $i ).
thf(func_def_520,type,
sK127: $i > $i > $i > $i > $i ).
thf(func_def_528,type,
sK135: $i > $i ).
thf(func_def_538,type,
sK145: ( $i > $o ) > $i > $i > $i ).
thf(func_def_539,type,
sK146: ( $i > $o ) > $i > $i > $i ).
thf(func_def_540,type,
sK147: $i > ( $i > $o ) > $i ).
thf(func_def_543,type,
sK150: $i > $o ).
thf(func_def_548,type,
sK155: $i > $i > $o ).
thf(func_def_550,type,
sK157: $i > ( $i > $i > $o ) > $i ).
thf(func_def_551,type,
sK158: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_552,type,
sK159: $i > ( $i > $i > $o ) > $i ).
thf(func_def_553,type,
sK160: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_576,type,
sK183: $i > $i ).
thf(func_def_577,type,
sK184: ( $i > $i ) > $i > $i > $i ).
thf(func_def_600,type,
sK207: $i > $i ).
thf(func_def_601,type,
sK208: $i > $i > $i > $i > $i ).
thf(func_def_602,type,
sK209: $i > $i > $i > $i > $i ).
thf(func_def_620,type,
sK227: $i > $i > $o ).
thf(func_def_623,type,
sK230: $i > $o ).
thf(func_def_624,type,
sK231: ( $i > $o ) > $i > $i ).
thf(func_def_625,type,
sK232: $i > ( $i > $o ) > $i ).
thf(func_def_634,type,
sK241: $i > $i > $i > $i ).
thf(func_def_635,type,
sK242: $i > $i > $i > $i ).
thf(func_def_639,type,
sK246: $i > $i > $i ).
thf(func_def_640,type,
sK247: $i > $i > $i ).
thf(func_def_641,type,
sK248: $i > $i > $i > $i ).
thf(func_def_642,type,
sK249: $i > $i > $i ).
thf(func_def_643,type,
sK250: $i > $i > $i ).
thf(func_def_644,type,
sK251: $i > $i > $i ).
thf(func_def_645,type,
sK252: $i > $i > $i ).
thf(func_def_646,type,
sK253: $i > $i > $i ).
thf(func_def_647,type,
sK254: $i > $i ).
thf(func_def_648,type,
sK255: $i > $i ).
thf(func_def_649,type,
sK256: $i > $i ).
thf(func_def_650,type,
sK257: $i > $i ).
thf(func_def_651,type,
sK258: $i > $i > $i ).
thf(func_def_652,type,
sK259: $i > $i > $i ).
thf(func_def_653,type,
sK260: $i > $i > $i ).
thf(func_def_654,type,
sK261: $i > $i > $i > $i ).
thf(func_def_655,type,
sK262: $i > $i > $i > $i ).
thf(func_def_657,type,
sK264: $i > $i ).
thf(func_def_663,type,
sK270: $i > $i > $i > $i ).
thf(func_def_664,type,
sK271: $i > $i > $i > $i ).
thf(func_def_665,type,
sK272: $i > $i > $i > $i ).
thf(func_def_666,type,
sK273: $i > $i > $i > $i ).
thf(func_def_691,type,
sK298: $i > $i > $o ).
thf(func_def_695,type,
sK302: $i > $o ).
thf(func_def_696,type,
sK303: ( $i > $o ) > $i > $i ).
thf(func_def_697,type,
sK304: $i > ( $i > $o ) > $i ).
thf(func_def_723,type,
sK330: $i > ( $i > $o ) > $i ).
thf(func_def_724,type,
sK331: $i > ( $i > $o ) > $i ).
thf(func_def_725,type,
sK332: $i > ( $i > $o ) > $i ).
thf(func_def_726,type,
sK333: $i > $o ).
thf(func_def_730,type,
sK337: $i > $i ).
thf(func_def_731,type,
sK338: ( $i > $i ) > $i > $i > $i ).
thf(func_def_739,type,
sK346: $i > ( $i > $o ) > $i > $i ).
thf(func_def_741,type,
sK348: $i > $o ).
thf(func_def_750,type,
sK357: $i > $i > $o ).
thf(func_def_760,type,
sK367: $i > $i > $o > $i > $i > $i > $i ).
thf(func_def_772,type,
sK379: $i > $o ).
thf(func_def_773,type,
sK380: ( $i > $o ) > $i > $i ).
thf(func_def_774,type,
sK381: $i > ( $i > $o ) > $i ).
thf(func_def_775,type,
sK382: $i > $i > $i ).
thf(func_def_782,type,
sK389: $i > $i > $i > $i > $i > $i ).
thf(func_def_796,type,
sK403: $i > $i ).
thf(func_def_798,type,
sK405: $i > ( $i > $i ) > $i > $i ).
thf(func_def_799,type,
sK406: $i > $o ).
thf(func_def_806,type,
sK413: $i > $o ).
thf(func_def_807,type,
sK414: ( $i > $o ) > $i > $i ).
thf(func_def_808,type,
sK415: $i > $o ).
thf(func_def_810,type,
sK417: $i > ( $i > $o ) > $i ).
thf(func_def_811,type,
sK418: $i > ( $i > $o ) > $i ).
thf(func_def_812,type,
sK419: $i > $o ).
thf(func_def_816,type,
sK423: $i > $o ).
thf(func_def_817,type,
sK424: $i > $o ).
thf(func_def_818,type,
sK425: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_819,type,
sK426: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_823,type,
sK430: $i > $o ).
thf(func_def_824,type,
sK431: ( $i > $o ) > $i > $i ).
thf(func_def_825,type,
sK432: $i > ( $i > $o ) > $i ).
thf(func_def_827,type,
sK434: $i > $i ).
thf(func_def_829,type,
sK436: $i > ( $i > $i ) > $i > $i ).
thf(func_def_830,type,
sK437: $i > $i > ( $i > $i ) > $i ).
thf(func_def_840,type,
sK447: ( $i > $o ) > $i ).
thf(func_def_841,type,
sK448: $i > ( $i > $o ) > $i ).
thf(func_def_842,type,
sK449: $i > $o ).
thf(func_def_883,type,
sK490: $i > $o ).
thf(func_def_890,type,
sK497: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_891,type,
sK498: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_892,type,
sK499: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_893,type,
sK500: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_896,type,
sK503: $i > $i > $o ).
thf(func_def_907,type,
sK514: $i > $i > $i ).
thf(func_def_917,type,
sK524: $i > $i > $o ).
thf(func_def_921,type,
sK528: $i > $o ).
thf(func_def_924,type,
sK531: $i > $i > $i > $i > $i ).
thf(func_def_925,type,
sK532: $i > $i > $i > $i > $i ).
thf(func_def_926,type,
sK533: $i > $i > $i > $i > $i ).
thf(func_def_927,type,
sK534: $i > $i > $i > $i > $i ).
thf(func_def_934,type,
sK541: $i > $i > $o ).
thf(func_def_939,type,
sK546: $i > $i > $o ).
thf(func_def_948,type,
sK555: $i > $i ).
thf(func_def_949,type,
sK556: ( $i > $i ) > $i > $i > $i ).
thf(func_def_950,type,
sK557: $i > $i > ( $i > $i ) > $i ).
thf(func_def_954,type,
sK561: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_955,type,
sK562: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_958,type,
sK565: $i > $o ).
thf(func_def_959,type,
sK566: $i > $o ).
thf(func_def_962,type,
sK569: $i > $i ).
thf(func_def_985,type,
sK592: $i > $i > $i ).
thf(func_def_988,type,
sK595: $i > $i > $i ).
thf(func_def_990,type,
sK597: $i > $o ).
thf(func_def_1000,type,
sK607: ( $i > $o ) > $i ).
thf(func_def_1001,type,
sK608: $i > $o ).
thf(func_def_1006,type,
sK613: $i > $o ).
thf(func_def_1011,type,
sK618: $i > $i > $o ).
thf(func_def_1025,type,
sK632: $i > $o ).
thf(func_def_1031,type,
sK638: $i > $i ).
thf(func_def_1034,type,
sK641: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1042,type,
sK649: $i > $i > $i > $i ).
thf(func_def_1043,type,
sK650: $i > $i > $i > $o ).
thf(func_def_1049,type,
sK656: $i > $o ).
thf(func_def_1055,type,
sK662: $i > $i ).
thf(func_def_1056,type,
sK663: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1057,type,
sK664: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1058,type,
sK665: $i > ( $i > $o ) > $i ).
thf(func_def_1062,type,
sK669: $i > $o ).
thf(func_def_1077,type,
sK684: $i > $i > $o ).
thf(func_def_1083,type,
sK690: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1084,type,
sK691: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1085,type,
sK692: $i > ( $i > $o ) > $i ).
thf(func_def_1088,type,
sK695: $i > $o ).
thf(func_def_1090,type,
sK697: $i > $i ).
thf(func_def_1111,type,
sK718: $i > $i ).
thf(func_def_1112,type,
sK719: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1113,type,
sK720: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1114,type,
sK721: $i > $i > $i > ( $i > $i ) > $i ).
thf(func_def_1163,type,
sK770: $i > $i ).
thf(func_def_1169,type,
sK776: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1170,type,
sK777: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1171,type,
sK778: ( $i > $o ) > $i > $i ).
thf(func_def_1175,type,
sK782: $i > $o ).
thf(func_def_1176,type,
sK783: $i > $i > $i > $i ).
thf(func_def_1177,type,
sK784: $i > $i > $i > $i ).
thf(func_def_1214,type,
sK821: $i > $i > $i ).
thf(func_def_1216,type,
sK823: $i > $o ).
thf(func_def_1218,type,
sK825: $i > ( $i > $o ) > $i ).
thf(func_def_1219,type,
sK826: ( $i > $o ) > $i ).
thf(func_def_1220,type,
sK827: $i > $o ).
thf(func_def_1229,type,
sK836: $i > $i > $o > $i ).
thf(func_def_1239,type,
sK846: ( $i > $o ) > $i ).
thf(func_def_1240,type,
sK847: $i > ( $i > $o ) > $i ).
thf(func_def_1241,type,
sK848: $i > $o ).
thf(func_def_1260,type,
sK867: $i > $o ).
thf(func_def_1268,type,
sK875: $i > $i > $i > $o ).
thf(func_def_1275,type,
sK882: $i > $i > $i > $i ).
thf(func_def_1276,type,
sK883: $i > $i > $i > $i > $i ).
thf(func_def_1287,type,
sK894: ( $i > $o ) > $i > $i ).
thf(func_def_1288,type,
sK895: ( $i > $o ) > $i > $i ).
thf(func_def_1289,type,
sK896: $i > ( $i > $o ) > $i ).
thf(func_def_1290,type,
sK897: $i > $o ).
thf(func_def_1304,type,
sK911: $i > $i > $i > $i > $i ).
thf(func_def_1318,type,
sK925: $i > $i > $i ).
thf(func_def_1319,type,
sK926: $i > $i ).
thf(func_def_1339,type,
sK946: $i > $i ).
thf(func_def_1340,type,
sK947: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1341,type,
sK948: $i > ( $i > $o ) > $i ).
thf(func_def_1342,type,
sK949: ( $i > $o ) > $i ).
thf(func_def_1343,type,
sK950: $i > $o ).
thf(func_def_1357,type,
sK964: $i > $i > $i > $i > $o ).
thf(func_def_1377,type,
sK984: $i > $i ).
thf(func_def_1378,type,
sK985: $i > $i ).
thf(func_def_1397,type,
sK1004: $i > $i > $i > $i ).
thf(func_def_1413,type,
sK1020: $i > $i ).
thf(func_def_1437,type,
sK1044: $i > $i > $i > $i ).
thf(func_def_1438,type,
sK1045: $i > $i > $i > $i > $i ).
thf(func_def_1452,type,
sK1059: $i > $o ).
thf(func_def_1453,type,
sK1060: ( $i > $o ) > $i > $i ).
thf(func_def_1454,type,
sK1061: $i > ( $i > $o ) > $i ).
thf(func_def_1459,type,
sK1066: $i > $o ).
thf(func_def_1473,type,
sK1080: $i > $i ).
thf(func_def_1476,type,
sK1083: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1484,type,
sK1091: $i > $i > $i > $i ).
thf(func_def_1485,type,
sK1092: $i > $i > $i > $i ).
thf(func_def_1495,type,
sK1102: $i > $o ).
thf(func_def_1508,type,
sK1115: $i > $i > $i > $i ).
thf(func_def_1518,type,
sK1125: $i > $i > $i ).
thf(func_def_1521,type,
sK1128: $i > $i > $i ).
thf(func_def_1533,type,
sK1140: $i > $o ).
thf(func_def_1534,type,
sK1141: $i > $o ).
thf(func_def_1535,type,
sK1142: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1536,type,
sK1143: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1548,type,
sK1155: $i > $i ).
thf(func_def_1549,type,
sK1156: $i > $i ).
thf(func_def_1550,type,
sK1157: ( $i > $i ) > $i > $i ).
thf(func_def_1551,type,
sK1158: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1558,type,
sK1165: $i > $i > $i ).
thf(func_def_1559,type,
sK1166: $i > $i > $i ).
thf(func_def_1560,type,
sK1167: $i > $o ).
thf(func_def_1561,type,
sK1168: ( $i > $o ) > $i ).
thf(func_def_1562,type,
sK1169: ( $i > $o ) > $i ).
thf(func_def_1563,type,
sK1170: $i > $o ).
thf(func_def_1567,type,
ph1174:
!>[X0: $tType] : X0 ).
thf(f8483,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f8477,f5192]) ).
thf(f5192,plain,
( $true
= ( in @ sK1162 @ omega ) ),
inference(cnf_transformation,[status(thm)],[f3438]) ).
thf(f3438,plain,
( ( lamp = $true )
& ( vacuousDall = $true )
& ( exu__Cong = $true )
& ( omegaIndAx = $true )
& ( cartprodfstpairEq = $true )
& ( dsetconstr__Cong = $true )
& ( binunionEcases = $true )
& ( emptyset__Cong = $true )
& ( demorgan2a2 = $true )
& ( setextsub = $true )
& ( setukpairinjL2 = $true )
& ( setoftrueEq = $true )
& ( descr__Cong = $true )
& ( setminusLsub = $true )
& ( quantDeMorgan2 = $true )
& ( complementImpComplementIntersect = $true )
& ( ex1I2 = $true )
& ( setminusERneg = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( breln1all2 = $true )
& ( omega0Ax = $true )
& ( doubleComplementSub2 = $true )
& ( cartprodpairin = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( ubforcartprodlem3 = $true )
& ( symdiffI2 = $true )
& ( woz1_2 = $true )
& ( setunionE = $true )
& ( setunionsingleton1 = $true )
& ( ap2apEq2 = $true )
& ( cartprodsndin = $true )
& ( bs114d = $true )
& ( exuI3 = $true )
& ( dpsetconstrERa = $true )
& ( symdiffIneg1 = $true )
& ( demorgan1b = $true )
& ( theprop = $true )
& ( binintersectSubset2 = $true )
& ( image1I = $true )
& ( setukpairIL = $true )
& ( ubforcartprodlem2 = $true )
& ( exuEu = $true )
& ( upairequniteq = $true )
& ( setadjoinOr = $true )
& ( surjCantorThm = $true )
& ( binintersectEL = $true )
& ( funcGraphProp4 = $true )
& ( breln1all1 = $true )
& ( emptysetAx = $true )
& ( iftrueProp2 = $true )
& ( emptyE1 = $true )
& ( notdexE = $true )
& ( ex1I = $true )
& ( symdiffE = $true )
& ( demorgan2a = $true )
& ( kpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( notdallE = $true )
& ( kpairiskpair = $true )
& ( powerset__Cong = $true )
& ( woz2A = $true )
& ( powersetI = $true )
& ( setadjoinSub = $true )
& ( setOfPairsIsBReln = $true )
& ( nonemptyImpWitness = $true )
& ( image1Ex1 = $true )
& ( eqbreln1 = $true )
& ( demorgan2a1 = $true )
& ( exuE3e = $true )
& ( cartprodpairmemEL = $true )
& ( setminusI = $true )
& ( binintersectSubset4 = $true )
& ( singletoninpowerset = $true )
& ( binunionT_lem = $true )
& ( cartprodpairmemER = $true )
& ( woz2W = $true )
& ( powersetAx = $true )
& ( setukpairinjR11 = $true )
& ( setadjoinIL = $true )
& ( powersetTE1 = $true )
& ( upairset2IR = $true )
& ( setextT = $true )
& ( woz1_3 = $true )
& ( contraSubsetComplement = $true )
& ( setunionI = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( secondinupair = $true )
& ( setminusIRneg = $true )
& ( funcext = $true )
& ( complementTcontraSubset = $true )
& ( uniqinunit = $true )
& ( choice2fnsingleton = $true )
& ( symdiffI1 = $true )
& ( woz13rule0 = $true )
& ( breln1unionE = $true )
& ( image1Equiv = $true )
& ( setunion__Cong = $true )
& ( image1E = $true )
& ( iffalse = $true )
& ( binintersectER = $true )
& ( singletonsswitch = $true )
& ( binintersectTERcontra = $true )
& ( iffalseProp1 = $true )
& ( ubforcartprodlem1 = $true )
& ( breln1SetBreln1 = $true )
& ( woz13rule4 = $true )
& ( inIntersectImpInUnion = $true )
& ( powersetE1 = $true )
& ( binunionRsub = $true )
& ( brelnall2 = $true )
& ( notinself2 = $true )
& ( doubleComplementI1 = $true )
& ( emptysetsubset = $true )
& ( upairsubunion = $true )
& ( injFuncSetFuncInj = $true )
& ( complementTI1 = $true )
& ( setminusSubset2 = $true )
& ( binunionTILcontra = $true )
& ( setext = $true )
& ( binunionIR = $true )
& ( setukpairinjR12 = $true )
& ( breln1compE = $true )
& ( setukpairinjL = $true )
& ( setukpairIR = $true )
& ( setukpairinjL1 = $true )
& ( subsetI2 = $true )
& ( setunionsingleton = $true )
& ( foundation2 = $true )
& ( funcinfuncset = $true )
& ( binunionE = $true )
& ( ex1E2 = $true )
& ( iftrueProp1 = $true )
& ( powersetI1 = $true )
& ( funcImageSingleton = $true )
& ( cartprodmempair = $true )
& ( wellorderingAx = $true )
& ( notequalI1 = $true )
& ( demorgan1a = $true )
& ( replAx = $true )
& ( dsetconstrI = $true )
& ( foundationAx = $true )
& ( kfstsingleton = $true )
& ( lam2p = $true )
& ( demorgan1 = $true )
& ( quantDeMorgan3 = $true )
& ( subsetE = $true )
& ( inIntersectImpInUnion2 = $true )
& ( complementTE1 = $true )
& ( iftrue = $true )
& ( woz13rule3 = $true )
& ( injFuncSetFuncIn = $true )
& ( breln1unionIL = $true )
& ( contrasubsetT3 = $true )
& ( setukpairinjR2 = $true )
& ( subset2powerset = $true )
& ( sepSubset = $true )
& ( complementUnionInPowersetComplement = $true )
& ( eqimpsubset1 = $true )
& ( doubleComplementE1 = $true )
& ( subsetI1 = $true )
& ( woz2B = $true )
& ( powersetT_lem = $true )
& ( setukpairinjR1 = $true )
& ( woz1_5 = $true )
& ( beta2 = $true )
& ( woz2Ex = $true )
& ( apProp = $true )
& ( cartprodmempair1 = $true )
& ( subsetE2 = $true )
& ( omega__Cong = $true )
& ( subbreln1 = $true )
& ( prop2setE = $true )
& ( omegaSAx = $true )
& ( powersetTI1 = $true )
& ( binintersectSubset1 = $true )
& ( image1Ex = $true )
& ( nonemptyI1 = $true )
& ( notinself = $true )
& ( doubleComplementSub1 = $true )
& ( binunionLsub = $true )
& ( app = $true )
& ( nonemptyE1 = $true )
& ( emptyinPowerset = $true )
& ( upairsetIL = $true )
& ( eta1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( notinemptyset = $true )
& ( setminusER = $true )
& ( emptyinunitempty = $true )
& ( quantDeMorgan4 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( symdiffIneg2 = $true )
& ( setminusSubset1 = $true )
& ( funcGraphProp2 = $true )
& ( subbreln = $true )
& ( doubleComplementEq = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( surjFuncSetFuncIn = $true )
& ( exuE3u = $true )
& ( dpsetconstrI = $true )
& ( demorgan2b = $true )
& ( breln1invE = $true )
& ( subsetTI = $true )
& ( cartprodpairsurjEq = $true )
& ( emptysetE = $true )
& ( binunionTIRcontra = $true )
& ( $true
= ( in @ sK1162 @ omega ) )
& ( $true
!= ( in @ ( omegaS @ sK1162 ) @ omega ) )
& ( setadjoinAx = $true )
& ( breln1unionprop = $true )
& ( dsetconstrER = $true )
& ( upairset2E = $true )
& ( prop2setI = $true )
& ( setadjoinSub2 = $true )
& ( demorgan2b2 = $true )
& ( theeq = $true )
& ( setminusEL = $true )
& ( upairsetE = $true )
& ( woz1_4 = $true )
& ( binintersectLsub = $true )
& ( emptyI = $true )
& ( complementTnotintersectT = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( funcGraphProp3 = $true )
& ( powersetE = $true )
& ( setadjoinIR = $true )
& ( emptysetimpfalse = $true )
& ( exuE2 = $true )
& ( setbeta = $true )
& ( contrasubsetT1 = $true )
& ( setadjoin__Cong = $true )
& ( woz13rule1 = $true )
& ( binunionTE = $true )
& ( breln1unionIR = $true )
& ( subPowSU = $true )
& ( binintersectTELcontra = $true )
& ( setminusELneg = $true )
& ( binintersectSubset5 = $true )
& ( quantDeMorgan1 = $true )
& ( exuI2 = $true )
& ( eqimpsubset2 = $true )
& ( disjointsetsI1 = $true )
& ( injFuncInInjFuncSet = $true )
& ( beta1 = $true )
& ( binintersectRsub = $true )
& ( setukpairinjR = $true )
& ( brelnall1 = $true )
& ( exuE1 = $true )
& ( exuI1 = $true )
& ( setadjoinE = $true )
& ( kpairp = $true )
& ( surjFuncSetFuncSurj = $true )
& ( setunionAx = $true )
& ( breln1compprop = $true )
& ( iftrueorfalse = $true )
& ( emptyInPowerset = $true )
& ( dpsetconstrEL1 = $true )
& ( inCongP = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( iffalseProp2 = $true )
& ( inPowerset = $true )
& ( sepInPowerset = $true )
& ( breln1invprop = $true )
& ( setminusILneg = $true )
& ( eta2 = $true )
& ( ap2apEq1 = $true )
& ( lamProp = $true )
& ( powersetsubset = $true )
& ( leftInvIsSurj = $true )
& ( dpsetconstrSub = $true )
& ( subsetRefl = $true )
& ( nonemptyI = $true )
& ( setunionE2 = $true )
& ( lam2lamEq = $true )
& ( funcext2 = $true )
& ( subsetTrans = $true )
& ( singletonprop = $true )
& ( ex1E1 = $true )
& ( singletonsuniq = $true )
& ( ksndpairEq = $true )
& ( breln1invI = $true )
& ( demorgan2 = $true )
& ( woz13rule2 = $true )
& ( binintersectI = $true )
& ( singletoninpowunion = $true )
& ( kfstpairEq = $true )
& ( notequalI2 = $true )
& ( funcextLem = $true )
& ( ifp = $true )
& ( cartprodfstin = $true )
& ( cartprodmempaircEq = $true )
& ( binintersectSubset3 = $true )
& ( prop2set2propI = $true )
& ( funcGraphProp1 = $true )
& ( descrp = $true )
& ( contrasubsetT = $true )
& ( ifSingleton = $true )
& ( binintersectT_lem = $true )
& ( ksndsingleton = $true )
& ( breln1unionI = $true )
& ( breln1unionEcases = $true )
& ( eqinunit = $true )
& ( notsubsetI = $true )
& ( setextAx = $true )
& ( contrasubsetT2 = $true )
& ( binunionTEcontra = $true )
& ( upairinpowunion = $true )
& ( infuncsetfunc = $true )
& ( breln1compEex = $true )
& ( dpsetconstrEL2 = $true )
& ( notinsingleton = $true )
& ( ap2p = $true )
& ( binunionIL = $true )
& ( eqbreln = $true )
& ( singletonsubset = $true )
& ( in__Cong = $true )
& ( woz1_1 = $true )
& ( setunionsingleton2 = $true )
& ( noeltsimpempty = $true )
& ( dpsetconstrER = $true )
& ( upairsetIR = $true )
& ( breln1unionCommutes = $true )
& ( breln1compI = $true )
& ( dsetconstrEL = $true )
& ( subsetemptysetimpeq = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1162])],[f1130,f3437]) ).
thf(f3437,plain,
( ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( ( in @ ( omegaS @ X0 ) @ omega )
!= $true ) )
=> ( ( $true
= ( in @ sK1162 @ omega ) )
& ( $true
!= ( in @ ( omegaS @ sK1162 ) @ omega ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1130,plain,
( ( lamp = $true )
& ( vacuousDall = $true )
& ( exu__Cong = $true )
& ( omegaIndAx = $true )
& ( cartprodfstpairEq = $true )
& ( dsetconstr__Cong = $true )
& ( binunionEcases = $true )
& ( emptyset__Cong = $true )
& ( demorgan2a2 = $true )
& ( setextsub = $true )
& ( setukpairinjL2 = $true )
& ( setoftrueEq = $true )
& ( descr__Cong = $true )
& ( setminusLsub = $true )
& ( quantDeMorgan2 = $true )
& ( complementImpComplementIntersect = $true )
& ( ex1I2 = $true )
& ( setminusERneg = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( breln1all2 = $true )
& ( omega0Ax = $true )
& ( doubleComplementSub2 = $true )
& ( cartprodpairin = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( ubforcartprodlem3 = $true )
& ( symdiffI2 = $true )
& ( woz1_2 = $true )
& ( setunionE = $true )
& ( setunionsingleton1 = $true )
& ( ap2apEq2 = $true )
& ( cartprodsndin = $true )
& ( bs114d = $true )
& ( exuI3 = $true )
& ( dpsetconstrERa = $true )
& ( symdiffIneg1 = $true )
& ( demorgan1b = $true )
& ( theprop = $true )
& ( binintersectSubset2 = $true )
& ( image1I = $true )
& ( setukpairIL = $true )
& ( ubforcartprodlem2 = $true )
& ( exuEu = $true )
& ( upairequniteq = $true )
& ( setadjoinOr = $true )
& ( surjCantorThm = $true )
& ( binintersectEL = $true )
& ( funcGraphProp4 = $true )
& ( breln1all1 = $true )
& ( emptysetAx = $true )
& ( iftrueProp2 = $true )
& ( emptyE1 = $true )
& ( notdexE = $true )
& ( ex1I = $true )
& ( symdiffE = $true )
& ( demorgan2a = $true )
& ( kpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( notdallE = $true )
& ( kpairiskpair = $true )
& ( powerset__Cong = $true )
& ( woz2A = $true )
& ( powersetI = $true )
& ( setadjoinSub = $true )
& ( setOfPairsIsBReln = $true )
& ( nonemptyImpWitness = $true )
& ( image1Ex1 = $true )
& ( eqbreln1 = $true )
& ( demorgan2a1 = $true )
& ( exuE3e = $true )
& ( cartprodpairmemEL = $true )
& ( setminusI = $true )
& ( binintersectSubset4 = $true )
& ( singletoninpowerset = $true )
& ( binunionT_lem = $true )
& ( cartprodpairmemER = $true )
& ( woz2W = $true )
& ( powersetAx = $true )
& ( setukpairinjR11 = $true )
& ( setadjoinIL = $true )
& ( powersetTE1 = $true )
& ( upairset2IR = $true )
& ( setextT = $true )
& ( woz1_3 = $true )
& ( contraSubsetComplement = $true )
& ( setunionI = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( secondinupair = $true )
& ( setminusIRneg = $true )
& ( funcext = $true )
& ( complementTcontraSubset = $true )
& ( uniqinunit = $true )
& ( choice2fnsingleton = $true )
& ( symdiffI1 = $true )
& ( woz13rule0 = $true )
& ( breln1unionE = $true )
& ( image1Equiv = $true )
& ( setunion__Cong = $true )
& ( image1E = $true )
& ( iffalse = $true )
& ( binintersectER = $true )
& ( singletonsswitch = $true )
& ( binintersectTERcontra = $true )
& ( iffalseProp1 = $true )
& ( ubforcartprodlem1 = $true )
& ( breln1SetBreln1 = $true )
& ( woz13rule4 = $true )
& ( inIntersectImpInUnion = $true )
& ( powersetE1 = $true )
& ( binunionRsub = $true )
& ( brelnall2 = $true )
& ( notinself2 = $true )
& ( doubleComplementI1 = $true )
& ( emptysetsubset = $true )
& ( upairsubunion = $true )
& ( injFuncSetFuncInj = $true )
& ( complementTI1 = $true )
& ( setminusSubset2 = $true )
& ( binunionTILcontra = $true )
& ( setext = $true )
& ( binunionIR = $true )
& ( setukpairinjR12 = $true )
& ( breln1compE = $true )
& ( setukpairinjL = $true )
& ( setukpairIR = $true )
& ( setukpairinjL1 = $true )
& ( subsetI2 = $true )
& ( setunionsingleton = $true )
& ( foundation2 = $true )
& ( funcinfuncset = $true )
& ( binunionE = $true )
& ( ex1E2 = $true )
& ( iftrueProp1 = $true )
& ( powersetI1 = $true )
& ( funcImageSingleton = $true )
& ( cartprodmempair = $true )
& ( wellorderingAx = $true )
& ( notequalI1 = $true )
& ( demorgan1a = $true )
& ( replAx = $true )
& ( dsetconstrI = $true )
& ( foundationAx = $true )
& ( kfstsingleton = $true )
& ( lam2p = $true )
& ( demorgan1 = $true )
& ( quantDeMorgan3 = $true )
& ( subsetE = $true )
& ( inIntersectImpInUnion2 = $true )
& ( complementTE1 = $true )
& ( iftrue = $true )
& ( woz13rule3 = $true )
& ( injFuncSetFuncIn = $true )
& ( breln1unionIL = $true )
& ( contrasubsetT3 = $true )
& ( setukpairinjR2 = $true )
& ( subset2powerset = $true )
& ( sepSubset = $true )
& ( complementUnionInPowersetComplement = $true )
& ( eqimpsubset1 = $true )
& ( doubleComplementE1 = $true )
& ( subsetI1 = $true )
& ( woz2B = $true )
& ( powersetT_lem = $true )
& ( setukpairinjR1 = $true )
& ( woz1_5 = $true )
& ( beta2 = $true )
& ( woz2Ex = $true )
& ( apProp = $true )
& ( cartprodmempair1 = $true )
& ( subsetE2 = $true )
& ( omega__Cong = $true )
& ( subbreln1 = $true )
& ( prop2setE = $true )
& ( omegaSAx = $true )
& ( powersetTI1 = $true )
& ( binintersectSubset1 = $true )
& ( image1Ex = $true )
& ( nonemptyI1 = $true )
& ( notinself = $true )
& ( doubleComplementSub1 = $true )
& ( binunionLsub = $true )
& ( app = $true )
& ( nonemptyE1 = $true )
& ( emptyinPowerset = $true )
& ( upairsetIL = $true )
& ( eta1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( notinemptyset = $true )
& ( setminusER = $true )
& ( emptyinunitempty = $true )
& ( quantDeMorgan4 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( symdiffIneg2 = $true )
& ( setminusSubset1 = $true )
& ( funcGraphProp2 = $true )
& ( subbreln = $true )
& ( doubleComplementEq = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( surjFuncSetFuncIn = $true )
& ( exuE3u = $true )
& ( dpsetconstrI = $true )
& ( demorgan2b = $true )
& ( breln1invE = $true )
& ( subsetTI = $true )
& ( cartprodpairsurjEq = $true )
& ( emptysetE = $true )
& ( binunionTIRcontra = $true )
& ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( ( in @ ( omegaS @ X0 ) @ omega )
!= $true ) )
& ( setadjoinAx = $true )
& ( breln1unionprop = $true )
& ( dsetconstrER = $true )
& ( upairset2E = $true )
& ( prop2setI = $true )
& ( setadjoinSub2 = $true )
& ( demorgan2b2 = $true )
& ( theeq = $true )
& ( setminusEL = $true )
& ( upairsetE = $true )
& ( woz1_4 = $true )
& ( binintersectLsub = $true )
& ( emptyI = $true )
& ( complementTnotintersectT = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( funcGraphProp3 = $true )
& ( powersetE = $true )
& ( setadjoinIR = $true )
& ( emptysetimpfalse = $true )
& ( exuE2 = $true )
& ( setbeta = $true )
& ( contrasubsetT1 = $true )
& ( setadjoin__Cong = $true )
& ( woz13rule1 = $true )
& ( binunionTE = $true )
& ( breln1unionIR = $true )
& ( subPowSU = $true )
& ( binintersectTELcontra = $true )
& ( setminusELneg = $true )
& ( binintersectSubset5 = $true )
& ( quantDeMorgan1 = $true )
& ( exuI2 = $true )
& ( eqimpsubset2 = $true )
& ( disjointsetsI1 = $true )
& ( injFuncInInjFuncSet = $true )
& ( beta1 = $true )
& ( binintersectRsub = $true )
& ( setukpairinjR = $true )
& ( brelnall1 = $true )
& ( exuE1 = $true )
& ( exuI1 = $true )
& ( setadjoinE = $true )
& ( kpairp = $true )
& ( surjFuncSetFuncSurj = $true )
& ( setunionAx = $true )
& ( breln1compprop = $true )
& ( iftrueorfalse = $true )
& ( emptyInPowerset = $true )
& ( dpsetconstrEL1 = $true )
& ( inCongP = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( iffalseProp2 = $true )
& ( inPowerset = $true )
& ( sepInPowerset = $true )
& ( breln1invprop = $true )
& ( setminusILneg = $true )
& ( eta2 = $true )
& ( ap2apEq1 = $true )
& ( lamProp = $true )
& ( powersetsubset = $true )
& ( leftInvIsSurj = $true )
& ( dpsetconstrSub = $true )
& ( subsetRefl = $true )
& ( nonemptyI = $true )
& ( setunionE2 = $true )
& ( lam2lamEq = $true )
& ( funcext2 = $true )
& ( subsetTrans = $true )
& ( singletonprop = $true )
& ( ex1E1 = $true )
& ( singletonsuniq = $true )
& ( ksndpairEq = $true )
& ( breln1invI = $true )
& ( demorgan2 = $true )
& ( woz13rule2 = $true )
& ( binintersectI = $true )
& ( singletoninpowunion = $true )
& ( kfstpairEq = $true )
& ( notequalI2 = $true )
& ( funcextLem = $true )
& ( ifp = $true )
& ( cartprodfstin = $true )
& ( cartprodmempaircEq = $true )
& ( binintersectSubset3 = $true )
& ( prop2set2propI = $true )
& ( funcGraphProp1 = $true )
& ( descrp = $true )
& ( contrasubsetT = $true )
& ( ifSingleton = $true )
& ( binintersectT_lem = $true )
& ( ksndsingleton = $true )
& ( breln1unionI = $true )
& ( breln1unionEcases = $true )
& ( eqinunit = $true )
& ( notsubsetI = $true )
& ( setextAx = $true )
& ( contrasubsetT2 = $true )
& ( binunionTEcontra = $true )
& ( upairinpowunion = $true )
& ( infuncsetfunc = $true )
& ( breln1compEex = $true )
& ( dpsetconstrEL2 = $true )
& ( notinsingleton = $true )
& ( ap2p = $true )
& ( binunionIL = $true )
& ( eqbreln = $true )
& ( singletonsubset = $true )
& ( in__Cong = $true )
& ( woz1_1 = $true )
& ( setunionsingleton2 = $true )
& ( noeltsimpempty = $true )
& ( dpsetconstrER = $true )
& ( upairsetIR = $true )
& ( breln1unionCommutes = $true )
& ( breln1compI = $true )
& ( dsetconstrEL = $true )
& ( subsetemptysetimpeq = $true ) ),
inference(flattening,[status(thm)],[f1129]) ).
thf(f1129,plain,
( ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( ( in @ ( omegaS @ X0 ) @ omega )
!= $true ) )
& ( notinself2 = $true )
& ( notinself = $true )
& ( foundation2 = $true )
& ( surjCantorThm = $true )
& ( leftInvIsSurj = $true )
& ( surjFuncSetFuncSurj = $true )
& ( surjFuncSetFuncIn = $true )
& ( injFuncSetFuncInj = $true )
& ( injFuncSetFuncIn = $true )
& ( injFuncInInjFuncSet = $true )
& ( image1I = $true )
& ( image1E = $true )
& ( image1Equiv = $true )
& ( image1Ex1 = $true )
& ( image1Ex = $true )
& ( woz2B = $true )
& ( woz2A = $true )
& ( woz2W = $true )
& ( woz2Ex = $true )
& ( breln1unionCommutes = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f812]) ).
thf(f812,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ( ( breln1unionE = $true )
=> ( ( breln1unionEcases = $true )
=> ( ( breln1unionCommutes = $true )
=> ( ( woz2Ex = $true )
=> ( ( woz2W = $true )
=> ( ( woz2A = $true )
=> ( ( woz2B = $true )
=> ( ( image1Ex = $true )
=> ( ( image1Ex1 = $true )
=> ( ( image1Equiv = $true )
=> ( ( image1E = $true )
=> ( ( image1I = $true )
=> ( ( injFuncInInjFuncSet = $true )
=> ( ( injFuncSetFuncIn = $true )
=> ( ( injFuncSetFuncInj = $true )
=> ( ( surjFuncSetFuncIn = $true )
=> ( ( surjFuncSetFuncSurj = $true )
=> ( ( leftInvIsSurj = $true )
=> ( ( surjCantorThm = $true )
=> ( ( foundation2 = $true )
=> ( ( notinself = $true )
=> ( ( notinself2 = $true )
=> ! [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
=> ( ( in @ ( omegaS @ X0 ) @ omega )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f811]) ).
thf(f811,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( in @ ( omegaS @ X0 ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f329]) ).
thf(f329,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( omegaS @ X1 ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f328]) ).
thf(f328,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( omegaS @ X1 ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.KvMWiWE9BC/DTF2THF_28453.p',omegaSp) ).
thf(f8477,plain,
( $true
!= ( in @ sK1162 @ omega ) ),
inference(trivial_inequality_removal,[status(thm)],[f8471]) ).
thf(f8471,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK1162 @ omega ) ) ),
inference(superposition,[status(thm)],[f7003,f6780]) ).
thf(f6780,plain,
! [X1: $i] :
( ( ( in @ ( setadjoin @ X1 @ X1 ) @ omega )
= $true )
| ( ( in @ X1 @ omega )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f6187]) ).
thf(f6187,plain,
! [X1: $i] :
( ( ( in @ ( setadjoin @ X1 @ X1 ) @ omega )
= $true )
| ( $true != $true )
| ( ( in @ X1 @ omega )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f4525,f5226]) ).
thf(f5226,plain,
omegaSAx = $true,
inference(cnf_transformation,[status(thm)],[f3438]) ).
thf(f4525,plain,
! [X1: $i] :
( ( ( in @ X1 @ omega )
!= $true )
| ( ( in @ ( setadjoin @ X1 @ X1 ) @ omega )
= $true )
| ( omegaSAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f2820]) ).
thf(f2820,plain,
( ( ( omegaSAx = $true )
| ( ( $true
= ( in @ sK808 @ omega ) )
& ( $true
!= ( in @ ( setadjoin @ sK808 @ sK808 ) @ omega ) ) ) )
& ( ! [X1: $i] :
( ( ( in @ X1 @ omega )
!= $true )
| ( ( in @ ( setadjoin @ X1 @ X1 ) @ omega )
= $true ) )
| ( omegaSAx != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK808])],[f2818,f2819]) ).
thf(f2819,plain,
( ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( $true
!= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
=> ( ( $true
= ( in @ sK808 @ omega ) )
& ( $true
!= ( in @ ( setadjoin @ sK808 @ sK808 ) @ omega ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2818,plain,
( ( ( omegaSAx = $true )
| ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( $true
!= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ) )
& ( ! [X1: $i] :
( ( ( in @ X1 @ omega )
!= $true )
| ( ( in @ ( setadjoin @ X1 @ X1 ) @ omega )
= $true ) )
| ( omegaSAx != $true ) ) ),
inference(rectify,[status(thm)],[f2817]) ).
thf(f2817,plain,
( ( ( omegaSAx = $true )
| ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( $true
!= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ) )
& ( ! [X0: $i] :
( ( ( in @ X0 @ omega )
!= $true )
| ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
| ( omegaSAx != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1314]) ).
thf(f1314,plain,
( ( omegaSAx = $true )
<=> ! [X0: $i] :
( ( ( in @ X0 @ omega )
!= $true )
| ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ) ),
inference(ennf_transformation,[status(thm)],[f788]) ).
thf(f788,plain,
( ( omegaSAx = $true )
<=> ! [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
=> ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ) ),
inference(fool_elimination,[status(thm)],[f787]) ).
thf(f787,plain,
! [X0: $i] :
( ( ( in @ X0 @ omega )
=> ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) )
= omegaSAx ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,axiom,
! [X1: $i] :
( ( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
= omegaSAx ),
file('/export/starexec/sandbox/tmp/tmp.KvMWiWE9BC/DTF2THF_28453.p',omegaSAx) ).
thf(f7003,plain,
( $true
!= ( in @ ( setadjoin @ sK1162 @ sK1162 ) @ omega ) ),
inference(beta_eta_normalization,[status(thm)],[f6602]) ).
thf(f6602,plain,
( $true
!= ( in
@ ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 )
@ sK1162 )
@ omega ) ),
inference(definition_unfolding,[status(thm)],[f5191,f3935]) ).
thf(f3935,plain,
( omegaS
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f800]) ).
thf(f800,plain,
( omegaS
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f799]) ).
thf(f799,plain,
( omegaS
= ( ^ [X0: $i] : ( setadjoin @ X0 @ X0 ) ) ),
inference(rectify,[status(thm)],[f327]) ).
thf(f327,axiom,
( omegaS
= ( ^ [X1: $i] : ( setadjoin @ X1 @ X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.KvMWiWE9BC/DTF2THF_28453.p',omegaS) ).
thf(f5191,plain,
( $true
!= ( in @ ( omegaS @ sK1162 ) @ omega ) ),
inference(cnf_transformation,[status(thm)],[f3438]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU808^1 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n026.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:34:55 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.47 ---- Original DTF file ---
% 0.32/0.47 thf(spec,logic,$$dhol).
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % File : SEU808^1 : TPTP v9.2.1. Released v3.7.0.
% 0.32/0.47 % Domain : Set Theory
% 0.32/0.47 % Problem : Omega and Peano
% 0.32/0.47 % Version : Especial.
% 0.32/0.47 % English : (! x:i.in x omega -> in (omegaS x) omega)
% 0.32/0.47
% 0.32/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.32/0.47 % Source : [Bro08]
% 0.32/0.47 % Names : ZFC310g [Bro08]
% 0.32/0.47
% 0.32/0.47 % Status : Theorem
% 0.32/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.80 v5.1.0, 1.00 v5.0.0, 0.80 v4.1.0, 0.67 v3.7.0
% 0.32/0.47 % Syntax : Number of formulae : 706 ( 327 unt; 378 typ; 327 def)
% 0.32/0.47 % Number of atoms : 2009 ( 469 equ; 0 cnn)
% 0.32/0.47 % Maximal formula atoms : 326 ( 6 avg)
% 0.32/0.47 % Number of connectives : 4403 ( 76 ~; 14 |; 54 &;3205 @)
% 0.32/0.47 % ( 17 <=>;1037 =>; 0 <=; 0 <~>)
% 0.32/0.47 % Maximal formula depth : 330 ( 2 avg)
% 0.32/0.47 % Number of types : 2 ( 0 usr)
% 0.32/0.47 % Number of type conns : 177 ( 177 >; 0 *; 0 +; 0 <<)
% 0.32/0.47 % Number of symbols : 381 ( 378 usr; 328 con; 0-4 aty)
% 0.32/0.47 % Number of variables : 1148 ( 76 ^;1028 !; 44 ?;1148 :)
% 0.32/0.47 % SPC : TH0_THM_EQU_NAR
% 0.32/0.47
% 0.32/0.47 % Comments : http://mathgate.info/detsetitem.php?id=441
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 thf(in_type,type,
% 0.32/0.47 in: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu_type,type,
% 0.32/0.47 exu: ( $i > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu,definition,
% 0.32/0.47 ( exu
% 0.32/0.47 = ( ^ [Xphi: $i > $o] :
% 0.32/0.47 ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setextAx_type,type,
% 0.32/0.47 setextAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setextAx,definition,
% 0.32/0.47 ( setextAx
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 <=> ( in @ Xx @ B ) )
% 0.32/0.47 => ( A = B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset_type,type,
% 0.32/0.47 emptyset: $i ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetAx_type,type,
% 0.32/0.47 emptysetAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetAx,definition,
% 0.32/0.47 ( emptysetAx
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin_type,type,
% 0.32/0.47 setadjoin: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinAx_type,type,
% 0.32/0.47 setadjoinAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinAx,definition,
% 0.32/0.47 ( setadjoinAx
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 <=> ( ( Xy = Xx )
% 0.32/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powerset_type,type,
% 0.32/0.47 powerset: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(powersetAx_type,type,
% 0.32/0.47 powersetAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetAx,definition,
% 0.32/0.47 ( powersetAx
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.47 <=> ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunion_type,type,
% 0.32/0.47 setunion: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(setunionAx_type,type,
% 0.32/0.47 setunionAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionAx,definition,
% 0.32/0.47 ( setunionAx
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.47 <=> ? [B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omega_type,type,
% 0.32/0.47 omega: $i ).
% 0.32/0.47
% 0.32/0.47 thf(omega0Ax_type,type,
% 0.32/0.47 omega0Ax: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omega0Ax,definition,
% 0.32/0.47 ( omega0Ax
% 0.32/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omegaSAx_type,type,
% 0.32/0.47 omegaSAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omegaSAx,definition,
% 0.32/0.47 ( omegaSAx
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ omega )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omegaIndAx_type,type,
% 0.32/0.47 omegaIndAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omegaIndAx,definition,
% 0.32/0.47 ( omegaIndAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( ( in @ emptyset @ A )
% 0.32/0.47 & ! [Xx: $i] :
% 0.32/0.47 ( ( ( in @ Xx @ omega )
% 0.32/0.47 & ( in @ Xx @ A ) )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ omega )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(replAx_type,type,
% 0.32/0.47 replAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(replAx,definition,
% 0.32/0.47 ( replAx
% 0.32/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.32/0.47 => ? [B: $i] :
% 0.32/0.47 ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 <=> ? [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(foundationAx_type,type,
% 0.32/0.47 foundationAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(foundationAx,definition,
% 0.32/0.47 ( foundationAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/0.47 => ? [B: $i] :
% 0.32/0.47 ( ( in @ B @ A )
% 0.32/0.47 & ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(wellorderingAx_type,type,
% 0.32/0.47 wellorderingAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(wellorderingAx,definition,
% 0.32/0.47 ( wellorderingAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ? [B: $i] :
% 0.32/0.47 ( ! [C: $i] :
% 0.32/0.47 ( ( in @ C @ B )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ A ) ) )
% 0.32/0.47 & ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( ( in @ Xx @ A )
% 0.32/0.47 & ( in @ Xy @ A ) )
% 0.32/0.47 => ( ! [C: $i] :
% 0.32/0.47 ( ( in @ C @ B )
% 0.32/0.47 => ( ( in @ Xx @ C )
% 0.32/0.47 <=> ( in @ Xy @ C ) ) )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 & ! [C: $i,D: $i] :
% 0.32/0.47 ( ( ( in @ C @ B )
% 0.32/0.47 & ( in @ D @ B ) )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ D ) )
% 0.32/0.47 | ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ D )
% 0.32/0.47 => ( in @ Xx @ C ) ) ) )
% 0.32/0.47 & ! [C: $i] :
% 0.32/0.47 ( ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.32/0.47 => ? [D: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ D @ B )
% 0.32/0.47 & ( in @ Xx @ C )
% 0.32/0.47 & ~ ? [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ D )
% 0.32/0.47 & ( in @ Xy @ C ) )
% 0.32/0.47 & ! [E: $i] :
% 0.32/0.47 ( ( in @ E @ B )
% 0.32/0.47 => ( ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ E )
% 0.32/0.47 => ( in @ Xy @ D ) )
% 0.32/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(descr_type,type,
% 0.32/0.47 descr: ( $i > $o ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(descrp_type,type,
% 0.32/0.47 descrp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(descrp,definition,
% 0.32/0.47 ( descrp
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ( Xphi
% 0.32/0.47 @ ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr_type,type,
% 0.32/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrI_type,type,
% 0.32/0.47 dsetconstrI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrI,definition,
% 0.32/0.47 ( dsetconstrI
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrEL_type,type,
% 0.32/0.47 dsetconstrEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrEL,definition,
% 0.32/0.47 ( dsetconstrEL
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrER_type,type,
% 0.32/0.47 dsetconstrER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrER,definition,
% 0.32/0.47 ( dsetconstrER
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE1_type,type,
% 0.32/0.47 exuE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE1,definition,
% 0.32/0.47 ( exuE1
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set_type,type,
% 0.32/0.47 prop2set: $o > $i ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setE_type,type,
% 0.32/0.47 prop2setE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setE,definition,
% 0.32/0.47 ( prop2setE
% 0.32/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.32/0.47 => Xphi ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetE_type,type,
% 0.32/0.47 emptysetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetE,definition,
% 0.32/0.47 ( emptysetE
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetimpfalse_type,type,
% 0.32/0.47 emptysetimpfalse: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetimpfalse,definition,
% 0.32/0.47 ( emptysetimpfalse
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => $false ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notinemptyset_type,type,
% 0.32/0.47 notinemptyset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notinemptyset,definition,
% 0.32/0.47 ( notinemptyset
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3e_type,type,
% 0.32/0.47 exuE3e: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3e,definition,
% 0.32/0.47 ( exuE3e
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setext_type,type,
% 0.32/0.47 setext: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setext,definition,
% 0.32/0.47 ( setext
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 => ( A = B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyI_type,type,
% 0.32/0.47 emptyI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyI,definition,
% 0.32/0.47 ( emptyI
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ A )
% 0.32/0.47 => ( A = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(noeltsimpempty_type,type,
% 0.32/0.47 noeltsimpempty: $o ).
% 0.32/0.47
% 0.32/0.47 thf(noeltsimpempty,definition,
% 0.32/0.47 ( noeltsimpempty
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ A )
% 0.32/0.47 => ( A = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setbeta_type,type,
% 0.32/0.47 setbeta: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setbeta,definition,
% 0.32/0.47 ( setbeta
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonempty_type,type,
% 0.32/0.47 nonempty: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonempty,definition,
% 0.32/0.47 ( nonempty
% 0.32/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyE1_type,type,
% 0.32/0.47 nonemptyE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyE1,definition,
% 0.32/0.47 ( nonemptyE1
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( nonempty @ A )
% 0.32/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI_type,type,
% 0.32/0.47 nonemptyI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI,definition,
% 0.32/0.47 ( nonemptyI
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( nonempty
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI1_type,type,
% 0.32/0.47 nonemptyI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI1,definition,
% 0.32/0.47 ( nonemptyI1
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIL_type,type,
% 0.32/0.47 setadjoinIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIL,definition,
% 0.32/0.47 ( setadjoinIL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinunitempty_type,type,
% 0.32/0.47 emptyinunitempty: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinunitempty,definition,
% 0.32/0.47 ( emptyinunitempty
% 0.32/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIR_type,type,
% 0.32/0.47 setadjoinIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIR,definition,
% 0.32/0.47 ( setadjoinIR
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinE_type,type,
% 0.32/0.47 setadjoinE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinE,definition,
% 0.32/0.47 ( setadjoinE
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 => ! [Xphi: $o] :
% 0.32/0.47 ( ( ( Xy = Xx )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => ( ( ( in @ Xy @ A )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinOr_type,type,
% 0.32/0.47 setadjoinOr: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinOr,definition,
% 0.32/0.47 ( setadjoinOr
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 => ( ( Xy = Xx )
% 0.32/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setoftrueEq_type,type,
% 0.32/0.47 setoftrueEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setoftrueEq,definition,
% 0.32/0.47 ( setoftrueEq
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : $true )
% 0.32/0.47 = A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI_type,type,
% 0.32/0.47 powersetI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI,definition,
% 0.32/0.47 ( powersetI
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinPowerset_type,type,
% 0.32/0.47 emptyinPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinPowerset,definition,
% 0.32/0.47 ( emptyinPowerset
% 0.32/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyInPowerset_type,type,
% 0.32/0.47 emptyInPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyInPowerset,definition,
% 0.32/0.47 ( emptyInPowerset
% 0.32/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE_type,type,
% 0.32/0.47 powersetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE,definition,
% 0.32/0.47 ( powersetE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionI_type,type,
% 0.32/0.47 setunionI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionI,definition,
% 0.32/0.47 ( setunionI
% 0.32/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( ( in @ B @ A )
% 0.32/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE_type,type,
% 0.32/0.47 setunionE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE,definition,
% 0.32/0.47 ( setunionE
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.47 => ! [Xphi: $o] :
% 0.32/0.47 ( ! [B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( ( in @ B @ A )
% 0.32/0.47 => Xphi ) )
% 0.32/0.47 => Xphi ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subPowSU_type,type,
% 0.32/0.47 subPowSU: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subPowSU,definition,
% 0.32/0.47 ( subPowSU
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE2_type,type,
% 0.32/0.47 exuE2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE2,definition,
% 0.32/0.47 ( exuE2
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyImpWitness_type,type,
% 0.32/0.47 nonemptyImpWitness: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyImpWitness,definition,
% 0.32/0.47 ( nonemptyImpWitness
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( nonempty @ A )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & $true ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(uniqinunit_type,type,
% 0.32/0.47 uniqinunit: $o ).
% 0.32/0.47
% 0.32/0.47 thf(uniqinunit,definition,
% 0.32/0.47 ( uniqinunit
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notinsingleton_type,type,
% 0.32/0.47 notinsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notinsingleton,definition,
% 0.32/0.47 ( notinsingleton
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx != Xy )
% 0.32/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqinunit_type,type,
% 0.32/0.47 eqinunit: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqinunit,definition,
% 0.32/0.47 ( eqinunit
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsswitch_type,type,
% 0.32/0.47 singletonsswitch: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsswitch,definition,
% 0.32/0.47 ( singletonsswitch
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetE_type,type,
% 0.32/0.47 upairsetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetE,definition,
% 0.32/0.47 ( upairsetE
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.47 => ( ( Xz = Xx )
% 0.32/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIL_type,type,
% 0.32/0.47 upairsetIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIL,definition,
% 0.32/0.47 ( upairsetIL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIR_type,type,
% 0.32/0.47 upairsetIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIR,definition,
% 0.32/0.47 ( upairsetIR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyE1_type,type,
% 0.32/0.47 emptyE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyE1,definition,
% 0.32/0.47 ( emptyE1
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ( ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = emptyset )
% 0.32/0.47 => $false ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(vacuousDall_type,type,
% 0.32/0.47 vacuousDall: $o ).
% 0.32/0.47
% 0.32/0.47 thf(vacuousDall,definition,
% 0.32/0.47 ( vacuousDall
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan1_type,type,
% 0.32/0.47 quantDeMorgan1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan1,definition,
% 0.32/0.47 ( quantDeMorgan1
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan2_type,type,
% 0.32/0.47 quantDeMorgan2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan2,definition,
% 0.32/0.47 ( quantDeMorgan2
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) )
% 0.32/0.47 => ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan3_type,type,
% 0.32/0.47 quantDeMorgan3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan3,definition,
% 0.32/0.47 ( quantDeMorgan3
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan4_type,type,
% 0.32/0.47 quantDeMorgan4: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan4,definition,
% 0.32/0.47 ( quantDeMorgan4
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) )
% 0.32/0.47 => ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setI_type,type,
% 0.32/0.47 prop2setI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setI,definition,
% 0.32/0.47 ( prop2setI
% 0.32/0.47 = ( ! [Xphi: $o] :
% 0.32/0.47 ( Xphi
% 0.32/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(set2prop_type,type,
% 0.32/0.47 set2prop: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set2propI_type,type,
% 0.32/0.47 prop2set2propI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set2propI,definition,
% 0.32/0.47 ( prop2set2propI
% 0.32/0.47 = ( ! [Xphi: $o] :
% 0.32/0.47 ( Xphi
% 0.32/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notdexE_type,type,
% 0.32/0.47 notdexE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notdexE,definition,
% 0.32/0.47 ( notdexE
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notdallE_type,type,
% 0.32/0.47 notdallE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notdallE,definition,
% 0.32/0.47 ( notdallE
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI1_type,type,
% 0.32/0.47 exuI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI1,definition,
% 0.32/0.47 ( exuI1
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI3_type,type,
% 0.32/0.47 exuI3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI3,definition,
% 0.32/0.47 ( exuI3
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.32/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI2_type,type,
% 0.32/0.47 exuI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI2,definition,
% 0.32/0.47 ( exuI2
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 <=> ( Xy = Xx ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(inCongP_type,type,
% 0.32/0.47 inCongP: $o ).
% 0.32/0.47
% 0.32/0.47 thf(inCongP,definition,
% 0.32/0.47 ( inCongP
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(in__Cong_type,type,
% 0.32/0.47 in__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(in__Cong,definition,
% 0.32/0.47 ( in__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3u_type,type,
% 0.32/0.47 exuE3u: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3u,definition,
% 0.32/0.47 ( exuE3u
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exu__Cong_type,type,
% 0.32/0.47 exu__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu__Cong,definition,
% 0.32/0.47 ( exu__Cong
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 <=> ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset__Cong_type,type,
% 0.32/0.47 emptyset__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset__Cong,definition,
% 0.32/0.47 ( emptyset__Cong
% 0.32/0.47 = ( emptyset = emptyset ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin__Cong_type,type,
% 0.32/0.47 setadjoin__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin__Cong,definition,
% 0.32/0.47 ( setadjoin__Cong
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ! [Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( Xz = Xu )
% 0.32/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.32/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powerset__Cong_type,type,
% 0.32/0.47 powerset__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powerset__Cong,definition,
% 0.32/0.47 ( powerset__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( powerset @ A )
% 0.32/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunion__Cong_type,type,
% 0.32/0.47 setunion__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunion__Cong,definition,
% 0.32/0.47 ( setunion__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( setunion @ A )
% 0.32/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omega__Cong_type,type,
% 0.32/0.47 omega__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omega__Cong,definition,
% 0.32/0.47 ( omega__Cong
% 0.32/0.47 = ( omega = omega ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuEu_type,type,
% 0.32/0.47 exuEu: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuEu,definition,
% 0.32/0.47 ( exuEu
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(descr__Cong_type,type,
% 0.32/0.47 descr__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(descr__Cong,definition,
% 0.32/0.47 ( descr__Cong
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.32/0.47 => ( ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr__Cong_type,type,
% 0.32/0.47 dsetconstr__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr__Cong,definition,
% 0.32/0.47 ( dsetconstr__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.32/0.47 => ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = ( dsetconstr @ B
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subset_type,type,
% 0.32/0.47 subset: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(disjoint_type,type,
% 0.32/0.47 disjoint: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(setsmeet_type,type,
% 0.32/0.47 setsmeet: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI1_type,type,
% 0.32/0.47 subsetI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI1,definition,
% 0.32/0.47 ( subsetI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset2_type,type,
% 0.32/0.47 eqimpsubset2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset2,definition,
% 0.32/0.47 ( eqimpsubset2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset1_type,type,
% 0.32/0.47 eqimpsubset1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset1,definition,
% 0.32/0.47 ( eqimpsubset1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI2_type,type,
% 0.32/0.47 subsetI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI2,definition,
% 0.32/0.47 ( subsetI2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetsubset_type,type,
% 0.32/0.47 emptysetsubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetsubset,definition,
% 0.32/0.47 ( emptysetsubset
% 0.32/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE_type,type,
% 0.32/0.47 subsetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE,definition,
% 0.32/0.47 ( subsetE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE2_type,type,
% 0.32/0.47 subsetE2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE2,definition,
% 0.32/0.47 ( subsetE2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notsubsetI_type,type,
% 0.32/0.47 notsubsetI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notsubsetI,definition,
% 0.32/0.47 ( notsubsetI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI1_type,type,
% 0.32/0.47 notequalI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI1,definition,
% 0.32/0.47 ( notequalI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ~ ( subset @ A @ B )
% 0.32/0.47 => ( A != B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI2_type,type,
% 0.32/0.47 notequalI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI2,definition,
% 0.32/0.47 ( notequalI2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ( A != B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetRefl_type,type,
% 0.32/0.47 subsetRefl: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetRefl,definition,
% 0.32/0.47 ( subsetRefl
% 0.32/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetTrans_type,type,
% 0.32/0.47 subsetTrans: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetTrans,definition,
% 0.32/0.47 ( subsetTrans
% 0.32/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( subset @ B @ C )
% 0.32/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub_type,type,
% 0.32/0.47 setadjoinSub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub,definition,
% 0.32/0.47 ( setadjoinSub
% 0.32/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub2_type,type,
% 0.32/0.47 setadjoinSub2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub2,definition,
% 0.32/0.47 ( setadjoinSub2
% 0.32/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subset2powerset_type,type,
% 0.32/0.47 subset2powerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subset2powerset,definition,
% 0.32/0.47 ( subset2powerset
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setextsub_type,type,
% 0.32/0.47 setextsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setextsub,definition,
% 0.32/0.47 ( setextsub
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( subset @ B @ A )
% 0.32/0.47 => ( A = B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetemptysetimpeq_type,type,
% 0.32/0.47 subsetemptysetimpeq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetemptysetimpeq,definition,
% 0.32/0.47 ( subsetemptysetimpeq
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( subset @ A @ emptyset )
% 0.32/0.47 => ( A = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI1_type,type,
% 0.32/0.47 powersetI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI1,definition,
% 0.32/0.47 ( powersetI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ B @ A )
% 0.32/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE1_type,type,
% 0.32/0.47 powersetE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE1,definition,
% 0.32/0.47 ( powersetE1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(inPowerset_type,type,
% 0.32/0.47 inPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(inPowerset,definition,
% 0.32/0.47 ( inPowerset
% 0.32/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetsubset_type,type,
% 0.32/0.47 powersetsubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetsubset,definition,
% 0.32/0.47 ( powersetsubset
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sepInPowerset_type,type,
% 0.32/0.47 sepInPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(sepInPowerset,definition,
% 0.32/0.47 ( sepInPowerset
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( in
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sepSubset_type,type,
% 0.32/0.47 sepSubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(sepSubset,definition,
% 0.32/0.47 ( sepSubset
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( subset
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunion_type,type,
% 0.32/0.47 binunion: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(binunionIL_type,type,
% 0.32/0.47 binunionIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionIL,definition,
% 0.32/0.47 ( binunionIL
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairset2IR_type,type,
% 0.32/0.47 upairset2IR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairset2IR,definition,
% 0.32/0.47 ( upairset2IR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionIR_type,type,
% 0.32/0.47 binunionIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionIR,definition,
% 0.32/0.47 ( binunionIR
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionEcases_type,type,
% 0.32/0.47 binunionEcases: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionEcases,definition,
% 0.32/0.47 ( binunionEcases
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.32/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/0.47 => ( ( ( in @ Xx @ A )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => ( ( ( in @ Xx @ B )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionE_type,type,
% 0.32/0.47 binunionE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionE,definition,
% 0.32/0.47 ( binunionE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionLsub_type,type,
% 0.32/0.47 binunionLsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionLsub,definition,
% 0.32/0.47 ( binunionLsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionRsub_type,type,
% 0.32/0.47 binunionRsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionRsub,definition,
% 0.32/0.47 ( binunionRsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersect_type,type,
% 0.32/0.47 binintersect: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectI_type,type,
% 0.32/0.47 binintersectI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectI,definition,
% 0.32/0.47 ( binintersectI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset5_type,type,
% 0.32/0.47 binintersectSubset5: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset5,definition,
% 0.32/0.47 ( binintersectSubset5
% 0.32/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.47 ( ( subset @ C @ A )
% 0.32/0.47 => ( ( subset @ C @ B )
% 0.32/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectEL_type,type,
% 0.32/0.47 binintersectEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectEL,definition,
% 0.32/0.47 ( binintersectEL
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectLsub_type,type,
% 0.32/0.47 binintersectLsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectLsub,definition,
% 0.32/0.47 ( binintersectLsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset2_type,type,
% 0.32/0.47 binintersectSubset2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset2,definition,
% 0.32/0.47 ( binintersectSubset2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( binintersect @ A @ B )
% 0.32/0.47 = A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset3_type,type,
% 0.32/0.47 binintersectSubset3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset3,definition,
% 0.32/0.47 ( binintersectSubset3
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( ( binintersect @ A @ B )
% 0.32/0.47 = B )
% 0.32/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectER_type,type,
% 0.32/0.47 binintersectER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectER,definition,
% 0.32/0.47 ( binintersectER
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(disjointsetsI1_type,type,
% 0.32/0.47 disjointsetsI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(disjointsetsI1,definition,
% 0.32/0.47 ( disjointsetsI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( in @ Xx @ B ) )
% 0.32/0.47 => ( ( binintersect @ A @ B )
% 0.32/0.47 = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectRsub_type,type,
% 0.32/0.47 binintersectRsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectRsub,definition,
% 0.32/0.47 ( binintersectRsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset4_type,type,
% 0.32/0.47 binintersectSubset4: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset4,definition,
% 0.32/0.47 ( binintersectSubset4
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ B @ A )
% 0.32/0.47 => ( ( binintersect @ A @ B )
% 0.32/0.47 = B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset1_type,type,
% 0.32/0.47 binintersectSubset1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset1,definition,
% 0.32/0.47 ( binintersectSubset1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( ( binintersect @ A @ B )
% 0.32/0.47 = A )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(bs114d_type,type,
% 0.32/0.47 bs114d: $o ).
% 0.32/0.47
% 0.32/0.47 thf(bs114d,definition,
% 0.32/0.47 ( bs114d
% 0.32/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.32/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(regular_type,type,
% 0.32/0.47 regular: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminus_type,type,
% 0.32/0.47 setminus: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(setminusI_type,type,
% 0.32/0.47 setminusI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusI,definition,
% 0.32/0.47 ( setminusI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusEL_type,type,
% 0.32/0.47 setminusEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusEL,definition,
% 0.32/0.47 ( setminusEL
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusER_type,type,
% 0.32/0.47 setminusER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusER,definition,
% 0.32/0.47 ( setminusER
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusSubset2_type,type,
% 0.32/0.47 setminusSubset2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusSubset2,definition,
% 0.32/0.47 ( setminusSubset2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( setminus @ A @ B )
% 0.32/0.47 = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusERneg_type,type,
% 0.32/0.47 setminusERneg: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusERneg,definition,
% 0.32/0.47 ( setminusERneg
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusELneg_type,type,
% 0.32/0.47 setminusELneg: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusELneg,definition,
% 0.32/0.47 ( setminusELneg
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusILneg_type,type,
% 0.32/0.47 setminusILneg: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusILneg,definition,
% 0.32/0.47 ( setminusILneg
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ A )
% 0.32/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusIRneg_type,type,
% 0.32/0.47 setminusIRneg: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusIRneg,definition,
% 0.32/0.47 ( setminusIRneg
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusLsub_type,type,
% 0.32/0.47 setminusLsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusLsub,definition,
% 0.32/0.47 ( setminusLsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusSubset1_type,type,
% 0.32/0.47 setminusSubset1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusSubset1,definition,
% 0.32/0.47 ( setminusSubset1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( ( setminus @ A @ B )
% 0.32/0.47 = emptyset )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiff_type,type,
% 0.32/0.47 symdiff: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffE_type,type,
% 0.32/0.47 symdiffE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffE,definition,
% 0.32/0.47 ( symdiffE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.32/0.47 => ! [Xphi: $o] :
% 0.32/0.47 ( ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => Xphi ) )
% 0.32/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => Xphi ) )
% 0.32/0.47 => Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffI1_type,type,
% 0.32/0.47 symdiffI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffI1,definition,
% 0.32/0.47 ( symdiffI1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffI2_type,type,
% 0.32/0.47 symdiffI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffI2,definition,
% 0.32/0.47 ( symdiffI2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffIneg1_type,type,
% 0.32/0.47 symdiffIneg1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffIneg1,definition,
% 0.32/0.47 ( symdiffIneg1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffIneg2_type,type,
% 0.32/0.47 symdiffIneg2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffIneg2,definition,
% 0.32/0.47 ( symdiffIneg2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(iskpair_type,type,
% 0.32/0.47 iskpair: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(secondinupair_type,type,
% 0.32/0.47 secondinupair: $o ).
% 0.32/0.47
% 0.32/0.47 thf(secondinupair,definition,
% 0.32/0.47 ( secondinupair
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairIL_type,type,
% 0.32/0.47 setukpairIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairIL,definition,
% 0.32/0.47 ( setukpairIL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairIR_type,type,
% 0.32/0.47 setukpairIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairIR,definition,
% 0.32/0.47 ( setukpairIR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kpairiskpair_type,type,
% 0.32/0.47 kpairiskpair: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kpairiskpair,definition,
% 0.32/0.47 ( kpairiskpair
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kpair_type,type,
% 0.32/0.47 kpair: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(kpairp_type,type,
% 0.32/0.47 kpairp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kpairp,definition,
% 0.32/0.47 ( kpairp
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprod_type,type,
% 0.32/0.47 cartprod: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsubset_type,type,
% 0.32/0.47 singletonsubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsubset,definition,
% 0.32/0.47 ( singletonsubset
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletoninpowerset_type,type,
% 0.32/0.47 singletoninpowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletoninpowerset,definition,
% 0.32/0.47 ( singletoninpowerset
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletoninpowunion_type,type,
% 0.32/0.47 singletoninpowunion: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletoninpowunion,definition,
% 0.32/0.47 ( singletoninpowunion
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairset2E_type,type,
% 0.32/0.47 upairset2E: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairset2E,definition,
% 0.32/0.47 ( upairset2E
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.47 => ( ( Xz = Xx )
% 0.32/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsubunion_type,type,
% 0.32/0.47 upairsubunion: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsubunion,definition,
% 0.32/0.47 ( upairsubunion
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairinpowunion_type,type,
% 0.32/0.47 upairinpowunion: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairinpowunion,definition,
% 0.32/0.47 ( upairinpowunion
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem1_type,type,
% 0.32/0.47 ubforcartprodlem1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem1,definition,
% 0.32/0.47 ( ubforcartprodlem1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem2_type,type,
% 0.32/0.47 ubforcartprodlem2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem2,definition,
% 0.32/0.47 ( ubforcartprodlem2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem3_type,type,
% 0.32/0.47 ubforcartprodlem3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem3,definition,
% 0.32/0.47 ( ubforcartprodlem3
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairin_type,type,
% 0.32/0.47 cartprodpairin: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairin,definition,
% 0.32/0.47 ( cartprodpairin
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempair1_type,type,
% 0.32/0.47 cartprodmempair1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempair1,definition,
% 0.32/0.47 ( cartprodmempair1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ? [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 & ( Xu
% 0.32/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempair_type,type,
% 0.32/0.47 cartprodmempair: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempair,definition,
% 0.32/0.47 ( cartprodmempair
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE2_type,type,
% 0.32/0.47 setunionE2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE2,definition,
% 0.32/0.47 ( setunionE2
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.47 => ? [X: $i] :
% 0.32/0.47 ( ( in @ X @ A )
% 0.32/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton1_type,type,
% 0.32/0.47 setunionsingleton1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton1,definition,
% 0.32/0.47 ( setunionsingleton1
% 0.32/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton2_type,type,
% 0.32/0.47 setunionsingleton2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton2,definition,
% 0.32/0.47 ( setunionsingleton2
% 0.32/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton_type,type,
% 0.32/0.47 setunionsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton,definition,
% 0.32/0.47 ( setunionsingleton
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.32/0.47 = Xx ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singleton_type,type,
% 0.32/0.47 singleton: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonprop_type,type,
% 0.32/0.47 singletonprop: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonprop,definition,
% 0.32/0.47 ( singletonprop
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) )
% 0.32/0.47 => ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ( singleton
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ex1_type,type,
% 0.32/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1E1_type,type,
% 0.32/0.47 ex1E1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1E1,definition,
% 0.32/0.47 ( ex1E1
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ( ex1 @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ex1I_type,type,
% 0.32/0.47 ex1I: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1I,definition,
% 0.32/0.47 ( ex1I
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xy = Xx ) ) )
% 0.32/0.47 => ( ex1 @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ex1I2_type,type,
% 0.32/0.47 ex1I2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1I2,definition,
% 0.32/0.47 ( ex1I2
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) )
% 0.32/0.47 => ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ( ex1 @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsuniq_type,type,
% 0.32/0.47 singletonsuniq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsuniq,definition,
% 0.32/0.47 ( singletonsuniq
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.32/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(atmost1p_type,type,
% 0.32/0.47 atmost1p: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(atleast2p_type,type,
% 0.32/0.47 atleast2p: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(atmost2p_type,type,
% 0.32/0.47 atmost2p: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetp_type,type,
% 0.32/0.47 upairsetp: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL1_type,type,
% 0.32/0.47 setukpairinjL1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL1,definition,
% 0.32/0.47 ( setukpairinjL1
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kfstsingleton_type,type,
% 0.32/0.47 kfstsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kfstsingleton,definition,
% 0.32/0.47 ( kfstsingleton
% 0.32/0.47 = ( ! [Xu: $i] :
% 0.32/0.47 ( ( iskpair @ Xu )
% 0.32/0.47 => ( singleton
% 0.32/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(theprop_type,type,
% 0.32/0.47 theprop: $o ).
% 0.32/0.47
% 0.32/0.47 thf(theprop,definition,
% 0.32/0.47 ( theprop
% 0.32/0.47 = ( ! [X: $i] :
% 0.32/0.47 ( ( singleton @ X )
% 0.32/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kfst_type,type,
% 0.32/0.47 kfst: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(kfstpairEq_type,type,
% 0.32/0.47 kfstpairEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kfstpairEq,definition,
% 0.32/0.47 ( kfstpairEq
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/0.47 = Xx ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodfstin_type,type,
% 0.32/0.47 cartprodfstin: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodfstin,definition,
% 0.32/0.47 ( cartprodfstin
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL2_type,type,
% 0.32/0.47 setukpairinjL2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL2,definition,
% 0.32/0.47 ( setukpairinjL2
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL_type,type,
% 0.32/0.47 setukpairinjL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL,definition,
% 0.32/0.47 ( setukpairinjL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.47 = ( kpair @ Xz @ Xu ) )
% 0.32/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR11_type,type,
% 0.32/0.47 setukpairinjR11: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR11,definition,
% 0.32/0.47 ( setukpairinjR11
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR12_type,type,
% 0.32/0.47 setukpairinjR12: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR12,definition,
% 0.32/0.47 ( setukpairinjR12
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR1_type,type,
% 0.32/0.47 setukpairinjR1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR1,definition,
% 0.32/0.47 ( setukpairinjR1
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.47 => ( ( Xz = Xu )
% 0.32/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairequniteq_type,type,
% 0.32/0.47 upairequniteq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairequniteq,definition,
% 0.32/0.47 ( upairequniteq
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR2_type,type,
% 0.32/0.47 setukpairinjR2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR2,definition,
% 0.32/0.47 ( setukpairinjR2
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR_type,type,
% 0.32/0.47 setukpairinjR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR,definition,
% 0.32/0.47 ( setukpairinjR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.47 = ( kpair @ Xz @ Xu ) )
% 0.32/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ksndsingleton_type,type,
% 0.32/0.47 ksndsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ksndsingleton,definition,
% 0.32/0.47 ( ksndsingleton
% 0.32/0.47 = ( ! [Xu: $i] :
% 0.32/0.47 ( ( iskpair @ Xu )
% 0.32/0.47 => ( singleton
% 0.32/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/0.47 @ ^ [Xx: $i] :
% 0.32/0.47 ( Xu
% 0.32/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ksnd_type,type,
% 0.32/0.47 ksnd: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(ksndpairEq_type,type,
% 0.32/0.47 ksndpairEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ksndpairEq,definition,
% 0.32/0.47 ( ksndpairEq
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/0.47 = Xy ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kpairsurjEq_type,type,
% 0.32/0.47 kpairsurjEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kpairsurjEq,definition,
% 0.32/0.47 ( kpairsurjEq
% 0.32/0.47 = ( ! [Xu: $i] :
% 0.32/0.47 ( ( iskpair @ Xu )
% 0.32/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/0.47 = Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodsndin_type,type,
% 0.32/0.47 cartprodsndin: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodsndin,definition,
% 0.32/0.47 ( cartprodsndin
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairmemEL_type,type,
% 0.32/0.47 cartprodpairmemEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairmemEL,definition,
% 0.32/0.47 ( cartprodpairmemEL
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairmemER_type,type,
% 0.32/0.47 cartprodpairmemER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairmemER,definition,
% 0.32/0.47 ( cartprodpairmemER
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempaircEq_type,type,
% 0.32/0.47 cartprodmempaircEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempaircEq,definition,
% 0.32/0.47 ( cartprodmempaircEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( kpair @ Xx @ Xy )
% 0.32/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodfstpairEq_type,type,
% 0.32/0.47 cartprodfstpairEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodfstpairEq,definition,
% 0.32/0.47 ( cartprodfstpairEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/0.47 = Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodsndpairEq_type,type,
% 0.32/0.47 cartprodsndpairEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodsndpairEq,definition,
% 0.32/0.47 ( cartprodsndpairEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/0.47 = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairsurjEq_type,type,
% 0.32/0.47 cartprodpairsurjEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairsurjEq,definition,
% 0.32/0.47 ( cartprodpairsurjEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/0.47 = Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(breln_type,type,
% 0.32/0.47 breln: $i > $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstr_type,type,
% 0.32/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrI_type,type,
% 0.32/0.47 dpsetconstrI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrI,definition,
% 0.32/0.47 ( dpsetconstrI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrSub_type,type,
% 0.32/0.47 dpsetconstrSub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrSub,definition,
% 0.32/0.47 ( dpsetconstrSub
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.47 ( subset
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.32/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setOfPairsIsBReln_type,type,
% 0.32/0.47 setOfPairsIsBReln: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setOfPairsIsBReln,definition,
% 0.32/0.47 ( setOfPairsIsBReln
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.47 ( breln @ A @ B
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrERa_type,type,
% 0.32/0.47 dpsetconstrERa: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrERa,definition,
% 0.32/0.47 ( dpsetconstrERa
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrEL1_type,type,
% 0.32/0.47 dpsetconstrEL1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrEL1,definition,
% 0.32/0.47 ( dpsetconstrEL1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrEL2_type,type,
% 0.32/0.47 dpsetconstrEL2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrEL2,definition,
% 0.32/0.47 ( dpsetconstrEL2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrER_type,type,
% 0.32/0.47 dpsetconstrER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrER,definition,
% 0.32/0.47 ( dpsetconstrER
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(func_type,type,
% 0.32/0.47 func: $i > $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcSet_type,type,
% 0.32/0.47 funcSet: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(funcImageSingleton_type,type,
% 0.32/0.47 funcImageSingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcImageSingleton,definition,
% 0.32/0.47 ( funcImageSingleton
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( singleton
% 0.32/0.47 @ ( dsetconstr @ B
% 0.32/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(apProp_type,type,
% 0.32/0.47 apProp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(apProp,definition,
% 0.32/0.47 ( apProp
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in
% 0.32/0.47 @ ( setunion
% 0.32/0.47 @ ( dsetconstr @ B
% 0.32/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.32/0.47 @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ap_type,type,
% 0.32/0.47 ap: $i > $i > $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(app_type,type,
% 0.32/0.47 app: $o ).
% 0.32/0.47
% 0.32/0.47 thf(app,definition,
% 0.32/0.47 ( app
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(infuncsetfunc_type,type,
% 0.32/0.47 infuncsetfunc: $o ).
% 0.32/0.47
% 0.32/0.47 thf(infuncsetfunc,definition,
% 0.32/0.47 ( infuncsetfunc
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ap2p_type,type,
% 0.32/0.47 ap2p: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ap2p,definition,
% 0.32/0.47 ( ap2p
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcinfuncset_type,type,
% 0.32/0.47 funcinfuncset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcinfuncset,definition,
% 0.32/0.47 ( funcinfuncset
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(lamProp_type,type,
% 0.32/0.47 lamProp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(lamProp,definition,
% 0.32/0.47 ( lamProp
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.47 => ( func @ A @ B
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xf @ Xx )
% 0.32/0.47 = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(lam_type,type,
% 0.32/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(lamp_type,type,
% 0.32/0.47 lamp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(lamp,definition,
% 0.32/0.47 ( lamp
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.47 => ( func @ A @ B
% 0.32/0.47 @ ( lam @ A @ B
% 0.32/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(lam2p_type,type,
% 0.32/0.47 lam2p: $o ).
% 0.32/0.47
% 0.32/0.47 thf(lam2p,definition,
% 0.32/0.47 ( lam2p
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.47 => ( in
% 0.32/0.47 @ ( lam @ A @ B
% 0.32/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.32/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(brelnall1_type,type,
% 0.32/0.47 brelnall1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(brelnall1,definition,
% 0.32/0.47 ( brelnall1
% 0.32/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.47 ( ( breln @ A @ B @ R )
% 0.32/0.47 => ! [Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ R )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(brelnall2_type,type,
% 0.32/0.47 brelnall2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(brelnall2,definition,
% 0.32/0.47 ( brelnall2
% 0.32/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.47 ( ( breln @ A @ B @ R )
% 0.32/0.47 => ! [Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ R )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ex1E2_type,type,
% 0.32/0.47 ex1E2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1E2,definition,
% 0.32/0.47 ( ex1E2
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ( ex1 @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcGraphProp1_type,type,
% 0.32/0.47 funcGraphProp1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcGraphProp1,definition,
% 0.32/0.47 ( funcGraphProp1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcGraphProp3_type,type,
% 0.32/0.47 funcGraphProp3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcGraphProp3,definition,
% 0.32/0.47 ( funcGraphProp3
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcGraphProp2_type,type,
% 0.32/0.47 funcGraphProp2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcGraphProp2,definition,
% 0.32/0.47 ( funcGraphProp2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.32/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcextLem_type,type,
% 0.32/0.47 funcextLem: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcextLem,definition,
% 0.32/0.47 ( funcextLem
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xg: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xg )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcGraphProp4_type,type,
% 0.32/0.47 funcGraphProp4: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcGraphProp4,definition,
% 0.32/0.47 ( funcGraphProp4
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.32/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subbreln_type,type,
% 0.32/0.47 subbreln: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subbreln,definition,
% 0.32/0.47 ( subbreln
% 0.32/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.47 ( ( breln @ A @ B @ R )
% 0.32/0.47 => ! [S: $i] :
% 0.32/0.47 ( ( breln @ A @ B @ S )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqbreln_type,type,
% 0.32/0.47 eqbreln: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqbreln,definition,
% 0.32/0.47 ( eqbreln
% 0.32/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.47 ( ( breln @ A @ B @ R )
% 0.32/0.47 => ! [S: $i] :
% 0.32/0.47 ( ( breln @ A @ B @ S )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.32/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcext_type,type,
% 0.32/0.47 funcext: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcext,definition,
% 0.32/0.47 ( funcext
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xg: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xg )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcext2_type,type,
% 0.32/0.47 funcext2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcext2,definition,
% 0.32/0.47 ( funcext2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.47 => ! [Xg: $i] :
% 0.32/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ap2apEq1_type,type,
% 0.32/0.47 ap2apEq1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ap2apEq1,definition,
% 0.32/0.47 ( ap2apEq1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ap2apEq2_type,type,
% 0.32/0.47 ap2apEq2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ap2apEq2,definition,
% 0.32/0.47 ( ap2apEq2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(beta1_type,type,
% 0.32/0.47 beta1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(beta1,definition,
% 0.32/0.47 ( beta1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( ap @ A @ B
% 0.32/0.47 @ ( lam @ A @ B
% 0.32/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.32/0.47 @ Xx )
% 0.32/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eta1_type,type,
% 0.32/0.47 eta1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eta1,definition,
% 0.32/0.47 ( eta1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ( ( lam @ A @ B
% 0.32/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.32/0.47 = Xf ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(lam2lamEq_type,type,
% 0.32/0.47 lam2lamEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(lam2lamEq,definition,
% 0.32/0.47 ( lam2lamEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.47 => ( ( lam @ A @ B
% 0.32/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.32/0.47 = ( lam @ A @ B
% 0.32/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(beta2_type,type,
% 0.32/0.47 beta2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(beta2,definition,
% 0.32/0.47 ( beta2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( ap @ A @ B
% 0.32/0.48 @ ( lam @ A @ B
% 0.32/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.32/0.48 @ Xx )
% 0.32/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(eta2_type,type,
% 0.32/0.48 eta2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(eta2,definition,
% 0.32/0.48 ( eta2
% 0.32/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.48 => ( ( lam @ A @ B
% 0.32/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.32/0.48 = Xf ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(iffalseProp1_type,type,
% 0.32/0.48 iffalseProp1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(iffalseProp1,definition,
% 0.32/0.48 ( iffalseProp1
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ~ Xphi
% 0.32/0.48 => ( in @ Xy
% 0.32/0.48 @ ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xz: $i] :
% 0.32/0.48 ( ( Xphi
% 0.32/0.48 & ( Xz = Xx ) )
% 0.32/0.48 | ( ~ Xphi
% 0.32/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(iffalseProp2_type,type,
% 0.32/0.48 iffalseProp2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(iffalseProp2,definition,
% 0.32/0.48 ( iffalseProp2
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ~ Xphi
% 0.32/0.48 => ( ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xz: $i] :
% 0.32/0.48 ( ( Xphi
% 0.32/0.48 & ( Xz = Xx ) )
% 0.32/0.48 | ( ~ Xphi
% 0.32/0.48 & ( Xz = Xy ) ) ) )
% 0.32/0.48 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(iftrueProp1_type,type,
% 0.32/0.48 iftrueProp1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(iftrueProp1,definition,
% 0.32/0.48 ( iftrueProp1
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( Xphi
% 0.32/0.48 => ( in @ Xx
% 0.32/0.48 @ ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xz: $i] :
% 0.32/0.48 ( ( Xphi
% 0.32/0.48 & ( Xz = Xx ) )
% 0.32/0.48 | ( ~ Xphi
% 0.32/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(iftrueProp2_type,type,
% 0.32/0.48 iftrueProp2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(iftrueProp2,definition,
% 0.32/0.48 ( iftrueProp2
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( Xphi
% 0.32/0.48 => ( ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xz: $i] :
% 0.32/0.48 ( ( Xphi
% 0.32/0.48 & ( Xz = Xx ) )
% 0.32/0.48 | ( ~ Xphi
% 0.32/0.48 & ( Xz = Xy ) ) ) )
% 0.32/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(ifSingleton_type,type,
% 0.32/0.48 ifSingleton: $o ).
% 0.32/0.48
% 0.32/0.48 thf(ifSingleton,definition,
% 0.32/0.48 ( ifSingleton
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( singleton
% 0.32/0.48 @ ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xz: $i] :
% 0.32/0.48 ( ( Xphi
% 0.32/0.48 & ( Xz = Xx ) )
% 0.32/0.48 | ( ~ Xphi
% 0.32/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(if_type,type,
% 0.32/0.48 if: $i > $o > $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(ifp_type,type,
% 0.32/0.48 ifp: $o ).
% 0.32/0.48
% 0.32/0.48 thf(ifp,definition,
% 0.32/0.48 ( ifp
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(theeq_type,type,
% 0.32/0.48 theeq: $o ).
% 0.32/0.48
% 0.32/0.48 thf(theeq,definition,
% 0.32/0.48 ( theeq
% 0.32/0.48 = ( ! [X: $i] :
% 0.32/0.48 ( ( singleton @ X )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ X )
% 0.32/0.48 => ( ( setunion @ X )
% 0.32/0.48 = Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(iftrue_type,type,
% 0.32/0.48 iftrue: $o ).
% 0.32/0.48
% 0.32/0.48 thf(iftrue,definition,
% 0.32/0.48 ( iftrue
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( Xphi
% 0.32/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.32/0.48 = Xx ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(iffalse_type,type,
% 0.32/0.48 iffalse: $o ).
% 0.32/0.48
% 0.32/0.48 thf(iffalse,definition,
% 0.32/0.48 ( iffalse
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ~ Xphi
% 0.32/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.32/0.48 = Xy ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(iftrueorfalse_type,type,
% 0.32/0.48 iftrueorfalse: $o ).
% 0.32/0.48
% 0.32/0.48 thf(iftrueorfalse,definition,
% 0.32/0.48 ( iftrueorfalse
% 0.32/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectT_lem_type,type,
% 0.32/0.48 binintersectT_lem: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectT_lem,definition,
% 0.32/0.48 ( binintersectT_lem
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionT_lem_type,type,
% 0.32/0.48 binunionT_lem: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionT_lem,definition,
% 0.32/0.48 ( binunionT_lem
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powersetT_lem_type,type,
% 0.32/0.48 powersetT_lem: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powersetT_lem,definition,
% 0.32/0.48 ( powersetT_lem
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusT_lem_type,type,
% 0.32/0.48 setminusT_lem: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusT_lem,definition,
% 0.32/0.48 ( setminusT_lem
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementT_lem_type,type,
% 0.32/0.48 complementT_lem: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementT_lem,definition,
% 0.32/0.48 ( complementT_lem
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setextT_type,type,
% 0.32/0.48 setextT: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setextT,definition,
% 0.32/0.48 ( setextT
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ X )
% 0.32/0.48 => ( in @ Xx @ Y ) ) )
% 0.32/0.48 => ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ Y )
% 0.32/0.48 => ( in @ Xx @ X ) ) )
% 0.32/0.48 => ( X = Y ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subsetTI_type,type,
% 0.32/0.48 subsetTI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetTI,definition,
% 0.32/0.48 ( subsetTI
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ X )
% 0.32/0.48 => ( in @ Xx @ Y ) ) )
% 0.32/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powersetTI1_type,type,
% 0.32/0.48 powersetTI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powersetTI1,definition,
% 0.32/0.48 ( powersetTI1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ X )
% 0.32/0.48 => ( in @ Xx @ Y ) ) )
% 0.32/0.48 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powersetTE1_type,type,
% 0.32/0.48 powersetTE1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powersetTE1,definition,
% 0.32/0.48 ( powersetTE1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ X @ ( powerset @ Y ) )
% 0.32/0.48 => ( ( in @ Xx @ X )
% 0.32/0.48 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementTI1_type,type,
% 0.32/0.48 complementTI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementTI1,definition,
% 0.32/0.48 ( complementTI1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ X )
% 0.32/0.48 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementTE1_type,type,
% 0.32/0.48 complementTE1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementTE1,definition,
% 0.32/0.48 ( complementTE1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectTELcontra_type,type,
% 0.32/0.48 binintersectTELcontra: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectTELcontra,definition,
% 0.32/0.48 ( binintersectTELcontra
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ X )
% 0.32/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectTERcontra_type,type,
% 0.32/0.48 binintersectTERcontra: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectTERcontra,definition,
% 0.32/0.48 ( binintersectTERcontra
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ Y )
% 0.32/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(contrasubsetT_type,type,
% 0.32/0.48 contrasubsetT: $o ).
% 0.32/0.48
% 0.32/0.48 thf(contrasubsetT,definition,
% 0.32/0.48 ( contrasubsetT
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.48 => ( ( in @ Xx @ Y )
% 0.32/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(contrasubsetT1_type,type,
% 0.32/0.48 contrasubsetT1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(contrasubsetT1,definition,
% 0.32/0.48 ( contrasubsetT1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( subset @ X @ Y )
% 0.32/0.48 => ( ~ ( in @ Xx @ Y )
% 0.32/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(contrasubsetT2_type,type,
% 0.32/0.48 contrasubsetT2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(contrasubsetT2,definition,
% 0.32/0.48 ( contrasubsetT2
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ X @ Y )
% 0.32/0.48 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(contrasubsetT3_type,type,
% 0.32/0.48 contrasubsetT3: $o ).
% 0.32/0.48
% 0.32/0.48 thf(contrasubsetT3,definition,
% 0.32/0.48 ( contrasubsetT3
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.32/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementI1_type,type,
% 0.32/0.48 doubleComplementI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementI1,definition,
% 0.32/0.48 ( doubleComplementI1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ X )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementE1_type,type,
% 0.32/0.48 doubleComplementE1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementE1,definition,
% 0.32/0.48 ( doubleComplementE1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.32/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementSub1_type,type,
% 0.32/0.48 doubleComplementSub1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementSub1,definition,
% 0.32/0.48 ( doubleComplementSub1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementSub2_type,type,
% 0.32/0.48 doubleComplementSub2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementSub2,definition,
% 0.32/0.48 ( doubleComplementSub2
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementEq_type,type,
% 0.32/0.48 doubleComplementEq: $o ).
% 0.32/0.48
% 0.32/0.48 thf(doubleComplementEq,definition,
% 0.32/0.48 ( doubleComplementEq
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ( X
% 0.32/0.48 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementTnotintersectT_type,type,
% 0.32/0.48 complementTnotintersectT: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementTnotintersectT,definition,
% 0.32/0.48 ( complementTnotintersectT
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementImpComplementIntersect_type,type,
% 0.32/0.48 complementImpComplementIntersect: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementImpComplementIntersect,definition,
% 0.32/0.48 ( complementImpComplementIntersect
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementSubsetComplementIntersect_type,type,
% 0.32/0.48 complementSubsetComplementIntersect: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementSubsetComplementIntersect,definition,
% 0.32/0.48 ( complementSubsetComplementIntersect
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementInPowersetComplementIntersect_type,type,
% 0.32/0.48 complementInPowersetComplementIntersect: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementInPowersetComplementIntersect,definition,
% 0.32/0.48 ( complementInPowersetComplementIntersect
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(contraSubsetComplement_type,type,
% 0.32/0.48 contraSubsetComplement: $o ).
% 0.32/0.48
% 0.32/0.48 thf(contraSubsetComplement,definition,
% 0.32/0.48 ( contraSubsetComplement
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ Y )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementTcontraSubset_type,type,
% 0.32/0.48 complementTcontraSubset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementTcontraSubset,definition,
% 0.32/0.48 ( complementTcontraSubset
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionTILcontra_type,type,
% 0.32/0.48 binunionTILcontra: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionTILcontra,definition,
% 0.32/0.48 ( binunionTILcontra
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.32/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionTIRcontra_type,type,
% 0.32/0.48 binunionTIRcontra: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionTIRcontra,definition,
% 0.32/0.48 ( binunionTIRcontra
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.32/0.48 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(inIntersectImpInUnion_type,type,
% 0.32/0.48 inIntersectImpInUnion: $o ).
% 0.32/0.48
% 0.32/0.48 thf(inIntersectImpInUnion,definition,
% 0.32/0.48 ( inIntersectImpInUnion
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.48 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(inIntersectImpInUnion2_type,type,
% 0.32/0.48 inIntersectImpInUnion2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(inIntersectImpInUnion2,definition,
% 0.32/0.48 ( inIntersectImpInUnion2
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.48 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(inIntersectImpInIntersectUnions_type,type,
% 0.32/0.48 inIntersectImpInIntersectUnions: $o ).
% 0.32/0.48
% 0.32/0.48 thf(inIntersectImpInIntersectUnions,definition,
% 0.32/0.48 ( inIntersectImpInIntersectUnions
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.48 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(intersectInPowersetIntersectUnions_type,type,
% 0.32/0.48 intersectInPowersetIntersectUnions: $o ).
% 0.32/0.48
% 0.32/0.48 thf(intersectInPowersetIntersectUnions,definition,
% 0.32/0.48 ( intersectInPowersetIntersectUnions
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(inComplementUnionImpNotIn1_type,type,
% 0.32/0.48 inComplementUnionImpNotIn1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(inComplementUnionImpNotIn1,definition,
% 0.32/0.48 ( inComplementUnionImpNotIn1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(inComplementUnionImpInComplement1_type,type,
% 0.32/0.48 inComplementUnionImpInComplement1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(inComplementUnionImpInComplement1,definition,
% 0.32/0.48 ( inComplementUnionImpInComplement1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionTE_type,type,
% 0.32/0.48 binunionTE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionTE,definition,
% 0.32/0.48 ( binunionTE
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xphi: $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.32/0.48 => ( ( ( in @ Xx @ X )
% 0.32/0.48 => Xphi )
% 0.32/0.48 => ( ( ( in @ Xx @ Y )
% 0.32/0.48 => Xphi )
% 0.32/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionTEcontra_type,type,
% 0.32/0.48 binunionTEcontra: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionTEcontra,definition,
% 0.32/0.48 ( binunionTEcontra
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ X )
% 0.32/0.48 => ( ~ ( in @ Xx @ Y )
% 0.32/0.48 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2a1_type,type,
% 0.32/0.48 demorgan2a1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2a1,definition,
% 0.32/0.48 ( demorgan2a1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(complementUnionInPowersetComplement_type,type,
% 0.32/0.48 complementUnionInPowersetComplement: $o ).
% 0.32/0.48
% 0.32/0.48 thf(complementUnionInPowersetComplement,definition,
% 0.32/0.48 ( complementUnionInPowersetComplement
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2a2_type,type,
% 0.32/0.48 demorgan2a2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2a2,definition,
% 0.32/0.48 ( demorgan2a2
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan1a_type,type,
% 0.32/0.48 demorgan1a: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan1a,definition,
% 0.32/0.48 ( demorgan1a
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.32/0.48 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan1b_type,type,
% 0.32/0.48 demorgan1b: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan1b,definition,
% 0.32/0.48 ( demorgan1b
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan1_type,type,
% 0.32/0.48 demorgan1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan1,definition,
% 0.32/0.48 ( demorgan1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.32/0.48 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2a_type,type,
% 0.32/0.48 demorgan2a: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2a,definition,
% 0.32/0.48 ( demorgan2a
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.48 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2b2_type,type,
% 0.32/0.48 demorgan2b2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2b2,definition,
% 0.32/0.48 ( demorgan2b2
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.48 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2b_type,type,
% 0.32/0.48 demorgan2b: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2b,definition,
% 0.32/0.48 ( demorgan2b
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2_type,type,
% 0.32/0.48 demorgan2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(demorgan2,definition,
% 0.32/0.48 ( demorgan2
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.32/0.48 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule0_type,type,
% 0.32/0.48 woz13rule0: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule0,definition,
% 0.32/0.48 ( woz13rule0
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.48 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule1_type,type,
% 0.32/0.48 woz13rule1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule1,definition,
% 0.32/0.48 ( woz13rule1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ X @ Z )
% 0.32/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule2_type,type,
% 0.32/0.48 woz13rule2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule2,definition,
% 0.32/0.48 ( woz13rule2
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ Y @ Z )
% 0.32/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule3_type,type,
% 0.32/0.48 woz13rule3: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule3,definition,
% 0.32/0.48 ( woz13rule3
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ X @ Y )
% 0.32/0.48 => ( ( subset @ X @ Z )
% 0.32/0.48 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule4_type,type,
% 0.32/0.48 woz13rule4: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz13rule4,definition,
% 0.32/0.48 ( woz13rule4
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ! [W: $i] :
% 0.32/0.48 ( ( in @ W @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ X @ Z )
% 0.32/0.48 => ( ( subset @ Y @ W )
% 0.32/0.48 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_1_type,type,
% 0.32/0.48 woz1_1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_1,definition,
% 0.32/0.48 ( woz1_1
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_2_type,type,
% 0.32/0.48 woz1_2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_2,definition,
% 0.32/0.48 ( woz1_2
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ! [W: $i] :
% 0.32/0.48 ( ( in @ W @ ( powerset @ A ) )
% 0.32/0.48 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.32/0.48 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_3_type,type,
% 0.32/0.48 woz1_3: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_3,definition,
% 0.32/0.48 ( woz1_3
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ! [Z: $i] :
% 0.32/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_4_type,type,
% 0.32/0.48 woz1_4: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_4,definition,
% 0.32/0.48 ( woz1_4
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_5_type,type,
% 0.32/0.48 woz1_5: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz1_5,definition,
% 0.32/0.48 ( woz1_5
% 0.32/0.48 = ( ! [A: $i,X: $i] :
% 0.32/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.48 => ! [Y: $i] :
% 0.32/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1_type,type,
% 0.32/0.48 breln1: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1all2_type,type,
% 0.32/0.48 breln1all2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1all2,definition,
% 0.32/0.48 ( breln1all2
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [Xphi: $i > $o] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ R )
% 0.32/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1Set_type,type,
% 0.32/0.48 breln1Set: $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(breln1SetBreln1_type,type,
% 0.32/0.48 breln1SetBreln1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1SetBreln1,definition,
% 0.32/0.48 ( breln1SetBreln1
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( in @ R @ ( breln1Set @ A ) )
% 0.32/0.48 => ( breln1 @ A @ R ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(transitive_type,type,
% 0.32/0.48 transitive: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(antisymmetric_type,type,
% 0.32/0.48 antisymmetric: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(reflexive_type,type,
% 0.32/0.48 reflexive: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(refltransitive_type,type,
% 0.32/0.48 refltransitive: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(refllinearorder_type,type,
% 0.32/0.48 refllinearorder: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(reflwellordering_type,type,
% 0.32/0.48 reflwellordering: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(choice2fnsingleton_type,type,
% 0.32/0.48 choice2fnsingleton: $o ).
% 0.32/0.48
% 0.32/0.48 thf(choice2fnsingleton,definition,
% 0.32/0.48 ( choice2fnsingleton
% 0.32/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ? [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ B )
% 0.32/0.48 & ( Xphi @ Xx @ Xy ) ) )
% 0.32/0.48 => ! [R: $i] :
% 0.32/0.48 ( ( in @ R @ ( breln1Set @ B ) )
% 0.32/0.48 => ( ( reflwellordering @ B @ R )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( singleton
% 0.32/0.48 @ ( dsetconstr @ B
% 0.32/0.48 @ ^ [Xy: $i] :
% 0.32/0.48 ( ( Xphi @ Xx @ Xy )
% 0.32/0.48 & ! [Xz: $i] :
% 0.32/0.48 ( ( in @ Xz @ B )
% 0.32/0.48 => ( ( Xphi @ Xx @ Xz )
% 0.32/0.48 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setOfPairsIsBReln1_type,type,
% 0.32/0.48 setOfPairsIsBReln1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setOfPairsIsBReln1,definition,
% 0.32/0.48 ( setOfPairsIsBReln1
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.32/0.48 ( breln1 @ A
% 0.32/0.48 @ ( dpsetconstr @ A @ A
% 0.32/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1all1_type,type,
% 0.32/0.48 breln1all1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1all1,definition,
% 0.32/0.48 ( breln1all1
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [Xphi: $i > $o] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ R )
% 0.32/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subbreln1_type,type,
% 0.32/0.48 subbreln1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subbreln1,definition,
% 0.32/0.48 ( subbreln1
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(eqbreln1_type,type,
% 0.32/0.48 eqbreln1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(eqbreln1,definition,
% 0.32/0.48 ( eqbreln1
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.48 => ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.32/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1invset_type,type,
% 0.32/0.48 breln1invset: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(breln1invprop_type,type,
% 0.32/0.48 breln1invprop: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1invprop,definition,
% 0.32/0.48 ( breln1invprop
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1invI_type,type,
% 0.32/0.48 breln1invI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1invI,definition,
% 0.32/0.48 ( breln1invI
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1invE_type,type,
% 0.32/0.48 breln1invE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1invE,definition,
% 0.32/0.48 ( breln1invE
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.32/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compset_type,type,
% 0.32/0.48 breln1compset: $i > $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compprop_type,type,
% 0.32/0.48 breln1compprop: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compprop,definition,
% 0.32/0.48 ( breln1compprop
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compI_type,type,
% 0.32/0.48 breln1compI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compI,definition,
% 0.32/0.48 ( breln1compI
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ! [Xz: $i] :
% 0.32/0.48 ( ( in @ Xz @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.32/0.48 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.32/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compE_type,type,
% 0.32/0.48 breln1compE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compE,definition,
% 0.32/0.48 ( breln1compE
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.32/0.48 => ? [Xz: $i] :
% 0.32/0.48 ( ( in @ Xz @ A )
% 0.32/0.48 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.32/0.48 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compEex_type,type,
% 0.32/0.48 breln1compEex: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1compEex,definition,
% 0.32/0.48 ( breln1compEex
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.32/0.48 => ! [Xphi: $o] :
% 0.32/0.48 ( ! [Xz: $i] :
% 0.32/0.48 ( ( in @ Xz @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.32/0.48 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.32/0.48 => Xphi ) ) )
% 0.32/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionprop_type,type,
% 0.32/0.48 breln1unionprop: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionprop,definition,
% 0.32/0.48 ( breln1unionprop
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionIL_type,type,
% 0.32/0.48 breln1unionIL: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionIL,definition,
% 0.32/0.48 ( breln1unionIL
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionIR_type,type,
% 0.32/0.48 breln1unionIR: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionIR,definition,
% 0.32/0.48 ( breln1unionIR
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionI_type,type,
% 0.32/0.48 breln1unionI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionI,definition,
% 0.32/0.48 ( breln1unionI
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.32/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionE_type,type,
% 0.32/0.48 breln1unionE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionE,definition,
% 0.32/0.48 ( breln1unionE
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionEcases_type,type,
% 0.32/0.48 breln1unionEcases: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionEcases,definition,
% 0.32/0.48 ( breln1unionEcases
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.32/0.48 => ! [Xphi: $o] :
% 0.32/0.48 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.48 => Xphi )
% 0.32/0.48 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.48 => Xphi )
% 0.32/0.48 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionCommutes_type,type,
% 0.32/0.48 breln1unionCommutes: $o ).
% 0.32/0.48
% 0.32/0.48 thf(breln1unionCommutes,definition,
% 0.32/0.48 ( breln1unionCommutes
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ( ( binunion @ R @ S )
% 0.32/0.48 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz2Ex_type,type,
% 0.32/0.48 woz2Ex: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz2Ex,definition,
% 0.32/0.48 ( woz2Ex
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ( R
% 0.32/0.48 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz2W_type,type,
% 0.32/0.48 woz2W: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz2W,definition,
% 0.32/0.48 ( woz2W
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.32/0.48 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz2A_type,type,
% 0.32/0.48 woz2A: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz2A,definition,
% 0.32/0.48 ( woz2A
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [T: $i] :
% 0.32/0.48 ( ( breln1 @ A @ T )
% 0.32/0.48 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.32/0.48 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(woz2B_type,type,
% 0.32/0.48 woz2B: $o ).
% 0.32/0.48
% 0.32/0.48 thf(woz2B,definition,
% 0.32/0.48 ( woz2B
% 0.32/0.48 = ( ! [A: $i,R: $i] :
% 0.32/0.48 ( ( breln1 @ A @ R )
% 0.32/0.48 => ! [S: $i] :
% 0.32/0.48 ( ( breln1 @ A @ S )
% 0.32/0.48 => ! [T: $i] :
% 0.32/0.48 ( ( breln1 @ A @ T )
% 0.32/0.48 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.32/0.48 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(image1Ex_type,type,
% 0.32/0.48 image1Ex: $o ).
% 0.32/0.48
% 0.32/0.48 thf(image1Ex,definition,
% 0.32/0.48 ( image1Ex
% 0.32/0.48 = ( ! [A: $i,Xf: $i > $i] :
% 0.32/0.48 ? [B: $i] :
% 0.32/0.48 ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ B )
% 0.32/0.48 <=> ? [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 & ( Xx
% 0.32/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(image1Ex1_type,type,
% 0.32/0.48 image1Ex1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(image1Ex1,definition,
% 0.32/0.48 ( image1Ex1
% 0.32/0.48 = ( ! [A: $i,Xf: $i > $i] :
% 0.32/0.48 ( exu
% 0.32/0.48 @ ^ [B: $i] :
% 0.32/0.48 ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ B )
% 0.32/0.48 <=> ? [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 & ( Xx
% 0.32/0.48 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(image1_type,type,
% 0.32/0.48 image1: $i > ( $i > $i ) > $i ).
% 0.32/0.48
% 0.32/0.48 thf(image1Equiv_type,type,
% 0.32/0.48 image1Equiv: $o ).
% 0.32/0.48
% 0.32/0.48 thf(image1Equiv,definition,
% 0.32/0.48 ( image1Equiv
% 0.32/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx
% 0.32/0.48 @ ( image1 @ A
% 0.32/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.32/0.48 <=> ? [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 & ( Xx
% 0.32/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(image1E_type,type,
% 0.32/0.48 image1E: $o ).
% 0.32/0.48
% 0.32/0.48 thf(image1E,definition,
% 0.32/0.48 ( image1E
% 0.32/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx
% 0.32/0.48 @ ( image1 @ A
% 0.32/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.32/0.48 => ? [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 & ( Xx
% 0.32/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(image1I_type,type,
% 0.32/0.48 image1I: $o ).
% 0.32/0.48
% 0.32/0.48 thf(image1I,definition,
% 0.32/0.48 ( image1I
% 0.32/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.32/0.48 ( ? [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 & ( Xx
% 0.32/0.48 = ( Xf @ Xy ) ) )
% 0.32/0.48 => ( in @ Xx
% 0.32/0.48 @ ( image1 @ A
% 0.32/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(injective_type,type,
% 0.32/0.48 injective: $i > $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(injFuncSet_type,type,
% 0.32/0.48 injFuncSet: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(injFuncInInjFuncSet_type,type,
% 0.32/0.48 injFuncInInjFuncSet: $o ).
% 0.32/0.48
% 0.32/0.48 thf(injFuncInInjFuncSet,definition,
% 0.32/0.48 ( injFuncInInjFuncSet
% 0.32/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.48 => ( ( injective @ A @ B @ Xf )
% 0.32/0.48 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(injFuncSetFuncIn_type,type,
% 0.32/0.48 injFuncSetFuncIn: $o ).
% 0.32/0.48
% 0.32/0.48 thf(injFuncSetFuncIn,definition,
% 0.32/0.48 ( injFuncSetFuncIn
% 0.32/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.48 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.32/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(injFuncSetFuncInj_type,type,
% 0.32/0.48 injFuncSetFuncInj: $o ).
% 0.32/0.48
% 0.32/0.48 thf(injFuncSetFuncInj,definition,
% 0.32/0.48 ( injFuncSetFuncInj
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.32/0.48 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.32/0.48 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(surjective_type,type,
% 0.32/0.48 surjective: $i > $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(surjFuncSet_type,type,
% 0.32/0.48 surjFuncSet: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(surjFuncSetFuncIn_type,type,
% 0.32/0.48 surjFuncSetFuncIn: $o ).
% 0.32/0.48
% 0.32/0.48 thf(surjFuncSetFuncIn,definition,
% 0.32/0.48 ( surjFuncSetFuncIn
% 0.32/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.48 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.32/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(surjFuncSetFuncSurj_type,type,
% 0.32/0.48 surjFuncSetFuncSurj: $o ).
% 0.32/0.48
% 0.32/0.48 thf(surjFuncSetFuncSurj,definition,
% 0.32/0.48 ( surjFuncSetFuncSurj
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.32/0.48 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.32/0.48 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(leftInvIsSurj_type,type,
% 0.32/0.48 leftInvIsSurj: $o ).
% 0.32/0.48
% 0.32/0.48 thf(leftInvIsSurj,definition,
% 0.32/0.48 ( leftInvIsSurj
% 0.32/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.48 => ! [Xg: $i] :
% 0.32/0.48 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.32/0.48 => ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.32/0.48 = Xx ) )
% 0.32/0.48 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(surjCantorThm_type,type,
% 0.32/0.48 surjCantorThm: $o ).
% 0.32/0.48
% 0.32/0.48 thf(surjCantorThm,definition,
% 0.32/0.48 ( surjCantorThm
% 0.32/0.48 = ( ! [A: $i,Xf: $i] :
% 0.32/0.48 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.32/0.48 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(foundation2_type,type,
% 0.32/0.48 foundation2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(foundation2,definition,
% 0.32/0.48 ( foundation2
% 0.32/0.48 = ( ! [A: $i] :
% 0.32/0.48 ( ( nonempty @ A )
% 0.32/0.48 => ? [X: $i] :
% 0.32/0.48 ( ( in @ X @ A )
% 0.32/0.48 & ( ( binintersect @ X @ A )
% 0.32/0.48 = emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(notinself_type,type,
% 0.32/0.48 notinself: $o ).
% 0.32/0.48
% 0.32/0.48 thf(notinself,definition,
% 0.32/0.48 ( notinself
% 0.32/0.48 = ( ! [A: $i] :
% 0.32/0.48 ~ ( in @ A @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(notinself2_type,type,
% 0.32/0.48 notinself2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(notinself2,definition,
% 0.32/0.48 ( notinself2
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( in @ A @ B )
% 0.32/0.48 => ~ ( in @ B @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(omegaS_type,type,
% 0.32/0.48 omegaS: $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(omegaS,definition,
% 0.32/0.48 ( omegaS
% 0.32/0.48 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(omegaSp,conjecture,
% 0.32/0.48 ( setextAx
% 0.32/0.48 => ( emptysetAx
% 0.32/0.48 => ( setadjoinAx
% 0.32/0.48 => ( powersetAx
% 0.32/0.48 => ( setunionAx
% 0.32/0.48 => ( omega0Ax
% 0.32/0.48 => ( omegaSAx
% 0.32/0.48 => ( omegaIndAx
% 0.32/0.48 => ( replAx
% 0.32/0.48 => ( foundationAx
% 0.32/0.48 => ( wellorderingAx
% 0.32/0.48 => ( descrp
% 0.32/0.48 => ( dsetconstrI
% 0.32/0.48 => ( dsetconstrEL
% 0.32/0.48 => ( dsetconstrER
% 0.32/0.48 => ( exuE1
% 0.32/0.48 => ( prop2setE
% 0.32/0.48 => ( emptysetE
% 0.32/0.48 => ( emptysetimpfalse
% 0.32/0.48 => ( notinemptyset
% 0.32/0.48 => ( exuE3e
% 0.32/0.48 => ( setext
% 0.32/0.48 => ( emptyI
% 0.32/0.48 => ( noeltsimpempty
% 0.32/0.48 => ( setbeta
% 0.32/0.48 => ( nonemptyE1
% 0.32/0.48 => ( nonemptyI
% 0.32/0.48 => ( nonemptyI1
% 0.32/0.48 => ( setadjoinIL
% 0.32/0.48 => ( emptyinunitempty
% 0.32/0.48 => ( setadjoinIR
% 0.32/0.48 => ( setadjoinE
% 0.32/0.48 => ( setadjoinOr
% 0.32/0.48 => ( setoftrueEq
% 0.32/0.48 => ( powersetI
% 0.32/0.48 => ( emptyinPowerset
% 0.32/0.48 => ( emptyInPowerset
% 0.32/0.48 => ( powersetE
% 0.32/0.48 => ( setunionI
% 0.32/0.48 => ( setunionE
% 0.32/0.48 => ( subPowSU
% 0.32/0.48 => ( exuE2
% 0.32/0.48 => ( nonemptyImpWitness
% 0.32/0.48 => ( uniqinunit
% 0.32/0.48 => ( notinsingleton
% 0.32/0.48 => ( eqinunit
% 0.32/0.48 => ( singletonsswitch
% 0.32/0.48 => ( upairsetE
% 0.32/0.48 => ( upairsetIL
% 0.32/0.48 => ( upairsetIR
% 0.32/0.48 => ( emptyE1
% 0.32/0.48 => ( vacuousDall
% 0.32/0.48 => ( quantDeMorgan1
% 0.32/0.48 => ( quantDeMorgan2
% 0.32/0.48 => ( quantDeMorgan3
% 0.32/0.48 => ( quantDeMorgan4
% 0.32/0.48 => ( prop2setI
% 0.32/0.48 => ( prop2set2propI
% 0.32/0.48 => ( notdexE
% 0.32/0.48 => ( notdallE
% 0.32/0.48 => ( exuI1
% 0.32/0.48 => ( exuI3
% 0.32/0.48 => ( exuI2
% 0.32/0.48 => ( inCongP
% 0.32/0.48 => ( in__Cong
% 0.32/0.48 => ( exuE3u
% 0.32/0.48 => ( exu__Cong
% 0.32/0.48 => ( emptyset__Cong
% 0.32/0.48 => ( setadjoin__Cong
% 0.32/0.48 => ( powerset__Cong
% 0.32/0.48 => ( setunion__Cong
% 0.32/0.48 => ( omega__Cong
% 0.32/0.48 => ( exuEu
% 0.32/0.48 => ( descr__Cong
% 0.32/0.48 => ( dsetconstr__Cong
% 0.32/0.48 => ( subsetI1
% 0.32/0.48 => ( eqimpsubset2
% 0.32/0.48 => ( eqimpsubset1
% 0.32/0.48 => ( subsetI2
% 0.32/0.48 => ( emptysetsubset
% 0.32/0.48 => ( subsetE
% 0.32/0.48 => ( subsetE2
% 0.32/0.48 => ( notsubsetI
% 0.32/0.48 => ( notequalI1
% 0.32/0.48 => ( notequalI2
% 0.32/0.48 => ( subsetRefl
% 0.32/0.48 => ( subsetTrans
% 0.32/0.48 => ( setadjoinSub
% 0.32/0.48 => ( setadjoinSub2
% 0.32/0.48 => ( subset2powerset
% 0.32/0.48 => ( setextsub
% 0.32/0.48 => ( subsetemptysetimpeq
% 0.32/0.48 => ( powersetI1
% 0.32/0.48 => ( powersetE1
% 0.32/0.48 => ( inPowerset
% 0.32/0.48 => ( powersetsubset
% 0.32/0.48 => ( sepInPowerset
% 0.32/0.48 => ( sepSubset
% 0.32/0.48 => ( binunionIL
% 0.32/0.48 => ( upairset2IR
% 0.32/0.48 => ( binunionIR
% 0.32/0.48 => ( binunionEcases
% 0.32/0.48 => ( binunionE
% 0.32/0.48 => ( binunionLsub
% 0.32/0.48 => ( binunionRsub
% 0.32/0.48 => ( binintersectI
% 0.32/0.48 => ( binintersectSubset5
% 0.32/0.48 => ( binintersectEL
% 0.32/0.48 => ( binintersectLsub
% 0.32/0.48 => ( binintersectSubset2
% 0.32/0.48 => ( binintersectSubset3
% 0.32/0.48 => ( binintersectER
% 0.32/0.48 => ( disjointsetsI1
% 0.32/0.48 => ( binintersectRsub
% 0.32/0.48 => ( binintersectSubset4
% 0.32/0.48 => ( binintersectSubset1
% 0.32/0.48 => ( bs114d
% 0.32/0.48 => ( setminusI
% 0.32/0.48 => ( setminusEL
% 0.32/0.48 => ( setminusER
% 0.32/0.48 => ( setminusSubset2
% 0.32/0.48 => ( setminusERneg
% 0.32/0.48 => ( setminusELneg
% 0.32/0.48 => ( setminusILneg
% 0.32/0.48 => ( setminusIRneg
% 0.32/0.48 => ( setminusLsub
% 0.32/0.48 => ( setminusSubset1
% 0.32/0.48 => ( symdiffE
% 0.32/0.48 => ( symdiffI1
% 0.32/0.48 => ( symdiffI2
% 0.32/0.48 => ( symdiffIneg1
% 0.32/0.48 => ( symdiffIneg2
% 0.32/0.48 => ( secondinupair
% 0.32/0.48 => ( setukpairIL
% 0.32/0.48 => ( setukpairIR
% 0.32/0.48 => ( kpairiskpair
% 0.32/0.48 => ( kpairp
% 0.32/0.48 => ( singletonsubset
% 0.32/0.48 => ( singletoninpowerset
% 0.32/0.48 => ( singletoninpowunion
% 0.32/0.48 => ( upairset2E
% 0.32/0.48 => ( upairsubunion
% 0.32/0.48 => ( upairinpowunion
% 0.32/0.48 => ( ubforcartprodlem1
% 0.32/0.48 => ( ubforcartprodlem2
% 0.32/0.48 => ( ubforcartprodlem3
% 0.32/0.48 => ( cartprodpairin
% 0.32/0.48 => ( cartprodmempair1
% 0.32/0.48 => ( cartprodmempair
% 0.32/0.48 => ( setunionE2
% 0.32/0.48 => ( setunionsingleton1
% 0.32/0.48 => ( setunionsingleton2
% 0.32/0.48 => ( setunionsingleton
% 0.32/0.48 => ( singletonprop
% 0.32/0.48 => ( ex1E1
% 0.32/0.48 => ( ex1I
% 0.32/0.48 => ( ex1I2
% 0.32/0.48 => ( singletonsuniq
% 0.32/0.48 => ( setukpairinjL1
% 0.32/0.48 => ( kfstsingleton
% 0.32/0.48 => ( theprop
% 0.32/0.48 => ( kfstpairEq
% 0.32/0.48 => ( cartprodfstin
% 0.32/0.48 => ( setukpairinjL2
% 0.32/0.48 => ( setukpairinjL
% 0.32/0.48 => ( setukpairinjR11
% 0.32/0.48 => ( setukpairinjR12
% 0.32/0.48 => ( setukpairinjR1
% 0.32/0.48 => ( upairequniteq
% 0.32/0.48 => ( setukpairinjR2
% 0.32/0.48 => ( setukpairinjR
% 0.32/0.48 => ( ksndsingleton
% 0.32/0.48 => ( ksndpairEq
% 0.32/0.48 => ( kpairsurjEq
% 0.32/0.48 => ( cartprodsndin
% 0.32/0.48 => ( cartprodpairmemEL
% 0.32/0.48 => ( cartprodpairmemER
% 0.32/0.48 => ( cartprodmempaircEq
% 0.32/0.48 => ( cartprodfstpairEq
% 0.32/0.48 => ( cartprodsndpairEq
% 0.32/0.48 => ( cartprodpairsurjEq
% 0.32/0.48 => ( dpsetconstrI
% 0.32/0.48 => ( dpsetconstrSub
% 0.32/0.48 => ( setOfPairsIsBReln
% 0.32/0.48 => ( dpsetconstrERa
% 0.32/0.48 => ( dpsetconstrEL1
% 0.32/0.48 => ( dpsetconstrEL2
% 0.32/0.48 => ( dpsetconstrER
% 0.32/0.48 => ( funcImageSingleton
% 0.32/0.48 => ( apProp
% 0.32/0.48 => ( app
% 0.32/0.48 => ( infuncsetfunc
% 0.32/0.48 => ( ap2p
% 0.32/0.48 => ( funcinfuncset
% 0.32/0.48 => ( lamProp
% 0.32/0.48 => ( lamp
% 0.32/0.48 => ( lam2p
% 0.32/0.48 => ( brelnall1
% 0.32/0.48 => ( brelnall2
% 0.32/0.48 => ( ex1E2
% 0.32/0.48 => ( funcGraphProp1
% 0.32/0.48 => ( funcGraphProp3
% 0.32/0.48 => ( funcGraphProp2
% 0.32/0.48 => ( funcextLem
% 0.32/0.48 => ( funcGraphProp4
% 0.32/0.48 => ( subbreln
% 0.32/0.48 => ( eqbreln
% 0.32/0.48 => ( funcext
% 0.32/0.48 => ( funcext2
% 0.32/0.48 => ( ap2apEq1
% 0.32/0.48 => ( ap2apEq2
% 0.32/0.48 => ( beta1
% 0.32/0.48 => ( eta1
% 0.32/0.48 => ( lam2lamEq
% 0.32/0.48 => ( beta2
% 0.32/0.48 => ( eta2
% 0.32/0.48 => ( iffalseProp1
% 0.32/0.48 => ( iffalseProp2
% 0.32/0.48 => ( iftrueProp1
% 0.32/0.48 => ( iftrueProp2
% 0.32/0.48 => ( ifSingleton
% 0.32/0.48 => ( ifp
% 0.32/0.48 => ( theeq
% 0.32/0.48 => ( iftrue
% 0.32/0.48 => ( iffalse
% 0.32/0.48 => ( iftrueorfalse
% 0.32/0.48 => ( binintersectT_lem
% 0.32/0.48 => ( binunionT_lem
% 0.32/0.48 => ( powersetT_lem
% 0.32/0.48 => ( setminusT_lem
% 0.32/0.48 => ( complementT_lem
% 0.32/0.48 => ( setextT
% 0.32/0.48 => ( subsetTI
% 0.32/0.48 => ( powersetTI1
% 0.32/0.48 => ( powersetTE1
% 0.32/0.48 => ( complementTI1
% 0.32/0.48 => ( complementTE1
% 0.32/0.48 => ( binintersectTELcontra
% 0.32/0.48 => ( binintersectTERcontra
% 0.32/0.48 => ( contrasubsetT
% 0.32/0.48 => ( contrasubsetT1
% 0.32/0.48 => ( contrasubsetT2
% 0.32/0.48 => ( contrasubsetT3
% 0.32/0.48 => ( doubleComplementI1
% 0.32/0.48 => ( doubleComplementE1
% 0.32/0.48 => ( doubleComplementSub1
% 0.32/0.48 => ( doubleComplementSub2
% 0.32/0.48 => ( doubleComplementEq
% 0.32/0.48 => ( complementTnotintersectT
% 0.32/0.48 => ( complementImpComplementIntersect
% 0.32/0.48 => ( complementSubsetComplementIntersect
% 0.32/0.48 => ( complementInPowersetComplementIntersect
% 0.32/0.48 => ( contraSubsetComplement
% 0.32/0.48 => ( complementTcontraSubset
% 0.32/0.48 => ( binunionTILcontra
% 0.32/0.48 => ( binunionTIRcontra
% 0.32/0.48 => ( inIntersectImpInUnion
% 0.32/0.48 => ( inIntersectImpInUnion2
% 0.32/0.48 => ( inIntersectImpInIntersectUnions
% 0.32/0.48 => ( intersectInPowersetIntersectUnions
% 0.32/0.48 => ( inComplementUnionImpNotIn1
% 0.32/0.48 => ( inComplementUnionImpInComplement1
% 0.32/0.48 => ( binunionTE
% 0.32/0.48 => ( binunionTEcontra
% 0.32/0.48 => ( demorgan2a1
% 0.32/0.48 => ( complementUnionInPowersetComplement
% 0.32/0.48 => ( demorgan2a2
% 0.32/0.48 => ( demorgan1a
% 0.32/0.48 => ( demorgan1b
% 0.32/0.48 => ( demorgan1
% 0.32/0.48 => ( demorgan2a
% 0.32/0.48 => ( demorgan2b2
% 0.32/0.48 => ( demorgan2b
% 0.32/0.48 => ( demorgan2
% 0.32/0.48 => ( woz13rule0
% 0.32/0.48 => ( woz13rule1
% 0.32/0.48 => ( woz13rule2
% 0.32/0.48 => ( woz13rule3
% 0.32/0.48 => ( woz13rule4
% 0.32/0.48 => ( woz1_1
% 0.32/0.48 => ( woz1_2
% 0.32/0.48 => ( woz1_3
% 0.32/0.48 => ( woz1_4
% 0.32/0.48 => ( woz1_5
% 0.32/0.48 => ( breln1all2
% 0.32/0.48 => ( breln1SetBreln1
% 0.32/0.48 => ( choice2fnsingleton
% 0.32/0.48 => ( setOfPairsIsBReln1
% 0.32/0.48 => ( breln1all1
% 0.32/0.48 => ( subbreln1
% 0.32/0.48 => ( eqbreln1
% 0.32/0.48 => ( breln1invprop
% 0.32/0.48 => ( breln1invI
% 0.32/0.48 => ( breln1invE
% 0.32/0.48 => ( breln1compprop
% 0.32/0.48 => ( breln1compI
% 0.32/0.48 => ( breln1compE
% 0.32/0.48 => ( breln1compEex
% 0.32/0.48 => ( breln1unionprop
% 0.32/0.48 => ( breln1unionIL
% 0.32/0.48 => ( breln1unionIR
% 0.32/0.48 => ( breln1unionI
% 0.32/0.48 => ( breln1unionE
% 0.32/0.48 => ( breln1unionEcases
% 0.32/0.48 => ( breln1unionCommutes
% 0.32/0.48 => ( woz2Ex
% 0.32/0.48 => ( woz2W
% 0.32/0.48 => ( woz2A
% 0.32/0.48 => ( woz2B
% 0.32/0.48 => ( image1Ex
% 0.32/0.48 => ( image1Ex1
% 0.32/0.48 => ( image1Equiv
% 0.32/0.48 => ( image1E
% 0.32/0.48 => ( image1I
% 0.32/0.48 => ( injFuncInInjFuncSet
% 0.32/0.48 => ( injFuncSetFuncIn
% 0.32/0.48 => ( injFuncSetFuncInj
% 0.32/0.48 => ( surjFuncSetFuncIn
% 0.32/0.48 => ( surjFuncSetFuncSurj
% 0.32/0.48 => ( leftInvIsSurj
% 0.32/0.48 => ( surjCantorThm
% 0.32/0.48 => ( foundation2
% 0.32/0.48 => ( notinself
% 0.32/0.48 => ( notinself2
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ omega )
% 0.32/0.48 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 ------------------------
% 2.75/1.46 ---- Embedded in THF ---
% 2.75/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.75/1.46 %%% Generated on Tue Feb 24 12:34:56 EST 2026
% 2.75/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.75/1.46 %%% Logic specification used:
% 2.75/1.46 %%% thf(spec, logic, $$dhol).
% 2.75/1.46
% 2.75/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.KvMWiWE9BC/DTF2DTF28453.p
% See solution above
% 2.75/1.47 ------------------------
% 2.99/1.51 ---- Cleaned THF ---
% 2.99/1.51 thf(in_type,type,
% 2.99/1.51 in: $i > $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(exu_type,type,
% 2.99/1.51 exu: ( $i > $o ) > $o ).
% 2.99/1.51
% 2.99/1.51 thf(exu,definition,
% 2.99/1.51 ( exu
% 2.99/1.51 = ( ^ [Xphi: $i > $o] :
% 2.99/1.51 ? [Xx: $i] :
% 2.99/1.51 ( ( Xphi @ Xx )
% 2.99/1.51 & ! [Xy: $i] :
% 2.99/1.51 ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setextAx_type,type,
% 2.99/1.51 setextAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setextAx,definition,
% 2.99/1.51 ( setextAx
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 <=> ( in @ Xx @ B ) )
% 2.99/1.51 => ( A = B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptyset_type,type,
% 2.99/1.51 emptyset: $i ).
% 2.99/1.51
% 2.99/1.51 thf(emptysetAx_type,type,
% 2.99/1.51 emptysetAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptysetAx,definition,
% 2.99/1.51 ( emptysetAx
% 2.99/1.51 = ( ! [Xx: $i] :
% 2.99/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoin_type,type,
% 2.99/1.51 setadjoin: $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinAx_type,type,
% 2.99/1.51 setadjoinAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinAx,definition,
% 2.99/1.51 ( setadjoinAx
% 2.99/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.99/1.51 <=> ( ( Xy = Xx )
% 2.99/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(powerset_type,type,
% 2.99/1.51 powerset: $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(powersetAx_type,type,
% 2.99/1.51 powersetAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(powersetAx,definition,
% 2.99/1.51 ( powersetAx
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.99/1.51 <=> ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setunion_type,type,
% 2.99/1.51 setunion: $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(setunionAx_type,type,
% 2.99/1.51 setunionAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setunionAx,definition,
% 2.99/1.51 ( setunionAx
% 2.99/1.51 = ( ! [A: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.99/1.51 <=> ? [B: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 & ( in @ B @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(omega_type,type,
% 2.99/1.51 omega: $i ).
% 2.99/1.51
% 2.99/1.51 thf(omega0Ax_type,type,
% 2.99/1.51 omega0Ax: $o ).
% 2.99/1.51
% 2.99/1.51 thf(omega0Ax,definition,
% 2.99/1.51 ( omega0Ax
% 2.99/1.51 = ( in @ emptyset @ omega ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(omegaSAx_type,type,
% 2.99/1.51 omegaSAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(omegaSAx,definition,
% 2.99/1.51 ( omegaSAx
% 2.99/1.51 = ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ omega )
% 2.99/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(omegaIndAx_type,type,
% 2.99/1.51 omegaIndAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(omegaIndAx,definition,
% 2.99/1.51 ( omegaIndAx
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ( ( in @ emptyset @ A )
% 2.99/1.51 & ! [Xx: $i] :
% 2.99/1.51 ( ( ( in @ Xx @ omega )
% 2.99/1.51 & ( in @ Xx @ A ) )
% 2.99/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ omega )
% 2.99/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(replAx_type,type,
% 2.99/1.51 replAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(replAx,definition,
% 2.99/1.51 ( replAx
% 2.99/1.51 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( exu
% 2.99/1.51 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.99/1.51 => ? [B: $i] :
% 2.99/1.51 ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 <=> ? [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ A )
% 2.99/1.51 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(foundationAx_type,type,
% 2.99/1.51 foundationAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(foundationAx,definition,
% 2.99/1.51 ( foundationAx
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.99/1.51 => ? [B: $i] :
% 2.99/1.51 ( ( in @ B @ A )
% 2.99/1.51 & ~ ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(wellorderingAx_type,type,
% 2.99/1.51 wellorderingAx: $o ).
% 2.99/1.51
% 2.99/1.51 thf(wellorderingAx,definition,
% 2.99/1.51 ( wellorderingAx
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ? [B: $i] :
% 2.99/1.51 ( ! [C: $i] :
% 2.99/1.51 ( ( in @ C @ B )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ C )
% 2.99/1.51 => ( in @ Xx @ A ) ) )
% 2.99/1.51 & ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( ( in @ Xx @ A )
% 2.99/1.51 & ( in @ Xy @ A ) )
% 2.99/1.51 => ( ! [C: $i] :
% 2.99/1.51 ( ( in @ C @ B )
% 2.99/1.51 => ( ( in @ Xx @ C )
% 2.99/1.51 <=> ( in @ Xy @ C ) ) )
% 2.99/1.51 => ( Xx = Xy ) ) )
% 2.99/1.51 & ! [C: $i,D: $i] :
% 2.99/1.51 ( ( ( in @ C @ B )
% 2.99/1.51 & ( in @ D @ B ) )
% 2.99/1.51 => ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ C )
% 2.99/1.51 => ( in @ Xx @ D ) )
% 2.99/1.51 | ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ D )
% 2.99/1.51 => ( in @ Xx @ C ) ) ) )
% 2.99/1.51 & ! [C: $i] :
% 2.99/1.51 ( ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ C )
% 2.99/1.51 => ( in @ Xx @ A ) )
% 2.99/1.51 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.99/1.51 => ? [D: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ D @ B )
% 2.99/1.51 & ( in @ Xx @ C )
% 2.99/1.51 & ~ ? [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ D )
% 2.99/1.51 & ( in @ Xy @ C ) )
% 2.99/1.51 & ! [E: $i] :
% 2.99/1.51 ( ( in @ E @ B )
% 2.99/1.51 => ( ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ E )
% 2.99/1.51 => ( in @ Xy @ D ) )
% 2.99/1.51 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(descr_type,type,
% 2.99/1.51 descr: ( $i > $o ) > $i ).
% 2.99/1.51
% 2.99/1.51 thf(descrp_type,type,
% 2.99/1.51 descrp: $o ).
% 2.99/1.51
% 2.99/1.51 thf(descrp,definition,
% 2.99/1.51 ( descrp
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ( Xphi
% 2.99/1.51 @ ( descr
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstr_type,type,
% 2.99/1.51 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstrI_type,type,
% 2.99/1.51 dsetconstrI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstrI,definition,
% 2.99/1.51 ( dsetconstrI
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 => ( in @ Xx
% 2.99/1.51 @ ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstrEL_type,type,
% 2.99/1.51 dsetconstrEL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstrEL,definition,
% 2.99/1.51 ( dsetconstrEL
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx
% 2.99/1.51 @ ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.99/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstrER_type,type,
% 2.99/1.51 dsetconstrER: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstrER,definition,
% 2.99/1.51 ( dsetconstrER
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx
% 2.99/1.51 @ ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.99/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exuE1_type,type,
% 2.99/1.51 exuE1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exuE1,definition,
% 2.99/1.51 ( exuE1
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ? [Xx: $i] :
% 2.99/1.51 ( ( Xphi @ Xx )
% 2.99/1.51 & ! [Xy: $i] :
% 2.99/1.51 ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(prop2set_type,type,
% 2.99/1.51 prop2set: $o > $i ).
% 2.99/1.51
% 2.99/1.51 thf(prop2setE_type,type,
% 2.99/1.51 prop2setE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(prop2setE,definition,
% 2.99/1.51 ( prop2setE
% 2.99/1.51 = ( ! [Xphi: $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.99/1.51 => Xphi ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptysetE_type,type,
% 2.99/1.51 emptysetE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptysetE,definition,
% 2.99/1.51 ( emptysetE
% 2.99/1.51 = ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ emptyset )
% 2.99/1.51 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptysetimpfalse_type,type,
% 2.99/1.51 emptysetimpfalse: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptysetimpfalse,definition,
% 2.99/1.51 ( emptysetimpfalse
% 2.99/1.51 = ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ emptyset )
% 2.99/1.51 => $false ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(notinemptyset_type,type,
% 2.99/1.51 notinemptyset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(notinemptyset,definition,
% 2.99/1.51 ( notinemptyset
% 2.99/1.51 = ( ! [Xx: $i] :
% 2.99/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exuE3e_type,type,
% 2.99/1.51 exuE3e: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exuE3e,definition,
% 2.99/1.51 ( exuE3e
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setext_type,type,
% 2.99/1.51 setext: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setext,definition,
% 2.99/1.51 ( setext
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ Xx @ B ) )
% 2.99/1.51 => ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ A ) )
% 2.99/1.51 => ( A = B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptyI_type,type,
% 2.99/1.51 emptyI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptyI,definition,
% 2.99/1.51 ( emptyI
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ~ ( in @ Xx @ A )
% 2.99/1.51 => ( A = emptyset ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(noeltsimpempty_type,type,
% 2.99/1.51 noeltsimpempty: $o ).
% 2.99/1.51
% 2.99/1.51 thf(noeltsimpempty,definition,
% 2.99/1.51 ( noeltsimpempty
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ~ ( in @ Xx @ A )
% 2.99/1.51 => ( A = emptyset ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setbeta_type,type,
% 2.99/1.51 setbeta: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setbeta,definition,
% 2.99/1.51 ( setbeta
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( in @ Xx
% 2.99/1.51 @ ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.99/1.51 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(nonempty_type,type,
% 2.99/1.51 nonempty: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(nonempty,definition,
% 2.99/1.51 ( nonempty
% 2.99/1.51 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(nonemptyE1_type,type,
% 2.99/1.51 nonemptyE1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(nonemptyE1,definition,
% 2.99/1.51 ( nonemptyE1
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ( nonempty @ A )
% 2.99/1.51 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(nonemptyI_type,type,
% 2.99/1.51 nonemptyI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(nonemptyI,definition,
% 2.99/1.51 ( nonemptyI
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 => ( nonempty
% 2.99/1.51 @ ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(nonemptyI1_type,type,
% 2.99/1.51 nonemptyI1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(nonemptyI1,definition,
% 2.99/1.51 ( nonemptyI1
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.99/1.51 => ( nonempty @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinIL_type,type,
% 2.99/1.51 setadjoinIL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinIL,definition,
% 2.99/1.51 ( setadjoinIL
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptyinunitempty_type,type,
% 2.99/1.51 emptyinunitempty: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptyinunitempty,definition,
% 2.99/1.51 ( emptyinunitempty
% 2.99/1.51 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinIR_type,type,
% 2.99/1.51 setadjoinIR: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinIR,definition,
% 2.99/1.51 ( setadjoinIR
% 2.99/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ A )
% 2.99/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinE_type,type,
% 2.99/1.51 setadjoinE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinE,definition,
% 2.99/1.51 ( setadjoinE
% 2.99/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.99/1.51 => ! [Xphi: $o] :
% 2.99/1.51 ( ( ( Xy = Xx )
% 2.99/1.51 => Xphi )
% 2.99/1.51 => ( ( ( in @ Xy @ A )
% 2.99/1.51 => Xphi )
% 2.99/1.51 => Xphi ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinOr_type,type,
% 2.99/1.51 setadjoinOr: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinOr,definition,
% 2.99/1.51 ( setadjoinOr
% 2.99/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.99/1.51 => ( ( Xy = Xx )
% 2.99/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setoftrueEq_type,type,
% 2.99/1.51 setoftrueEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setoftrueEq,definition,
% 2.99/1.51 ( setoftrueEq
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xx: $i] : $true )
% 2.99/1.51 = A ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(powersetI_type,type,
% 2.99/1.51 powersetI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(powersetI,definition,
% 2.99/1.51 ( powersetI
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ A ) )
% 2.99/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptyinPowerset_type,type,
% 2.99/1.51 emptyinPowerset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptyinPowerset,definition,
% 2.99/1.51 ( emptyinPowerset
% 2.99/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptyInPowerset_type,type,
% 2.99/1.51 emptyInPowerset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptyInPowerset,definition,
% 2.99/1.51 ( emptyInPowerset
% 2.99/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(powersetE_type,type,
% 2.99/1.51 powersetE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(powersetE,definition,
% 2.99/1.51 ( powersetE
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.99/1.51 => ( ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setunionI_type,type,
% 2.99/1.51 setunionI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setunionI,definition,
% 2.99/1.51 ( setunionI
% 2.99/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 => ( ( in @ B @ A )
% 2.99/1.51 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setunionE_type,type,
% 2.99/1.51 setunionE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setunionE,definition,
% 2.99/1.51 ( setunionE
% 2.99/1.51 = ( ! [A: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.99/1.51 => ! [Xphi: $o] :
% 2.99/1.51 ( ! [B: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 => ( ( in @ B @ A )
% 2.99/1.51 => Xphi ) )
% 2.99/1.51 => Xphi ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subPowSU_type,type,
% 2.99/1.51 subPowSU: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subPowSU,definition,
% 2.99/1.51 ( subPowSU
% 2.99/1.51 = ( ! [A: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exuE2_type,type,
% 2.99/1.51 exuE2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exuE2,definition,
% 2.99/1.51 ( exuE2
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ? [Xx: $i] :
% 2.99/1.51 ! [Xy: $i] :
% 2.99/1.51 ( ( Xphi @ Xy )
% 2.99/1.51 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(nonemptyImpWitness_type,type,
% 2.99/1.51 nonemptyImpWitness: $o ).
% 2.99/1.51
% 2.99/1.51 thf(nonemptyImpWitness,definition,
% 2.99/1.51 ( nonemptyImpWitness
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ( nonempty @ A )
% 2.99/1.51 => ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & $true ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(uniqinunit_type,type,
% 2.99/1.51 uniqinunit: $o ).
% 2.99/1.51
% 2.99/1.51 thf(uniqinunit,definition,
% 2.99/1.51 ( uniqinunit
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(notinsingleton_type,type,
% 2.99/1.51 notinsingleton: $o ).
% 2.99/1.51
% 2.99/1.51 thf(notinsingleton,definition,
% 2.99/1.51 ( notinsingleton
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx != Xy )
% 2.99/1.51 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(eqinunit_type,type,
% 2.99/1.51 eqinunit: $o ).
% 2.99/1.51
% 2.99/1.51 thf(eqinunit,definition,
% 2.99/1.51 ( eqinunit
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx = Xy )
% 2.99/1.51 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(singletonsswitch_type,type,
% 2.99/1.51 singletonsswitch: $o ).
% 2.99/1.51
% 2.99/1.51 thf(singletonsswitch,definition,
% 2.99/1.51 ( singletonsswitch
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.99/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(upairsetE_type,type,
% 2.99/1.51 upairsetE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairsetE,definition,
% 2.99/1.51 ( upairsetE
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.99/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.99/1.51 => ( ( Xz = Xx )
% 2.99/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(upairsetIL_type,type,
% 2.99/1.51 upairsetIL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairsetIL,definition,
% 2.99/1.51 ( upairsetIL
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(upairsetIR_type,type,
% 2.99/1.51 upairsetIR: $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairsetIR,definition,
% 2.99/1.51 ( upairsetIR
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptyE1_type,type,
% 2.99/1.51 emptyE1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptyE1,definition,
% 2.99/1.51 ( emptyE1
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ( Xphi @ Xx ) )
% 2.99/1.51 => ( ( ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 = emptyset )
% 2.99/1.51 => $false ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(vacuousDall_type,type,
% 2.99/1.51 vacuousDall: $o ).
% 2.99/1.51
% 2.99/1.51 thf(vacuousDall,definition,
% 2.99/1.51 ( vacuousDall
% 2.99/1.51 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ emptyset )
% 2.99/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(quantDeMorgan1_type,type,
% 2.99/1.51 quantDeMorgan1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(quantDeMorgan1,definition,
% 2.99/1.51 ( quantDeMorgan1
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ~ ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( Xphi @ Xx ) )
% 2.99/1.51 => ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(quantDeMorgan2_type,type,
% 2.99/1.51 quantDeMorgan2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(quantDeMorgan2,definition,
% 2.99/1.51 ( quantDeMorgan2
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ~ ( Xphi @ Xx ) )
% 2.99/1.51 => ~ ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(quantDeMorgan3_type,type,
% 2.99/1.51 quantDeMorgan3: $o ).
% 2.99/1.51
% 2.99/1.51 thf(quantDeMorgan3,definition,
% 2.99/1.51 ( quantDeMorgan3
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ~ ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ( Xphi @ Xx ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(quantDeMorgan4_type,type,
% 2.99/1.51 quantDeMorgan4: $o ).
% 2.99/1.51
% 2.99/1.51 thf(quantDeMorgan4,definition,
% 2.99/1.51 ( quantDeMorgan4
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ~ ( Xphi @ Xx ) )
% 2.99/1.51 => ~ ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(prop2setI_type,type,
% 2.99/1.51 prop2setI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(prop2setI,definition,
% 2.99/1.51 ( prop2setI
% 2.99/1.51 = ( ! [Xphi: $o] :
% 2.99/1.51 ( Xphi
% 2.99/1.51 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(set2prop_type,type,
% 2.99/1.51 set2prop: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(prop2set2propI_type,type,
% 2.99/1.51 prop2set2propI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(prop2set2propI,definition,
% 2.99/1.51 ( prop2set2propI
% 2.99/1.51 = ( ! [Xphi: $o] :
% 2.99/1.51 ( Xphi
% 2.99/1.51 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(notdexE_type,type,
% 2.99/1.51 notdexE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(notdexE,definition,
% 2.99/1.51 ( notdexE
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ~ ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ( Xphi @ Xx ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(notdallE_type,type,
% 2.99/1.51 notdallE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(notdallE,definition,
% 2.99/1.51 ( notdallE
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ~ ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( Xphi @ Xx ) )
% 2.99/1.51 => ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exuI1_type,type,
% 2.99/1.51 exuI1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exuI1,definition,
% 2.99/1.51 ( exuI1
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ? [Xx: $i] :
% 2.99/1.51 ( ( Xphi @ Xx )
% 2.99/1.51 & ! [Xy: $i] :
% 2.99/1.51 ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) )
% 2.99/1.51 => ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exuI3_type,type,
% 2.99/1.51 exuI3: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exuI3,definition,
% 2.99/1.51 ( exuI3
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.99/1.51 => ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xphi @ Xx )
% 2.99/1.51 => ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) )
% 2.99/1.51 => ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exuI2_type,type,
% 2.99/1.51 exuI2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exuI2,definition,
% 2.99/1.51 ( exuI2
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ? [Xx: $i] :
% 2.99/1.51 ! [Xy: $i] :
% 2.99/1.51 ( ( Xphi @ Xy )
% 2.99/1.51 <=> ( Xy = Xx ) )
% 2.99/1.51 => ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(inCongP_type,type,
% 2.99/1.51 inCongP: $o ).
% 2.99/1.51
% 2.99/1.51 thf(inCongP,definition,
% 2.99/1.51 ( inCongP
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( A = B )
% 2.99/1.51 => ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx = Xy )
% 2.99/1.51 => ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(in__Cong_type,type,
% 2.99/1.51 in__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(in__Cong,definition,
% 2.99/1.51 ( in__Cong
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( A = B )
% 2.99/1.51 => ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx = Xy )
% 2.99/1.51 => ( ( in @ Xx @ A )
% 2.99/1.51 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exuE3u_type,type,
% 2.99/1.51 exuE3u: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exuE3u,definition,
% 2.99/1.51 ( exuE3u
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xphi @ Xx )
% 2.99/1.51 => ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exu__Cong_type,type,
% 2.99/1.51 exu__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exu__Cong,definition,
% 2.99/1.51 ( exu__Cong
% 2.99/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.99/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx = Xy )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.99/1.51 => ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 <=> ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptyset__Cong_type,type,
% 2.99/1.51 emptyset__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptyset__Cong,definition,
% 2.99/1.51 ( emptyset__Cong
% 2.99/1.51 = ( emptyset = emptyset ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoin__Cong_type,type,
% 2.99/1.51 setadjoin__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoin__Cong,definition,
% 2.99/1.51 ( setadjoin__Cong
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx = Xy )
% 2.99/1.51 => ! [Xz: $i,Xu: $i] :
% 2.99/1.51 ( ( Xz = Xu )
% 2.99/1.51 => ( ( setadjoin @ Xx @ Xz )
% 2.99/1.51 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(powerset__Cong_type,type,
% 2.99/1.51 powerset__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(powerset__Cong,definition,
% 2.99/1.51 ( powerset__Cong
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( A = B )
% 2.99/1.51 => ( ( powerset @ A )
% 2.99/1.51 = ( powerset @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setunion__Cong_type,type,
% 2.99/1.51 setunion__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setunion__Cong,definition,
% 2.99/1.51 ( setunion__Cong
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( A = B )
% 2.99/1.51 => ( ( setunion @ A )
% 2.99/1.51 = ( setunion @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(omega__Cong_type,type,
% 2.99/1.51 omega__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(omega__Cong,definition,
% 2.99/1.51 ( omega__Cong
% 2.99/1.51 = ( omega = omega ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(exuEu_type,type,
% 2.99/1.51 exuEu: $o ).
% 2.99/1.51
% 2.99/1.51 thf(exuEu,definition,
% 2.99/1.51 ( exuEu
% 2.99/1.51 = ( ! [Xphi: $i > $o] :
% 2.99/1.51 ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xphi @ Xx )
% 2.99/1.51 => ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(descr__Cong_type,type,
% 2.99/1.51 descr__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(descr__Cong,definition,
% 2.99/1.51 ( descr__Cong
% 2.99/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.99/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx = Xy )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.99/1.51 => ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ( ( exu
% 2.99/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.99/1.51 => ( ( descr
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 = ( descr
% 2.99/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstr__Cong_type,type,
% 2.99/1.51 dsetconstr__Cong: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dsetconstr__Cong,definition,
% 2.99/1.51 ( dsetconstr__Cong
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( A = B )
% 2.99/1.51 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( Xx = Xy )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.99/1.51 => ( ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 = ( dsetconstr @ B
% 2.99/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subset_type,type,
% 2.99/1.51 subset: $i > $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(disjoint_type,type,
% 2.99/1.51 disjoint: $i > $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(setsmeet_type,type,
% 2.99/1.51 setsmeet: $i > $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(subsetI1_type,type,
% 2.99/1.51 subsetI1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subsetI1,definition,
% 2.99/1.51 ( subsetI1
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ Xx @ B ) )
% 2.99/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(eqimpsubset2_type,type,
% 2.99/1.51 eqimpsubset2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(eqimpsubset2,definition,
% 2.99/1.51 ( eqimpsubset2
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( A = B )
% 2.99/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(eqimpsubset1_type,type,
% 2.99/1.51 eqimpsubset1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(eqimpsubset1,definition,
% 2.99/1.51 ( eqimpsubset1
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( A = B )
% 2.99/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subsetI2_type,type,
% 2.99/1.51 subsetI2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subsetI2,definition,
% 2.99/1.51 ( subsetI2
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ Xx @ B ) )
% 2.99/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(emptysetsubset_type,type,
% 2.99/1.51 emptysetsubset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(emptysetsubset,definition,
% 2.99/1.51 ( emptysetsubset
% 2.99/1.51 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subsetE_type,type,
% 2.99/1.51 subsetE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subsetE,definition,
% 2.99/1.51 ( subsetE
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subsetE2_type,type,
% 2.99/1.51 subsetE2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subsetE2,definition,
% 2.99/1.51 ( subsetE2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( ~ ( in @ Xx @ B )
% 2.99/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(notsubsetI_type,type,
% 2.99/1.51 notsubsetI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(notsubsetI,definition,
% 2.99/1.51 ( notsubsetI
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ~ ( in @ Xx @ B )
% 2.99/1.51 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(notequalI1_type,type,
% 2.99/1.51 notequalI1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(notequalI1,definition,
% 2.99/1.51 ( notequalI1
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ~ ( subset @ A @ B )
% 2.99/1.51 => ( A != B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(notequalI2_type,type,
% 2.99/1.51 notequalI2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(notequalI2,definition,
% 2.99/1.51 ( notequalI2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ~ ( in @ Xx @ B )
% 2.99/1.51 => ( A != B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subsetRefl_type,type,
% 2.99/1.51 subsetRefl: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subsetRefl,definition,
% 2.99/1.51 ( subsetRefl
% 2.99/1.51 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subsetTrans_type,type,
% 2.99/1.51 subsetTrans: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subsetTrans,definition,
% 2.99/1.51 ( subsetTrans
% 2.99/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( ( subset @ B @ C )
% 2.99/1.51 => ( subset @ A @ C ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinSub_type,type,
% 2.99/1.51 setadjoinSub: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinSub,definition,
% 2.99/1.51 ( setadjoinSub
% 2.99/1.51 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinSub2_type,type,
% 2.99/1.51 setadjoinSub2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setadjoinSub2,definition,
% 2.99/1.51 ( setadjoinSub2
% 2.99/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subset2powerset_type,type,
% 2.99/1.51 subset2powerset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subset2powerset,definition,
% 2.99/1.51 ( subset2powerset
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setextsub_type,type,
% 2.99/1.51 setextsub: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setextsub,definition,
% 2.99/1.51 ( setextsub
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( ( subset @ B @ A )
% 2.99/1.51 => ( A = B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subsetemptysetimpeq_type,type,
% 2.99/1.51 subsetemptysetimpeq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subsetemptysetimpeq,definition,
% 2.99/1.51 ( subsetemptysetimpeq
% 2.99/1.51 = ( ! [A: $i] :
% 2.99/1.51 ( ( subset @ A @ emptyset )
% 2.99/1.51 => ( A = emptyset ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(powersetI1_type,type,
% 2.99/1.51 powersetI1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(powersetI1,definition,
% 2.99/1.51 ( powersetI1
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( subset @ B @ A )
% 2.99/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(powersetE1_type,type,
% 2.99/1.51 powersetE1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(powersetE1,definition,
% 2.99/1.51 ( powersetE1
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.99/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(inPowerset_type,type,
% 2.99/1.51 inPowerset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(inPowerset,definition,
% 2.99/1.51 ( inPowerset
% 2.99/1.51 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(powersetsubset_type,type,
% 2.99/1.51 powersetsubset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(powersetsubset,definition,
% 2.99/1.51 ( powersetsubset
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(sepInPowerset_type,type,
% 2.99/1.51 sepInPowerset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(sepInPowerset,definition,
% 2.99/1.51 ( sepInPowerset
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( in
% 2.99/1.51 @ ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 @ ( powerset @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(sepSubset_type,type,
% 2.99/1.51 sepSubset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(sepSubset,definition,
% 2.99/1.51 ( sepSubset
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( subset
% 2.99/1.51 @ ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 @ A ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binunion_type,type,
% 2.99/1.51 binunion: $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(binunionIL_type,type,
% 2.99/1.51 binunionIL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binunionIL,definition,
% 2.99/1.51 ( binunionIL
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(upairset2IR_type,type,
% 2.99/1.51 upairset2IR: $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairset2IR,definition,
% 2.99/1.51 ( upairset2IR
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binunionIR_type,type,
% 2.99/1.51 binunionIR: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binunionIR,definition,
% 2.99/1.51 ( binunionIR
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binunionEcases_type,type,
% 2.99/1.51 binunionEcases: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binunionEcases,definition,
% 2.99/1.51 ( binunionEcases
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.99/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.99/1.51 => ( ( ( in @ Xx @ A )
% 2.99/1.51 => Xphi )
% 2.99/1.51 => ( ( ( in @ Xx @ B )
% 2.99/1.51 => Xphi )
% 2.99/1.51 => Xphi ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binunionE_type,type,
% 2.99/1.51 binunionE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binunionE,definition,
% 2.99/1.51 ( binunionE
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.99/1.51 => ( ( in @ Xx @ A )
% 2.99/1.51 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binunionLsub_type,type,
% 2.99/1.51 binunionLsub: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binunionLsub,definition,
% 2.99/1.51 ( binunionLsub
% 2.99/1.51 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binunionRsub_type,type,
% 2.99/1.51 binunionRsub: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binunionRsub,definition,
% 2.99/1.51 ( binunionRsub
% 2.99/1.51 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersect_type,type,
% 2.99/1.51 binintersect: $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectI_type,type,
% 2.99/1.51 binintersectI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectI,definition,
% 2.99/1.51 ( binintersectI
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset5_type,type,
% 2.99/1.51 binintersectSubset5: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset5,definition,
% 2.99/1.51 ( binintersectSubset5
% 2.99/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.99/1.51 ( ( subset @ C @ A )
% 2.99/1.51 => ( ( subset @ C @ B )
% 2.99/1.51 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectEL_type,type,
% 2.99/1.51 binintersectEL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectEL,definition,
% 2.99/1.51 ( binintersectEL
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.99/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectLsub_type,type,
% 2.99/1.51 binintersectLsub: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectLsub,definition,
% 2.99/1.51 ( binintersectLsub
% 2.99/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset2_type,type,
% 2.99/1.51 binintersectSubset2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset2,definition,
% 2.99/1.51 ( binintersectSubset2
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( ( binintersect @ A @ B )
% 2.99/1.51 = A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset3_type,type,
% 2.99/1.51 binintersectSubset3: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset3,definition,
% 2.99/1.51 ( binintersectSubset3
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( ( binintersect @ A @ B )
% 2.99/1.51 = B )
% 2.99/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectER_type,type,
% 2.99/1.51 binintersectER: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectER,definition,
% 2.99/1.51 ( binintersectER
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.99/1.51 => ( in @ Xx @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(disjointsetsI1_type,type,
% 2.99/1.51 disjointsetsI1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(disjointsetsI1,definition,
% 2.99/1.51 ( disjointsetsI1
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ~ ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ( in @ Xx @ B ) )
% 2.99/1.51 => ( ( binintersect @ A @ B )
% 2.99/1.51 = emptyset ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectRsub_type,type,
% 2.99/1.51 binintersectRsub: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectRsub,definition,
% 2.99/1.51 ( binintersectRsub
% 2.99/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset4_type,type,
% 2.99/1.51 binintersectSubset4: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset4,definition,
% 2.99/1.51 ( binintersectSubset4
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( subset @ B @ A )
% 2.99/1.51 => ( ( binintersect @ A @ B )
% 2.99/1.51 = B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset1_type,type,
% 2.99/1.51 binintersectSubset1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(binintersectSubset1,definition,
% 2.99/1.51 ( binintersectSubset1
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( ( binintersect @ A @ B )
% 2.99/1.51 = A )
% 2.99/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(bs114d_type,type,
% 2.99/1.51 bs114d: $o ).
% 2.99/1.51
% 2.99/1.51 thf(bs114d,definition,
% 2.99/1.51 ( bs114d
% 2.99/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.99/1.51 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.99/1.51 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(regular_type,type,
% 2.99/1.51 regular: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminus_type,type,
% 2.99/1.51 setminus: $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(setminusI_type,type,
% 2.99/1.51 setminusI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusI,definition,
% 2.99/1.51 ( setminusI
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ~ ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusEL_type,type,
% 2.99/1.51 setminusEL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusEL,definition,
% 2.99/1.51 ( setminusEL
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.99/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusER_type,type,
% 2.99/1.51 setminusER: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusER,definition,
% 2.99/1.51 ( setminusER
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.99/1.51 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusSubset2_type,type,
% 2.99/1.51 setminusSubset2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusSubset2,definition,
% 2.99/1.51 ( setminusSubset2
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( subset @ A @ B )
% 2.99/1.51 => ( ( setminus @ A @ B )
% 2.99/1.51 = emptyset ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusERneg_type,type,
% 2.99/1.51 setminusERneg: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusERneg,definition,
% 2.99/1.51 ( setminusERneg
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.99/1.51 => ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusELneg_type,type,
% 2.99/1.51 setminusELneg: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusELneg,definition,
% 2.99/1.51 ( setminusELneg
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.99/1.51 => ( ~ ( in @ Xx @ B )
% 2.99/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusILneg_type,type,
% 2.99/1.51 setminusILneg: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusILneg,definition,
% 2.99/1.51 ( setminusILneg
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ~ ( in @ Xx @ A )
% 2.99/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusIRneg_type,type,
% 2.99/1.51 setminusIRneg: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusIRneg,definition,
% 2.99/1.51 ( setminusIRneg
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ B )
% 2.99/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusLsub_type,type,
% 2.99/1.51 setminusLsub: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusLsub,definition,
% 2.99/1.51 ( setminusLsub
% 2.99/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setminusSubset1_type,type,
% 2.99/1.51 setminusSubset1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setminusSubset1,definition,
% 2.99/1.51 ( setminusSubset1
% 2.99/1.51 = ( ! [A: $i,B: $i] :
% 2.99/1.51 ( ( ( setminus @ A @ B )
% 2.99/1.51 = emptyset )
% 2.99/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(symdiff_type,type,
% 2.99/1.51 symdiff: $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffE_type,type,
% 2.99/1.51 symdiffE: $o ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffE,definition,
% 2.99/1.51 ( symdiffE
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.99/1.51 => ! [Xphi: $o] :
% 2.99/1.51 ( ( ( in @ Xx @ A )
% 2.99/1.51 => ( ~ ( in @ Xx @ B )
% 2.99/1.51 => Xphi ) )
% 2.99/1.51 => ( ( ~ ( in @ Xx @ A )
% 2.99/1.51 => ( ( in @ Xx @ B )
% 2.99/1.51 => Xphi ) )
% 2.99/1.51 => Xphi ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffI1_type,type,
% 2.99/1.51 symdiffI1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffI1,definition,
% 2.99/1.51 ( symdiffI1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ~ ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffI2_type,type,
% 2.99/1.51 symdiffI2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffI2,definition,
% 2.99/1.51 ( symdiffI2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ~ ( in @ Xx @ A )
% 2.99/1.51 => ( ( in @ Xx @ B )
% 2.99/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffIneg1_type,type,
% 2.99/1.51 symdiffIneg1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffIneg1,definition,
% 2.99/1.51 ( symdiffIneg1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( in @ Xx @ B )
% 2.99/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffIneg2_type,type,
% 2.99/1.51 symdiffIneg2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(symdiffIneg2,definition,
% 2.99/1.51 ( symdiffIneg2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ~ ( in @ Xx @ A )
% 2.99/1.51 => ( ~ ( in @ Xx @ B )
% 2.99/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(iskpair_type,type,
% 2.99/1.51 iskpair: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(secondinupair_type,type,
% 2.99/1.51 secondinupair: $o ).
% 2.99/1.51
% 2.99/1.51 thf(secondinupair,definition,
% 2.99/1.51 ( secondinupair
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairIL_type,type,
% 2.99/1.51 setukpairIL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairIL,definition,
% 2.99/1.51 ( setukpairIL
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairIR_type,type,
% 2.99/1.51 setukpairIR: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairIR,definition,
% 2.99/1.51 ( setukpairIR
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(kpairiskpair_type,type,
% 2.99/1.51 kpairiskpair: $o ).
% 2.99/1.51
% 2.99/1.51 thf(kpairiskpair,definition,
% 2.99/1.51 ( kpairiskpair
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(kpair_type,type,
% 2.99/1.51 kpair: $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(kpairp_type,type,
% 2.99/1.51 kpairp: $o ).
% 2.99/1.51
% 2.99/1.51 thf(kpairp,definition,
% 2.99/1.51 ( kpairp
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprod_type,type,
% 2.99/1.51 cartprod: $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(singletonsubset_type,type,
% 2.99/1.51 singletonsubset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(singletonsubset,definition,
% 2.99/1.51 ( singletonsubset
% 2.99/1.51 = ( ! [A: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(singletoninpowerset_type,type,
% 2.99/1.51 singletoninpowerset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(singletoninpowerset,definition,
% 2.99/1.51 ( singletoninpowerset
% 2.99/1.51 = ( ! [A: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(singletoninpowunion_type,type,
% 2.99/1.51 singletoninpowunion: $o ).
% 2.99/1.51
% 2.99/1.51 thf(singletoninpowunion,definition,
% 2.99/1.51 ( singletoninpowunion
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(upairset2E_type,type,
% 2.99/1.51 upairset2E: $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairset2E,definition,
% 2.99/1.51 ( upairset2E
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.99/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.99/1.51 => ( ( Xz = Xx )
% 2.99/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(upairsubunion_type,type,
% 2.99/1.51 upairsubunion: $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairsubunion,definition,
% 2.99/1.51 ( upairsubunion
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(upairinpowunion_type,type,
% 2.99/1.51 upairinpowunion: $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairinpowunion,definition,
% 2.99/1.51 ( upairinpowunion
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ubforcartprodlem1_type,type,
% 2.99/1.51 ubforcartprodlem1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ubforcartprodlem1,definition,
% 2.99/1.51 ( ubforcartprodlem1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ubforcartprodlem2_type,type,
% 2.99/1.51 ubforcartprodlem2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ubforcartprodlem2,definition,
% 2.99/1.51 ( ubforcartprodlem2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ubforcartprodlem3_type,type,
% 2.99/1.51 ubforcartprodlem3: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ubforcartprodlem3,definition,
% 2.99/1.51 ( ubforcartprodlem3
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodpairin_type,type,
% 2.99/1.51 cartprodpairin: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodpairin,definition,
% 2.99/1.51 ( cartprodpairin
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodmempair1_type,type,
% 2.99/1.51 cartprodmempair1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodmempair1,definition,
% 2.99/1.51 ( cartprodmempair1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.99/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.99/1.51 => ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ? [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 & ( Xu
% 2.99/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodmempair_type,type,
% 2.99/1.51 cartprodmempair: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodmempair,definition,
% 2.99/1.51 ( cartprodmempair
% 2.99/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.99/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.99/1.51 => ( iskpair @ Xu ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setunionE2_type,type,
% 2.99/1.51 setunionE2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setunionE2,definition,
% 2.99/1.51 ( setunionE2
% 2.99/1.51 = ( ! [A: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.99/1.51 => ? [X: $i] :
% 2.99/1.51 ( ( in @ X @ A )
% 2.99/1.51 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setunionsingleton1_type,type,
% 2.99/1.51 setunionsingleton1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setunionsingleton1,definition,
% 2.99/1.51 ( setunionsingleton1
% 2.99/1.51 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setunionsingleton2_type,type,
% 2.99/1.51 setunionsingleton2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setunionsingleton2,definition,
% 2.99/1.51 ( setunionsingleton2
% 2.99/1.51 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setunionsingleton_type,type,
% 2.99/1.51 setunionsingleton: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setunionsingleton,definition,
% 2.99/1.51 ( setunionsingleton
% 2.99/1.51 = ( ! [Xx: $i] :
% 2.99/1.51 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.99/1.51 = Xx ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(singleton_type,type,
% 2.99/1.51 singleton: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(singletonprop_type,type,
% 2.99/1.51 singletonprop: $o ).
% 2.99/1.51
% 2.99/1.51 thf(singletonprop,definition,
% 2.99/1.51 ( singletonprop
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ A )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 => ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) )
% 2.99/1.51 => ( ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ( Xphi @ Xx ) )
% 2.99/1.51 => ( singleton
% 2.99/1.51 @ ( dsetconstr @ A
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ex1_type,type,
% 2.99/1.51 ex1: $i > ( $i > $o ) > $o ).
% 2.99/1.51
% 2.99/1.51 thf(ex1E1_type,type,
% 2.99/1.51 ex1E1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ex1E1,definition,
% 2.99/1.51 ( ex1E1
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ( ex1 @ A
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ex1I_type,type,
% 2.99/1.51 ex1I: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ex1I,definition,
% 2.99/1.51 ( ex1I
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 => ( ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ A )
% 2.99/1.51 => ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xy = Xx ) ) )
% 2.99/1.51 => ( ex1 @ A
% 2.99/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ex1I2_type,type,
% 2.99/1.51 ex1I2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ex1I2,definition,
% 2.99/1.51 ( ex1I2
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ A )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 => ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) )
% 2.99/1.51 => ( ? [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 & ( Xphi @ Xx ) )
% 2.99/1.51 => ( ex1 @ A
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(singletonsuniq_type,type,
% 2.99/1.51 singletonsuniq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(singletonsuniq,definition,
% 2.99/1.51 ( singletonsuniq
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( ( setadjoin @ Xx @ emptyset )
% 2.99/1.51 = ( setadjoin @ Xy @ emptyset ) )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(atmost1p_type,type,
% 2.99/1.51 atmost1p: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(atleast2p_type,type,
% 2.99/1.51 atleast2p: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(atmost2p_type,type,
% 2.99/1.51 atmost2p: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairsetp_type,type,
% 2.99/1.51 upairsetp: $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjL1_type,type,
% 2.99/1.51 setukpairinjL1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjL1,definition,
% 2.99/1.51 ( setukpairinjL1
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.99/1.51 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.99/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(kfstsingleton_type,type,
% 2.99/1.51 kfstsingleton: $o ).
% 2.99/1.51
% 2.99/1.51 thf(kfstsingleton,definition,
% 2.99/1.51 ( kfstsingleton
% 2.99/1.51 = ( ! [Xu: $i] :
% 2.99/1.51 ( ( iskpair @ Xu )
% 2.99/1.51 => ( singleton
% 2.99/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.99/1.51 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(theprop_type,type,
% 2.99/1.51 theprop: $o ).
% 2.99/1.51
% 2.99/1.51 thf(theprop,definition,
% 2.99/1.51 ( theprop
% 2.99/1.51 = ( ! [X: $i] :
% 2.99/1.51 ( ( singleton @ X )
% 2.99/1.51 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(kfst_type,type,
% 2.99/1.51 kfst: $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(kfstpairEq_type,type,
% 2.99/1.51 kfstpairEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(kfstpairEq,definition,
% 2.99/1.51 ( kfstpairEq
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.99/1.51 = Xx ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodfstin_type,type,
% 2.99/1.51 cartprodfstin: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodfstin,definition,
% 2.99/1.51 ( cartprodfstin
% 2.99/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.99/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.99/1.51 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjL2_type,type,
% 2.99/1.51 setukpairinjL2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjL2,definition,
% 2.99/1.51 ( setukpairinjL2
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.99/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.99/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.99/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjL_type,type,
% 2.99/1.51 setukpairinjL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjL,definition,
% 2.99/1.51 ( setukpairinjL
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.99/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.99/1.51 = ( kpair @ Xz @ Xu ) )
% 2.99/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR11_type,type,
% 2.99/1.51 setukpairinjR11: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR11,definition,
% 2.99/1.51 ( setukpairinjR11
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx = Xy )
% 2.99/1.51 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.99/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR12_type,type,
% 2.99/1.51 setukpairinjR12: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR12,definition,
% 2.99/1.51 ( setukpairinjR12
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xx = Xy )
% 2.99/1.51 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.99/1.51 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR1_type,type,
% 2.99/1.51 setukpairinjR1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR1,definition,
% 2.99/1.51 ( setukpairinjR1
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.99/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.99/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.99/1.51 => ( ( Xz = Xu )
% 2.99/1.51 => ( Xy = Xu ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(upairequniteq_type,type,
% 2.99/1.51 upairequniteq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(upairequniteq,definition,
% 2.99/1.51 ( upairequniteq
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.99/1.51 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.99/1.51 = ( setadjoin @ Xz @ emptyset ) )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR2_type,type,
% 2.99/1.51 setukpairinjR2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR2,definition,
% 2.99/1.51 ( setukpairinjR2
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.99/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.99/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.99/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR_type,type,
% 2.99/1.51 setukpairinjR: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setukpairinjR,definition,
% 2.99/1.51 ( setukpairinjR
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.99/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.99/1.51 = ( kpair @ Xz @ Xu ) )
% 2.99/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ksndsingleton_type,type,
% 2.99/1.51 ksndsingleton: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ksndsingleton,definition,
% 2.99/1.51 ( ksndsingleton
% 2.99/1.51 = ( ! [Xu: $i] :
% 2.99/1.51 ( ( iskpair @ Xu )
% 2.99/1.51 => ( singleton
% 2.99/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.99/1.51 @ ^ [Xx: $i] :
% 2.99/1.51 ( Xu
% 2.99/1.51 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ksnd_type,type,
% 2.99/1.51 ksnd: $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(ksndpairEq_type,type,
% 2.99/1.51 ksndpairEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ksndpairEq,definition,
% 2.99/1.51 ( ksndpairEq
% 2.99/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.99/1.51 = Xy ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(kpairsurjEq_type,type,
% 2.99/1.51 kpairsurjEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(kpairsurjEq,definition,
% 2.99/1.51 ( kpairsurjEq
% 2.99/1.51 = ( ! [Xu: $i] :
% 2.99/1.51 ( ( iskpair @ Xu )
% 2.99/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.99/1.51 = Xu ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodsndin_type,type,
% 2.99/1.51 cartprodsndin: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodsndin,definition,
% 2.99/1.51 ( cartprodsndin
% 2.99/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.99/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.99/1.51 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodpairmemEL_type,type,
% 2.99/1.51 cartprodpairmemEL: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodpairmemEL,definition,
% 2.99/1.51 ( cartprodpairmemEL
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.99/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodpairmemER_type,type,
% 2.99/1.51 cartprodpairmemER: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodpairmemER,definition,
% 2.99/1.51 ( cartprodpairmemER
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.99/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodmempaircEq_type,type,
% 2.99/1.51 cartprodmempaircEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodmempaircEq,definition,
% 2.99/1.51 ( cartprodmempaircEq
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( kpair @ Xx @ Xy )
% 2.99/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodfstpairEq_type,type,
% 2.99/1.51 cartprodfstpairEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodfstpairEq,definition,
% 2.99/1.51 ( cartprodfstpairEq
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.99/1.51 = Xx ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodsndpairEq_type,type,
% 2.99/1.51 cartprodsndpairEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodsndpairEq,definition,
% 2.99/1.51 ( cartprodsndpairEq
% 2.99/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.99/1.51 = Xy ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodpairsurjEq_type,type,
% 2.99/1.51 cartprodpairsurjEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(cartprodpairsurjEq,definition,
% 2.99/1.51 ( cartprodpairsurjEq
% 2.99/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.99/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.99/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.99/1.51 = Xu ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(breln_type,type,
% 2.99/1.51 breln: $i > $i > $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstr_type,type,
% 2.99/1.51 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrI_type,type,
% 2.99/1.51 dpsetconstrI: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrI,definition,
% 2.99/1.51 ( dpsetconstrI
% 2.99/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( Xphi @ Xx @ Xy )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ Xy )
% 2.99/1.51 @ ( dpsetconstr @ A @ B
% 2.99/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrSub_type,type,
% 2.99/1.51 dpsetconstrSub: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrSub,definition,
% 2.99/1.51 ( dpsetconstrSub
% 2.99/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.99/1.51 ( subset
% 2.99/1.51 @ ( dpsetconstr @ A @ B
% 2.99/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.99/1.51 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(setOfPairsIsBReln_type,type,
% 2.99/1.51 setOfPairsIsBReln: $o ).
% 2.99/1.51
% 2.99/1.51 thf(setOfPairsIsBReln,definition,
% 2.99/1.51 ( setOfPairsIsBReln
% 2.99/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.99/1.51 ( breln @ A @ B
% 2.99/1.51 @ ( dpsetconstr @ A @ B
% 2.99/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrERa_type,type,
% 2.99/1.51 dpsetconstrERa: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrERa,definition,
% 2.99/1.51 ( dpsetconstrERa
% 2.99/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.99/1.51 @ ( dpsetconstr @ A @ B
% 2.99/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.99/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrEL1_type,type,
% 2.99/1.51 dpsetconstrEL1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrEL1,definition,
% 2.99/1.51 ( dpsetconstrEL1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.99/1.51 @ ( dpsetconstr @ A @ B
% 2.99/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.99/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrEL2_type,type,
% 2.99/1.51 dpsetconstrEL2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrEL2,definition,
% 2.99/1.51 ( dpsetconstrEL2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.99/1.51 @ ( dpsetconstr @ A @ B
% 2.99/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.99/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrER_type,type,
% 2.99/1.51 dpsetconstrER: $o ).
% 2.99/1.51
% 2.99/1.51 thf(dpsetconstrER,definition,
% 2.99/1.51 ( dpsetconstrER
% 2.99/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.99/1.51 @ ( dpsetconstr @ A @ B
% 2.99/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.99/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(func_type,type,
% 2.99/1.51 func: $i > $i > $i > $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcSet_type,type,
% 2.99/1.51 funcSet: $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(funcImageSingleton_type,type,
% 2.99/1.51 funcImageSingleton: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcImageSingleton,definition,
% 2.99/1.51 ( funcImageSingleton
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( singleton
% 2.99/1.51 @ ( dsetconstr @ B
% 2.99/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(apProp_type,type,
% 2.99/1.51 apProp: $o ).
% 2.99/1.51
% 2.99/1.51 thf(apProp,definition,
% 2.99/1.51 ( apProp
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in
% 2.99/1.51 @ ( setunion
% 2.99/1.51 @ ( dsetconstr @ B
% 2.99/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.99/1.51 @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ap_type,type,
% 2.99/1.51 ap: $i > $i > $i > $i > $i ).
% 2.99/1.51
% 2.99/1.51 thf(app_type,type,
% 2.99/1.51 app: $o ).
% 2.99/1.51
% 2.99/1.51 thf(app,definition,
% 2.99/1.51 ( app
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(infuncsetfunc_type,type,
% 2.99/1.51 infuncsetfunc: $o ).
% 2.99/1.51
% 2.99/1.51 thf(infuncsetfunc,definition,
% 2.99/1.51 ( infuncsetfunc
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.99/1.51 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ap2p_type,type,
% 2.99/1.51 ap2p: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ap2p,definition,
% 2.99/1.51 ( ap2p
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(funcinfuncset_type,type,
% 2.99/1.51 funcinfuncset: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcinfuncset,definition,
% 2.99/1.51 ( funcinfuncset
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(lamProp_type,type,
% 2.99/1.51 lamProp: $o ).
% 2.99/1.51
% 2.99/1.51 thf(lamProp,definition,
% 2.99/1.51 ( lamProp
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.99/1.51 => ( func @ A @ B
% 2.99/1.51 @ ( dpsetconstr @ A @ B
% 2.99/1.51 @ ^ [Xx: $i,Xy: $i] :
% 2.99/1.51 ( ( Xf @ Xx )
% 2.99/1.51 = Xy ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(lam_type,type,
% 2.99/1.51 lam: $i > $i > ( $i > $i ) > $i ).
% 2.99/1.51
% 2.99/1.51 thf(lamp_type,type,
% 2.99/1.51 lamp: $o ).
% 2.99/1.51
% 2.99/1.51 thf(lamp,definition,
% 2.99/1.51 ( lamp
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.99/1.51 => ( func @ A @ B
% 2.99/1.51 @ ( lam @ A @ B
% 2.99/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(lam2p_type,type,
% 2.99/1.51 lam2p: $o ).
% 2.99/1.51
% 2.99/1.51 thf(lam2p,definition,
% 2.99/1.51 ( lam2p
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.99/1.51 => ( in
% 2.99/1.51 @ ( lam @ A @ B
% 2.99/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.99/1.51 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(brelnall1_type,type,
% 2.99/1.51 brelnall1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(brelnall1,definition,
% 2.99/1.51 ( brelnall1
% 2.99/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.99/1.51 ( ( breln @ A @ B @ R )
% 2.99/1.51 => ! [Xphi: $i > $o] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ R )
% 2.99/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(brelnall2_type,type,
% 2.99/1.51 brelnall2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(brelnall2,definition,
% 2.99/1.51 ( brelnall2
% 2.99/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.99/1.51 ( ( breln @ A @ B @ R )
% 2.99/1.51 => ! [Xphi: $i > $o] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ R )
% 2.99/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ex1E2_type,type,
% 2.99/1.51 ex1E2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ex1E2,definition,
% 2.99/1.51 ( ex1E2
% 2.99/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.99/1.51 ( ( ex1 @ A
% 2.99/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ A )
% 2.99/1.51 => ( ( Xphi @ Xx )
% 2.99/1.51 => ( ( Xphi @ Xy )
% 2.99/1.51 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(funcGraphProp1_type,type,
% 2.99/1.51 funcGraphProp1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcGraphProp1,definition,
% 2.99/1.51 ( funcGraphProp1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(funcGraphProp3_type,type,
% 2.99/1.51 funcGraphProp3: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcGraphProp3,definition,
% 2.99/1.51 ( funcGraphProp3
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(funcGraphProp2_type,type,
% 2.99/1.51 funcGraphProp2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcGraphProp2,definition,
% 2.99/1.51 ( funcGraphProp2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.99/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.99/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(funcextLem_type,type,
% 2.99/1.51 funcextLem: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcextLem,definition,
% 2.99/1.51 ( funcextLem
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ! [Xg: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xg )
% 2.99/1.51 => ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.99/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(funcGraphProp4_type,type,
% 2.99/1.51 funcGraphProp4: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcGraphProp4,definition,
% 2.99/1.51 ( funcGraphProp4
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.99/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.99/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(subbreln_type,type,
% 2.99/1.51 subbreln: $o ).
% 2.99/1.51
% 2.99/1.51 thf(subbreln,definition,
% 2.99/1.51 ( subbreln
% 2.99/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.99/1.51 ( ( breln @ A @ B @ R )
% 2.99/1.51 => ! [S: $i] :
% 2.99/1.51 ( ( breln @ A @ B @ S )
% 2.99/1.51 => ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.99/1.51 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(eqbreln_type,type,
% 2.99/1.51 eqbreln: $o ).
% 2.99/1.51
% 2.99/1.51 thf(eqbreln,definition,
% 2.99/1.51 ( eqbreln
% 2.99/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.99/1.51 ( ( breln @ A @ B @ R )
% 2.99/1.51 => ! [S: $i] :
% 2.99/1.51 ( ( breln @ A @ B @ S )
% 2.99/1.51 => ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.99/1.51 => ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ! [Xy: $i] :
% 2.99/1.51 ( ( in @ Xy @ B )
% 2.99/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.99/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.99/1.51 => ( R = S ) ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(funcext_type,type,
% 2.99/1.51 funcext: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcext,definition,
% 2.99/1.51 ( funcext
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ! [Xg: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xg )
% 2.99/1.51 => ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.99/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.99/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(funcext2_type,type,
% 2.99/1.51 funcext2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(funcext2,definition,
% 2.99/1.51 ( funcext2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.99/1.51 => ! [Xg: $i] :
% 2.99/1.51 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.99/1.51 => ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.99/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.99/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ap2apEq1_type,type,
% 2.99/1.51 ap2apEq1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ap2apEq1,definition,
% 2.99/1.51 ( ap2apEq1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.99/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(ap2apEq2_type,type,
% 2.99/1.51 ap2apEq2: $o ).
% 2.99/1.51
% 2.99/1.51 thf(ap2apEq2,definition,
% 2.99/1.51 ( ap2apEq2
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.99/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(beta1_type,type,
% 2.99/1.51 beta1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(beta1,definition,
% 2.99/1.51 ( beta1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.99/1.51 => ! [Xx: $i] :
% 2.99/1.51 ( ( in @ Xx @ A )
% 2.99/1.51 => ( ( ap @ A @ B
% 2.99/1.51 @ ( lam @ A @ B
% 2.99/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.99/1.51 @ Xx )
% 2.99/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(eta1_type,type,
% 2.99/1.51 eta1: $o ).
% 2.99/1.51
% 2.99/1.51 thf(eta1,definition,
% 2.99/1.51 ( eta1
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.51 ( ( func @ A @ B @ Xf )
% 2.99/1.51 => ( ( lam @ A @ B
% 2.99/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.99/1.51 = Xf ) ) ) ) ).
% 2.99/1.51
% 2.99/1.51 thf(lam2lamEq_type,type,
% 2.99/1.51 lam2lamEq: $o ).
% 2.99/1.51
% 2.99/1.51 thf(lam2lamEq,definition,
% 2.99/1.51 ( lam2lamEq
% 2.99/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.99/1.51 ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.99/1.52 => ( ( lam @ A @ B
% 2.99/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.99/1.52 = ( lam @ A @ B
% 2.99/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(beta2_type,type,
% 2.99/1.52 beta2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(beta2,definition,
% 2.99/1.52 ( beta2
% 2.99/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.99/1.52 ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( ap @ A @ B
% 2.99/1.52 @ ( lam @ A @ B
% 2.99/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.99/1.52 @ Xx )
% 2.99/1.52 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(eta2_type,type,
% 2.99/1.52 eta2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(eta2,definition,
% 2.99/1.52 ( eta2
% 2.99/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.99/1.52 => ( ( lam @ A @ B
% 2.99/1.52 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.99/1.52 = Xf ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(iffalseProp1_type,type,
% 2.99/1.52 iffalseProp1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(iffalseProp1,definition,
% 2.99/1.52 ( iffalseProp1
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ~ Xphi
% 2.99/1.52 => ( in @ Xy
% 2.99/1.52 @ ( dsetconstr @ A
% 2.99/1.52 @ ^ [Xz: $i] :
% 2.99/1.52 ( ( Xphi
% 2.99/1.52 & ( Xz = Xx ) )
% 2.99/1.52 | ( ~ Xphi
% 2.99/1.52 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(iffalseProp2_type,type,
% 2.99/1.52 iffalseProp2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(iffalseProp2,definition,
% 2.99/1.52 ( iffalseProp2
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ~ Xphi
% 2.99/1.52 => ( ( dsetconstr @ A
% 2.99/1.52 @ ^ [Xz: $i] :
% 2.99/1.52 ( ( Xphi
% 2.99/1.52 & ( Xz = Xx ) )
% 2.99/1.52 | ( ~ Xphi
% 2.99/1.52 & ( Xz = Xy ) ) ) )
% 2.99/1.52 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(iftrueProp1_type,type,
% 2.99/1.52 iftrueProp1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(iftrueProp1,definition,
% 2.99/1.52 ( iftrueProp1
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( Xphi
% 2.99/1.52 => ( in @ Xx
% 2.99/1.52 @ ( dsetconstr @ A
% 2.99/1.52 @ ^ [Xz: $i] :
% 2.99/1.52 ( ( Xphi
% 2.99/1.52 & ( Xz = Xx ) )
% 2.99/1.52 | ( ~ Xphi
% 2.99/1.52 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(iftrueProp2_type,type,
% 2.99/1.52 iftrueProp2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(iftrueProp2,definition,
% 2.99/1.52 ( iftrueProp2
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( Xphi
% 2.99/1.52 => ( ( dsetconstr @ A
% 2.99/1.52 @ ^ [Xz: $i] :
% 2.99/1.52 ( ( Xphi
% 2.99/1.52 & ( Xz = Xx ) )
% 2.99/1.52 | ( ~ Xphi
% 2.99/1.52 & ( Xz = Xy ) ) ) )
% 2.99/1.52 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(ifSingleton_type,type,
% 2.99/1.52 ifSingleton: $o ).
% 2.99/1.52
% 2.99/1.52 thf(ifSingleton,definition,
% 2.99/1.52 ( ifSingleton
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( singleton
% 2.99/1.52 @ ( dsetconstr @ A
% 2.99/1.52 @ ^ [Xz: $i] :
% 2.99/1.52 ( ( Xphi
% 2.99/1.52 & ( Xz = Xx ) )
% 2.99/1.52 | ( ~ Xphi
% 2.99/1.52 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(if_type,type,
% 2.99/1.52 if: $i > $o > $i > $i > $i ).
% 2.99/1.52
% 2.99/1.52 thf(ifp_type,type,
% 2.99/1.52 ifp: $o ).
% 2.99/1.52
% 2.99/1.52 thf(ifp,definition,
% 2.99/1.52 ( ifp
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(theeq_type,type,
% 2.99/1.52 theeq: $o ).
% 2.99/1.52
% 2.99/1.52 thf(theeq,definition,
% 2.99/1.52 ( theeq
% 2.99/1.52 = ( ! [X: $i] :
% 2.99/1.52 ( ( singleton @ X )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ X )
% 2.99/1.52 => ( ( setunion @ X )
% 2.99/1.52 = Xx ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(iftrue_type,type,
% 2.99/1.52 iftrue: $o ).
% 2.99/1.52
% 2.99/1.52 thf(iftrue,definition,
% 2.99/1.52 ( iftrue
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( Xphi
% 2.99/1.52 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.99/1.52 = Xx ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(iffalse_type,type,
% 2.99/1.52 iffalse: $o ).
% 2.99/1.52
% 2.99/1.52 thf(iffalse,definition,
% 2.99/1.52 ( iffalse
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ~ Xphi
% 2.99/1.52 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.99/1.52 = Xy ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(iftrueorfalse_type,type,
% 2.99/1.52 iftrueorfalse: $o ).
% 2.99/1.52
% 2.99/1.52 thf(iftrueorfalse,definition,
% 2.99/1.52 ( iftrueorfalse
% 2.99/1.52 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(binintersectT_lem_type,type,
% 2.99/1.52 binintersectT_lem: $o ).
% 2.99/1.52
% 2.99/1.52 thf(binintersectT_lem,definition,
% 2.99/1.52 ( binintersectT_lem
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(binunionT_lem_type,type,
% 2.99/1.52 binunionT_lem: $o ).
% 2.99/1.52
% 2.99/1.52 thf(binunionT_lem,definition,
% 2.99/1.52 ( binunionT_lem
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(powersetT_lem_type,type,
% 2.99/1.52 powersetT_lem: $o ).
% 2.99/1.52
% 2.99/1.52 thf(powersetT_lem,definition,
% 2.99/1.52 ( powersetT_lem
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(setminusT_lem_type,type,
% 2.99/1.52 setminusT_lem: $o ).
% 2.99/1.52
% 2.99/1.52 thf(setminusT_lem,definition,
% 2.99/1.52 ( setminusT_lem
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementT_lem_type,type,
% 2.99/1.52 complementT_lem: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementT_lem,definition,
% 2.99/1.52 ( complementT_lem
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(setextT_type,type,
% 2.99/1.52 setextT: $o ).
% 2.99/1.52
% 2.99/1.52 thf(setextT,definition,
% 2.99/1.52 ( setextT
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ X )
% 2.99/1.52 => ( in @ Xx @ Y ) ) )
% 2.99/1.52 => ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ Y )
% 2.99/1.52 => ( in @ Xx @ X ) ) )
% 2.99/1.52 => ( X = Y ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(subsetTI_type,type,
% 2.99/1.52 subsetTI: $o ).
% 2.99/1.52
% 2.99/1.52 thf(subsetTI,definition,
% 2.99/1.52 ( subsetTI
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ X )
% 2.99/1.52 => ( in @ Xx @ Y ) ) )
% 2.99/1.52 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(powersetTI1_type,type,
% 2.99/1.52 powersetTI1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(powersetTI1,definition,
% 2.99/1.52 ( powersetTI1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ X )
% 2.99/1.52 => ( in @ Xx @ Y ) ) )
% 2.99/1.52 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(powersetTE1_type,type,
% 2.99/1.52 powersetTE1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(powersetTE1,definition,
% 2.99/1.52 ( powersetTE1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ X @ ( powerset @ Y ) )
% 2.99/1.52 => ( ( in @ Xx @ X )
% 2.99/1.52 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementTI1_type,type,
% 2.99/1.52 complementTI1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementTI1,definition,
% 2.99/1.52 ( complementTI1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ X )
% 2.99/1.52 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementTE1_type,type,
% 2.99/1.52 complementTE1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementTE1,definition,
% 2.99/1.52 ( complementTE1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.99/1.52 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(binintersectTELcontra_type,type,
% 2.99/1.52 binintersectTELcontra: $o ).
% 2.99/1.52
% 2.99/1.52 thf(binintersectTELcontra,definition,
% 2.99/1.52 ( binintersectTELcontra
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ~ ( in @ Xx @ X )
% 2.99/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(binintersectTERcontra_type,type,
% 2.99/1.52 binintersectTERcontra: $o ).
% 2.99/1.52
% 2.99/1.52 thf(binintersectTERcontra,definition,
% 2.99/1.52 ( binintersectTERcontra
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ~ ( in @ Xx @ Y )
% 2.99/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(contrasubsetT_type,type,
% 2.99/1.52 contrasubsetT: $o ).
% 2.99/1.52
% 2.99/1.52 thf(contrasubsetT,definition,
% 2.99/1.52 ( contrasubsetT
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.99/1.52 => ( ( in @ Xx @ Y )
% 2.99/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(contrasubsetT1_type,type,
% 2.99/1.52 contrasubsetT1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(contrasubsetT1,definition,
% 2.99/1.52 ( contrasubsetT1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( subset @ X @ Y )
% 2.99/1.52 => ( ~ ( in @ Xx @ Y )
% 2.99/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(contrasubsetT2_type,type,
% 2.99/1.52 contrasubsetT2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(contrasubsetT2,definition,
% 2.99/1.52 ( contrasubsetT2
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ X @ Y )
% 2.99/1.52 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(contrasubsetT3_type,type,
% 2.99/1.52 contrasubsetT3: $o ).
% 2.99/1.52
% 2.99/1.52 thf(contrasubsetT3,definition,
% 2.99/1.52 ( contrasubsetT3
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.99/1.52 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementI1_type,type,
% 2.99/1.52 doubleComplementI1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementI1,definition,
% 2.99/1.52 ( doubleComplementI1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ X )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementE1_type,type,
% 2.99/1.52 doubleComplementE1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementE1,definition,
% 2.99/1.52 ( doubleComplementE1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.99/1.52 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementSub1_type,type,
% 2.99/1.52 doubleComplementSub1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementSub1,definition,
% 2.99/1.52 ( doubleComplementSub1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementSub2_type,type,
% 2.99/1.52 doubleComplementSub2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementSub2,definition,
% 2.99/1.52 ( doubleComplementSub2
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementEq_type,type,
% 2.99/1.52 doubleComplementEq: $o ).
% 2.99/1.52
% 2.99/1.52 thf(doubleComplementEq,definition,
% 2.99/1.52 ( doubleComplementEq
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ( X
% 2.99/1.52 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementTnotintersectT_type,type,
% 2.99/1.52 complementTnotintersectT: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementTnotintersectT,definition,
% 2.99/1.52 ( complementTnotintersectT
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.99/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementImpComplementIntersect_type,type,
% 2.99/1.52 complementImpComplementIntersect: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementImpComplementIntersect,definition,
% 2.99/1.52 ( complementImpComplementIntersect
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementSubsetComplementIntersect_type,type,
% 2.99/1.52 complementSubsetComplementIntersect: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementSubsetComplementIntersect,definition,
% 2.99/1.52 ( complementSubsetComplementIntersect
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementInPowersetComplementIntersect_type,type,
% 2.99/1.52 complementInPowersetComplementIntersect: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementInPowersetComplementIntersect,definition,
% 2.99/1.52 ( complementInPowersetComplementIntersect
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(contraSubsetComplement_type,type,
% 2.99/1.52 contraSubsetComplement: $o ).
% 2.99/1.52
% 2.99/1.52 thf(contraSubsetComplement,definition,
% 2.99/1.52 ( contraSubsetComplement
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ Y )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementTcontraSubset_type,type,
% 2.99/1.52 complementTcontraSubset: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementTcontraSubset,definition,
% 2.99/1.52 ( complementTcontraSubset
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.99/1.52 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(binunionTILcontra_type,type,
% 2.99/1.52 binunionTILcontra: $o ).
% 2.99/1.52
% 2.99/1.52 thf(binunionTILcontra,definition,
% 2.99/1.52 ( binunionTILcontra
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.99/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(binunionTIRcontra_type,type,
% 2.99/1.52 binunionTIRcontra: $o ).
% 2.99/1.52
% 2.99/1.52 thf(binunionTIRcontra,definition,
% 2.99/1.52 ( binunionTIRcontra
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.99/1.52 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(inIntersectImpInUnion_type,type,
% 2.99/1.52 inIntersectImpInUnion: $o ).
% 2.99/1.52
% 2.99/1.52 thf(inIntersectImpInUnion,definition,
% 2.99/1.52 ( inIntersectImpInUnion
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.99/1.52 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(inIntersectImpInUnion2_type,type,
% 2.99/1.52 inIntersectImpInUnion2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(inIntersectImpInUnion2,definition,
% 2.99/1.52 ( inIntersectImpInUnion2
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.99/1.52 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(inIntersectImpInIntersectUnions_type,type,
% 2.99/1.52 inIntersectImpInIntersectUnions: $o ).
% 2.99/1.52
% 2.99/1.52 thf(inIntersectImpInIntersectUnions,definition,
% 2.99/1.52 ( inIntersectImpInIntersectUnions
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.99/1.52 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(intersectInPowersetIntersectUnions_type,type,
% 2.99/1.52 intersectInPowersetIntersectUnions: $o ).
% 2.99/1.52
% 2.99/1.52 thf(intersectInPowersetIntersectUnions,definition,
% 2.99/1.52 ( intersectInPowersetIntersectUnions
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(inComplementUnionImpNotIn1_type,type,
% 2.99/1.52 inComplementUnionImpNotIn1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(inComplementUnionImpNotIn1,definition,
% 2.99/1.52 ( inComplementUnionImpNotIn1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.99/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(inComplementUnionImpInComplement1_type,type,
% 2.99/1.52 inComplementUnionImpInComplement1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(inComplementUnionImpInComplement1,definition,
% 2.99/1.52 ( inComplementUnionImpInComplement1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(binunionTE_type,type,
% 2.99/1.52 binunionTE: $o ).
% 2.99/1.52
% 2.99/1.52 thf(binunionTE,definition,
% 2.99/1.52 ( binunionTE
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xphi: $o,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.99/1.52 => ( ( ( in @ Xx @ X )
% 2.99/1.52 => Xphi )
% 2.99/1.52 => ( ( ( in @ Xx @ Y )
% 2.99/1.52 => Xphi )
% 2.99/1.52 => Xphi ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(binunionTEcontra_type,type,
% 2.99/1.52 binunionTEcontra: $o ).
% 2.99/1.52
% 2.99/1.52 thf(binunionTEcontra,definition,
% 2.99/1.52 ( binunionTEcontra
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ~ ( in @ Xx @ X )
% 2.99/1.52 => ( ~ ( in @ Xx @ Y )
% 2.99/1.52 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2a1_type,type,
% 2.99/1.52 demorgan2a1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2a1,definition,
% 2.99/1.52 ( demorgan2a1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(complementUnionInPowersetComplement_type,type,
% 2.99/1.52 complementUnionInPowersetComplement: $o ).
% 2.99/1.52
% 2.99/1.52 thf(complementUnionInPowersetComplement,definition,
% 2.99/1.52 ( complementUnionInPowersetComplement
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2a2_type,type,
% 2.99/1.52 demorgan2a2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2a2,definition,
% 2.99/1.52 ( demorgan2a2
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan1a_type,type,
% 2.99/1.52 demorgan1a: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan1a,definition,
% 2.99/1.52 ( demorgan1a
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.99/1.52 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan1b_type,type,
% 2.99/1.52 demorgan1b: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan1b,definition,
% 2.99/1.52 ( demorgan1b
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan1_type,type,
% 2.99/1.52 demorgan1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan1,definition,
% 2.99/1.52 ( demorgan1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.99/1.52 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2a_type,type,
% 2.99/1.52 demorgan2a: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2a,definition,
% 2.99/1.52 ( demorgan2a
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.99/1.52 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2b2_type,type,
% 2.99/1.52 demorgan2b2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2b2,definition,
% 2.99/1.52 ( demorgan2b2
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.99/1.52 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2b_type,type,
% 2.99/1.52 demorgan2b: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2b,definition,
% 2.99/1.52 ( demorgan2b
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.99/1.52 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2_type,type,
% 2.99/1.52 demorgan2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(demorgan2,definition,
% 2.99/1.52 ( demorgan2
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.99/1.52 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule0_type,type,
% 2.99/1.52 woz13rule0: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule0,definition,
% 2.99/1.52 ( woz13rule0
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.99/1.52 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule1_type,type,
% 2.99/1.52 woz13rule1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule1,definition,
% 2.99/1.52 ( woz13rule1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ X @ Z )
% 2.99/1.52 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule2_type,type,
% 2.99/1.52 woz13rule2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule2,definition,
% 2.99/1.52 ( woz13rule2
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ Y @ Z )
% 2.99/1.52 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule3_type,type,
% 2.99/1.52 woz13rule3: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule3,definition,
% 2.99/1.52 ( woz13rule3
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ X @ Y )
% 2.99/1.52 => ( ( subset @ X @ Z )
% 2.99/1.52 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule4_type,type,
% 2.99/1.52 woz13rule4: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz13rule4,definition,
% 2.99/1.52 ( woz13rule4
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ! [W: $i] :
% 2.99/1.52 ( ( in @ W @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ X @ Z )
% 2.99/1.52 => ( ( subset @ Y @ W )
% 2.99/1.52 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_1_type,type,
% 2.99/1.52 woz1_1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_1,definition,
% 2.99/1.52 ( woz1_1
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_2_type,type,
% 2.99/1.52 woz1_2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_2,definition,
% 2.99/1.52 ( woz1_2
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ! [W: $i] :
% 2.99/1.52 ( ( in @ W @ ( powerset @ A ) )
% 2.99/1.52 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.99/1.52 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_3_type,type,
% 2.99/1.52 woz1_3: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_3,definition,
% 2.99/1.52 ( woz1_3
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ! [Z: $i] :
% 2.99/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_4_type,type,
% 2.99/1.52 woz1_4: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_4,definition,
% 2.99/1.52 ( woz1_4
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.99/1.52 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_5_type,type,
% 2.99/1.52 woz1_5: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz1_5,definition,
% 2.99/1.52 ( woz1_5
% 2.99/1.52 = ( ! [A: $i,X: $i] :
% 2.99/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.99/1.52 => ! [Y: $i] :
% 2.99/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.99/1.52 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1_type,type,
% 2.99/1.52 breln1: $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1all2_type,type,
% 2.99/1.52 breln1all2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1all2,definition,
% 2.99/1.52 ( breln1all2
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [Xphi: $i > $o] :
% 2.99/1.52 ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ R )
% 2.99/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1Set_type,type,
% 2.99/1.52 breln1Set: $i > $i ).
% 2.99/1.52
% 2.99/1.52 thf(breln1SetBreln1_type,type,
% 2.99/1.52 breln1SetBreln1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1SetBreln1,definition,
% 2.99/1.52 ( breln1SetBreln1
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( in @ R @ ( breln1Set @ A ) )
% 2.99/1.52 => ( breln1 @ A @ R ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(transitive_type,type,
% 2.99/1.52 transitive: $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(antisymmetric_type,type,
% 2.99/1.52 antisymmetric: $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(reflexive_type,type,
% 2.99/1.52 reflexive: $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(refltransitive_type,type,
% 2.99/1.52 refltransitive: $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(refllinearorder_type,type,
% 2.99/1.52 refllinearorder: $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(reflwellordering_type,type,
% 2.99/1.52 reflwellordering: $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(choice2fnsingleton_type,type,
% 2.99/1.52 choice2fnsingleton: $o ).
% 2.99/1.52
% 2.99/1.52 thf(choice2fnsingleton,definition,
% 2.99/1.52 ( choice2fnsingleton
% 2.99/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.99/1.52 ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ? [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ B )
% 2.99/1.52 & ( Xphi @ Xx @ Xy ) ) )
% 2.99/1.52 => ! [R: $i] :
% 2.99/1.52 ( ( in @ R @ ( breln1Set @ B ) )
% 2.99/1.52 => ( ( reflwellordering @ B @ R )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( singleton
% 2.99/1.52 @ ( dsetconstr @ B
% 2.99/1.52 @ ^ [Xy: $i] :
% 2.99/1.52 ( ( Xphi @ Xx @ Xy )
% 2.99/1.52 & ! [Xz: $i] :
% 2.99/1.52 ( ( in @ Xz @ B )
% 2.99/1.52 => ( ( Xphi @ Xx @ Xz )
% 2.99/1.52 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(setOfPairsIsBReln1_type,type,
% 2.99/1.52 setOfPairsIsBReln1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(setOfPairsIsBReln1,definition,
% 2.99/1.52 ( setOfPairsIsBReln1
% 2.99/1.52 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 2.99/1.52 ( breln1 @ A
% 2.99/1.52 @ ( dpsetconstr @ A @ A
% 2.99/1.52 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1all1_type,type,
% 2.99/1.52 breln1all1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1all1,definition,
% 2.99/1.52 ( breln1all1
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [Xphi: $i > $o] :
% 2.99/1.52 ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ R )
% 2.99/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(subbreln1_type,type,
% 2.99/1.52 subbreln1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(subbreln1,definition,
% 2.99/1.52 ( subbreln1
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.99/1.52 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(eqbreln1_type,type,
% 2.99/1.52 eqbreln1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(eqbreln1,definition,
% 2.99/1.52 ( eqbreln1
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.99/1.52 => ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.99/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.99/1.52 => ( R = S ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1invset_type,type,
% 2.99/1.52 breln1invset: $i > $i > $i ).
% 2.99/1.52
% 2.99/1.52 thf(breln1invprop_type,type,
% 2.99/1.52 breln1invprop: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1invprop,definition,
% 2.99/1.52 ( breln1invprop
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1invI_type,type,
% 2.99/1.52 breln1invI: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1invI,definition,
% 2.99/1.52 ( breln1invI
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1invE_type,type,
% 2.99/1.52 breln1invE: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1invE,definition,
% 2.99/1.52 ( breln1invE
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 2.99/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compset_type,type,
% 2.99/1.52 breln1compset: $i > $i > $i > $i ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compprop_type,type,
% 2.99/1.52 breln1compprop: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compprop,definition,
% 2.99/1.52 ( breln1compprop
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compI_type,type,
% 2.99/1.52 breln1compI: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compI,definition,
% 2.99/1.52 ( breln1compI
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ! [Xz: $i] :
% 2.99/1.52 ( ( in @ Xz @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.99/1.52 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.99/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compE_type,type,
% 2.99/1.52 breln1compE: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compE,definition,
% 2.99/1.52 ( breln1compE
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.99/1.52 => ? [Xz: $i] :
% 2.99/1.52 ( ( in @ Xz @ A )
% 2.99/1.52 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.99/1.52 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compEex_type,type,
% 2.99/1.52 breln1compEex: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1compEex,definition,
% 2.99/1.52 ( breln1compEex
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.99/1.52 => ! [Xphi: $o] :
% 2.99/1.52 ( ! [Xz: $i] :
% 2.99/1.52 ( ( in @ Xz @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.99/1.52 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.99/1.52 => Xphi ) ) )
% 2.99/1.52 => Xphi ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionprop_type,type,
% 2.99/1.52 breln1unionprop: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionprop,definition,
% 2.99/1.52 ( breln1unionprop
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionIL_type,type,
% 2.99/1.52 breln1unionIL: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionIL,definition,
% 2.99/1.52 ( breln1unionIL
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionIR_type,type,
% 2.99/1.52 breln1unionIR: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionIR,definition,
% 2.99/1.52 ( breln1unionIR
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.99/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionI_type,type,
% 2.99/1.52 breln1unionI: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionI,definition,
% 2.99/1.52 ( breln1unionI
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 2.99/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionE_type,type,
% 2.99/1.52 breln1unionE: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionE,definition,
% 2.99/1.52 ( breln1unionE
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionEcases_type,type,
% 2.99/1.52 breln1unionEcases: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionEcases,definition,
% 2.99/1.52 ( breln1unionEcases
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ! [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.99/1.52 => ! [Xphi: $o] :
% 2.99/1.52 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.99/1.52 => Xphi )
% 2.99/1.52 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.99/1.52 => Xphi )
% 2.99/1.52 => Xphi ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionCommutes_type,type,
% 2.99/1.52 breln1unionCommutes: $o ).
% 2.99/1.52
% 2.99/1.52 thf(breln1unionCommutes,definition,
% 2.99/1.52 ( breln1unionCommutes
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ( ( binunion @ R @ S )
% 2.99/1.52 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz2Ex_type,type,
% 2.99/1.52 woz2Ex: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz2Ex,definition,
% 2.99/1.52 ( woz2Ex
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ( R
% 2.99/1.52 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz2W_type,type,
% 2.99/1.52 woz2W: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz2W,definition,
% 2.99/1.52 ( woz2W
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 2.99/1.52 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz2A_type,type,
% 2.99/1.52 woz2A: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz2A,definition,
% 2.99/1.52 ( woz2A
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [T: $i] :
% 2.99/1.52 ( ( breln1 @ A @ T )
% 2.99/1.52 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 2.99/1.52 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(woz2B_type,type,
% 2.99/1.52 woz2B: $o ).
% 2.99/1.52
% 2.99/1.52 thf(woz2B,definition,
% 2.99/1.52 ( woz2B
% 2.99/1.52 = ( ! [A: $i,R: $i] :
% 2.99/1.52 ( ( breln1 @ A @ R )
% 2.99/1.52 => ! [S: $i] :
% 2.99/1.52 ( ( breln1 @ A @ S )
% 2.99/1.52 => ! [T: $i] :
% 2.99/1.52 ( ( breln1 @ A @ T )
% 2.99/1.52 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 2.99/1.52 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(image1Ex_type,type,
% 2.99/1.52 image1Ex: $o ).
% 2.99/1.52
% 2.99/1.52 thf(image1Ex,definition,
% 2.99/1.52 ( image1Ex
% 2.99/1.52 = ( ! [A: $i,Xf: $i > $i] :
% 2.99/1.52 ? [B: $i] :
% 2.99/1.52 ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ B )
% 2.99/1.52 <=> ? [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 & ( Xx
% 2.99/1.52 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(image1Ex1_type,type,
% 2.99/1.52 image1Ex1: $o ).
% 2.99/1.52
% 2.99/1.52 thf(image1Ex1,definition,
% 2.99/1.52 ( image1Ex1
% 2.99/1.52 = ( ! [A: $i,Xf: $i > $i] :
% 2.99/1.52 ( exu
% 2.99/1.52 @ ^ [B: $i] :
% 2.99/1.52 ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ B )
% 2.99/1.52 <=> ? [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 & ( Xx
% 2.99/1.52 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(image1_type,type,
% 2.99/1.52 image1: $i > ( $i > $i ) > $i ).
% 2.99/1.52
% 2.99/1.52 thf(image1Equiv_type,type,
% 2.99/1.52 image1Equiv: $o ).
% 2.99/1.52
% 2.99/1.52 thf(image1Equiv,definition,
% 2.99/1.52 ( image1Equiv
% 2.99/1.52 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx
% 2.99/1.52 @ ( image1 @ A
% 2.99/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 2.99/1.52 <=> ? [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 & ( Xx
% 2.99/1.52 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(image1E_type,type,
% 2.99/1.52 image1E: $o ).
% 2.99/1.52
% 2.99/1.52 thf(image1E,definition,
% 2.99/1.52 ( image1E
% 2.99/1.52 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.99/1.52 ( ( in @ Xx
% 2.99/1.52 @ ( image1 @ A
% 2.99/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 2.99/1.52 => ? [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 & ( Xx
% 2.99/1.52 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(image1I_type,type,
% 2.99/1.52 image1I: $o ).
% 2.99/1.52
% 2.99/1.52 thf(image1I,definition,
% 2.99/1.52 ( image1I
% 2.99/1.52 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.99/1.52 ( ? [Xy: $i] :
% 2.99/1.52 ( ( in @ Xy @ A )
% 2.99/1.52 & ( Xx
% 2.99/1.52 = ( Xf @ Xy ) ) )
% 2.99/1.52 => ( in @ Xx
% 2.99/1.52 @ ( image1 @ A
% 2.99/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(injective_type,type,
% 2.99/1.52 injective: $i > $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(injFuncSet_type,type,
% 2.99/1.52 injFuncSet: $i > $i > $i ).
% 2.99/1.52
% 2.99/1.52 thf(injFuncInInjFuncSet_type,type,
% 2.99/1.52 injFuncInInjFuncSet: $o ).
% 2.99/1.52
% 2.99/1.52 thf(injFuncInInjFuncSet,definition,
% 2.99/1.52 ( injFuncInInjFuncSet
% 2.99/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.99/1.52 => ( ( injective @ A @ B @ Xf )
% 2.99/1.52 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(injFuncSetFuncIn_type,type,
% 2.99/1.52 injFuncSetFuncIn: $o ).
% 2.99/1.52
% 2.99/1.52 thf(injFuncSetFuncIn,definition,
% 2.99/1.52 ( injFuncSetFuncIn
% 2.99/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.52 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 2.99/1.52 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(injFuncSetFuncInj_type,type,
% 2.99/1.52 injFuncSetFuncInj: $o ).
% 2.99/1.52
% 2.99/1.52 thf(injFuncSetFuncInj,definition,
% 2.99/1.52 ( injFuncSetFuncInj
% 2.99/1.52 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 2.99/1.52 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 2.99/1.52 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(surjective_type,type,
% 2.99/1.52 surjective: $i > $i > $i > $o ).
% 2.99/1.52
% 2.99/1.52 thf(surjFuncSet_type,type,
% 2.99/1.52 surjFuncSet: $i > $i > $i ).
% 2.99/1.52
% 2.99/1.52 thf(surjFuncSetFuncIn_type,type,
% 2.99/1.52 surjFuncSetFuncIn: $o ).
% 2.99/1.52
% 2.99/1.52 thf(surjFuncSetFuncIn,definition,
% 2.99/1.52 ( surjFuncSetFuncIn
% 2.99/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.99/1.52 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 2.99/1.52 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(surjFuncSetFuncSurj_type,type,
% 2.99/1.52 surjFuncSetFuncSurj: $o ).
% 2.99/1.52
% 2.99/1.52 thf(surjFuncSetFuncSurj,definition,
% 2.99/1.52 ( surjFuncSetFuncSurj
% 2.99/1.52 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 2.99/1.52 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 2.99/1.52 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(leftInvIsSurj_type,type,
% 2.99/1.52 leftInvIsSurj: $o ).
% 2.99/1.52
% 2.99/1.52 thf(leftInvIsSurj,definition,
% 2.99/1.52 ( leftInvIsSurj
% 2.99/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.99/1.52 ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.99/1.52 => ! [Xg: $i] :
% 2.99/1.52 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 2.99/1.52 => ( ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ A )
% 2.99/1.52 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 2.99/1.52 = Xx ) )
% 2.99/1.52 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(surjCantorThm_type,type,
% 2.99/1.52 surjCantorThm: $o ).
% 2.99/1.52
% 2.99/1.52 thf(surjCantorThm,definition,
% 2.99/1.52 ( surjCantorThm
% 2.99/1.52 = ( ! [A: $i,Xf: $i] :
% 2.99/1.52 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 2.99/1.52 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(foundation2_type,type,
% 2.99/1.52 foundation2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(foundation2,definition,
% 2.99/1.52 ( foundation2
% 2.99/1.52 = ( ! [A: $i] :
% 2.99/1.52 ( ( nonempty @ A )
% 2.99/1.52 => ? [X: $i] :
% 2.99/1.52 ( ( in @ X @ A )
% 2.99/1.52 & ( ( binintersect @ X @ A )
% 2.99/1.52 = emptyset ) ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(notinself_type,type,
% 2.99/1.52 notinself: $o ).
% 2.99/1.52
% 2.99/1.52 thf(notinself,definition,
% 2.99/1.52 ( notinself
% 2.99/1.52 = ( ! [A: $i] :
% 2.99/1.52 ~ ( in @ A @ A ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(notinself2_type,type,
% 2.99/1.52 notinself2: $o ).
% 2.99/1.52
% 2.99/1.52 thf(notinself2,definition,
% 2.99/1.52 ( notinself2
% 2.99/1.52 = ( ! [A: $i,B: $i] :
% 2.99/1.52 ( ( in @ A @ B )
% 2.99/1.52 => ~ ( in @ B @ A ) ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(omegaS_type,type,
% 2.99/1.52 omegaS: $i > $i ).
% 2.99/1.52
% 2.99/1.52 thf(omegaS,definition,
% 2.99/1.52 ( omegaS
% 2.99/1.52 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 2.99/1.52
% 2.99/1.52 thf(omegaSp,conjecture,
% 2.99/1.52 ( setextAx
% 2.99/1.52 => ( emptysetAx
% 2.99/1.52 => ( setadjoinAx
% 2.99/1.52 => ( powersetAx
% 2.99/1.52 => ( setunionAx
% 2.99/1.52 => ( omega0Ax
% 2.99/1.52 => ( omegaSAx
% 2.99/1.52 => ( omegaIndAx
% 2.99/1.52 => ( replAx
% 2.99/1.52 => ( foundationAx
% 2.99/1.52 => ( wellorderingAx
% 2.99/1.52 => ( descrp
% 2.99/1.52 => ( dsetconstrI
% 2.99/1.52 => ( dsetconstrEL
% 2.99/1.52 => ( dsetconstrER
% 2.99/1.52 => ( exuE1
% 2.99/1.52 => ( prop2setE
% 2.99/1.52 => ( emptysetE
% 2.99/1.52 => ( emptysetimpfalse
% 2.99/1.52 => ( notinemptyset
% 2.99/1.52 => ( exuE3e
% 2.99/1.52 => ( setext
% 2.99/1.52 => ( emptyI
% 2.99/1.52 => ( noeltsimpempty
% 2.99/1.52 => ( setbeta
% 2.99/1.52 => ( nonemptyE1
% 2.99/1.52 => ( nonemptyI
% 2.99/1.52 => ( nonemptyI1
% 2.99/1.52 => ( setadjoinIL
% 2.99/1.52 => ( emptyinunitempty
% 2.99/1.52 => ( setadjoinIR
% 2.99/1.52 => ( setadjoinE
% 2.99/1.52 => ( setadjoinOr
% 2.99/1.52 => ( setoftrueEq
% 2.99/1.52 => ( powersetI
% 2.99/1.52 => ( emptyinPowerset
% 2.99/1.52 => ( emptyInPowerset
% 2.99/1.52 => ( powersetE
% 2.99/1.52 => ( setunionI
% 2.99/1.52 => ( setunionE
% 2.99/1.52 => ( subPowSU
% 2.99/1.52 => ( exuE2
% 2.99/1.52 => ( nonemptyImpWitness
% 2.99/1.52 => ( uniqinunit
% 2.99/1.52 => ( notinsingleton
% 2.99/1.52 => ( eqinunit
% 2.99/1.52 => ( singletonsswitch
% 2.99/1.52 => ( upairsetE
% 2.99/1.52 => ( upairsetIL
% 2.99/1.52 => ( upairsetIR
% 2.99/1.52 => ( emptyE1
% 2.99/1.52 => ( vacuousDall
% 2.99/1.52 => ( quantDeMorgan1
% 2.99/1.52 => ( quantDeMorgan2
% 2.99/1.52 => ( quantDeMorgan3
% 2.99/1.52 => ( quantDeMorgan4
% 2.99/1.52 => ( prop2setI
% 2.99/1.52 => ( prop2set2propI
% 2.99/1.52 => ( notdexE
% 2.99/1.52 => ( notdallE
% 2.99/1.52 => ( exuI1
% 2.99/1.52 => ( exuI3
% 2.99/1.52 => ( exuI2
% 2.99/1.52 => ( inCongP
% 2.99/1.52 => ( in__Cong
% 2.99/1.52 => ( exuE3u
% 2.99/1.52 => ( exu__Cong
% 2.99/1.52 => ( emptyset__Cong
% 2.99/1.52 => ( setadjoin__Cong
% 2.99/1.52 => ( powerset__Cong
% 2.99/1.52 => ( setunion__Cong
% 2.99/1.52 => ( omega__Cong
% 2.99/1.52 => ( exuEu
% 2.99/1.52 => ( descr__Cong
% 2.99/1.52 => ( dsetconstr__Cong
% 2.99/1.52 => ( subsetI1
% 2.99/1.52 => ( eqimpsubset2
% 2.99/1.52 => ( eqimpsubset1
% 2.99/1.52 => ( subsetI2
% 2.99/1.52 => ( emptysetsubset
% 2.99/1.52 => ( subsetE
% 2.99/1.52 => ( subsetE2
% 2.99/1.52 => ( notsubsetI
% 2.99/1.52 => ( notequalI1
% 2.99/1.52 => ( notequalI2
% 2.99/1.52 => ( subsetRefl
% 2.99/1.52 => ( subsetTrans
% 2.99/1.52 => ( setadjoinSub
% 2.99/1.52 => ( setadjoinSub2
% 2.99/1.52 => ( subset2powerset
% 2.99/1.52 => ( setextsub
% 2.99/1.52 => ( subsetemptysetimpeq
% 2.99/1.52 => ( powersetI1
% 2.99/1.52 => ( powersetE1
% 2.99/1.52 => ( inPowerset
% 2.99/1.52 => ( powersetsubset
% 2.99/1.52 => ( sepInPowerset
% 2.99/1.52 => ( sepSubset
% 2.99/1.52 => ( binunionIL
% 2.99/1.52 => ( upairset2IR
% 2.99/1.52 => ( binunionIR
% 2.99/1.52 => ( binunionEcases
% 2.99/1.52 => ( binunionE
% 2.99/1.52 => ( binunionLsub
% 2.99/1.52 => ( binunionRsub
% 2.99/1.52 => ( binintersectI
% 2.99/1.52 => ( binintersectSubset5
% 2.99/1.52 => ( binintersectEL
% 2.99/1.52 => ( binintersectLsub
% 2.99/1.52 => ( binintersectSubset2
% 2.99/1.52 => ( binintersectSubset3
% 2.99/1.52 => ( binintersectER
% 2.99/1.52 => ( disjointsetsI1
% 2.99/1.52 => ( binintersectRsub
% 2.99/1.52 => ( binintersectSubset4
% 2.99/1.52 => ( binintersectSubset1
% 2.99/1.52 => ( bs114d
% 2.99/1.52 => ( setminusI
% 2.99/1.52 => ( setminusEL
% 2.99/1.52 => ( setminusER
% 2.99/1.52 => ( setminusSubset2
% 2.99/1.52 => ( setminusERneg
% 2.99/1.52 => ( setminusELneg
% 2.99/1.52 => ( setminusILneg
% 2.99/1.52 => ( setminusIRneg
% 2.99/1.52 => ( setminusLsub
% 2.99/1.52 => ( setminusSubset1
% 2.99/1.52 => ( symdiffE
% 2.99/1.52 => ( symdiffI1
% 2.99/1.52 => ( symdiffI2
% 2.99/1.52 => ( symdiffIneg1
% 2.99/1.52 => ( symdiffIneg2
% 2.99/1.52 => ( secondinupair
% 2.99/1.52 => ( setukpairIL
% 2.99/1.52 => ( setukpairIR
% 2.99/1.52 => ( kpairiskpair
% 2.99/1.52 => ( kpairp
% 2.99/1.52 => ( singletonsubset
% 2.99/1.52 => ( singletoninpowerset
% 2.99/1.52 => ( singletoninpowunion
% 2.99/1.52 => ( upairset2E
% 2.99/1.52 => ( upairsubunion
% 2.99/1.52 => ( upairinpowunion
% 2.99/1.52 => ( ubforcartprodlem1
% 2.99/1.52 => ( ubforcartprodlem2
% 2.99/1.52 => ( ubforcartprodlem3
% 2.99/1.52 => ( cartprodpairin
% 2.99/1.52 => ( cartprodmempair1
% 2.99/1.52 => ( cartprodmempair
% 2.99/1.52 => ( setunionE2
% 2.99/1.52 => ( setunionsingleton1
% 2.99/1.52 => ( setunionsingleton2
% 2.99/1.52 => ( setunionsingleton
% 2.99/1.52 => ( singletonprop
% 2.99/1.52 => ( ex1E1
% 2.99/1.52 => ( ex1I
% 2.99/1.52 => ( ex1I2
% 2.99/1.52 => ( singletonsuniq
% 2.99/1.52 => ( setukpairinjL1
% 2.99/1.52 => ( kfstsingleton
% 2.99/1.52 => ( theprop
% 2.99/1.52 => ( kfstpairEq
% 2.99/1.52 => ( cartprodfstin
% 2.99/1.52 => ( setukpairinjL2
% 2.99/1.52 => ( setukpairinjL
% 2.99/1.52 => ( setukpairinjR11
% 2.99/1.52 => ( setukpairinjR12
% 2.99/1.52 => ( setukpairinjR1
% 2.99/1.52 => ( upairequniteq
% 2.99/1.52 => ( setukpairinjR2
% 2.99/1.52 => ( setukpairinjR
% 2.99/1.52 => ( ksndsingleton
% 2.99/1.52 => ( ksndpairEq
% 2.99/1.52 => ( kpairsurjEq
% 2.99/1.52 => ( cartprodsndin
% 2.99/1.52 => ( cartprodpairmemEL
% 2.99/1.52 => ( cartprodpairmemER
% 2.99/1.52 => ( cartprodmempaircEq
% 2.99/1.52 => ( cartprodfstpairEq
% 2.99/1.52 => ( cartprodsndpairEq
% 2.99/1.52 => ( cartprodpairsurjEq
% 2.99/1.52 => ( dpsetconstrI
% 2.99/1.52 => ( dpsetconstrSub
% 2.99/1.52 => ( setOfPairsIsBReln
% 2.99/1.52 => ( dpsetconstrERa
% 2.99/1.52 => ( dpsetconstrEL1
% 2.99/1.52 => ( dpsetconstrEL2
% 2.99/1.52 => ( dpsetconstrER
% 2.99/1.52 => ( funcImageSingleton
% 2.99/1.52 => ( apProp
% 2.99/1.52 => ( app
% 2.99/1.52 => ( infuncsetfunc
% 2.99/1.52 => ( ap2p
% 2.99/1.52 => ( funcinfuncset
% 2.99/1.52 => ( lamProp
% 2.99/1.52 => ( lamp
% 2.99/1.52 => ( lam2p
% 2.99/1.52 => ( brelnall1
% 2.99/1.52 => ( brelnall2
% 2.99/1.52 => ( ex1E2
% 2.99/1.52 => ( funcGraphProp1
% 2.99/1.52 => ( funcGraphProp3
% 2.99/1.52 => ( funcGraphProp2
% 2.99/1.52 => ( funcextLem
% 2.99/1.52 => ( funcGraphProp4
% 2.99/1.52 => ( subbreln
% 2.99/1.52 => ( eqbreln
% 2.99/1.52 => ( funcext
% 2.99/1.52 => ( funcext2
% 2.99/1.52 => ( ap2apEq1
% 2.99/1.52 => ( ap2apEq2
% 2.99/1.52 => ( beta1
% 2.99/1.52 => ( eta1
% 2.99/1.52 => ( lam2lamEq
% 2.99/1.52 => ( beta2
% 2.99/1.52 => ( eta2
% 2.99/1.52 => ( iffalseProp1
% 2.99/1.52 => ( iffalseProp2
% 2.99/1.52 => ( iftrueProp1
% 2.99/1.52 => ( iftrueProp2
% 2.99/1.52 => ( ifSingleton
% 2.99/1.52 => ( ifp
% 2.99/1.52 => ( theeq
% 2.99/1.52 => ( iftrue
% 2.99/1.52 => ( iffalse
% 2.99/1.52 => ( iftrueorfalse
% 2.99/1.52 => ( binintersectT_lem
% 2.99/1.52 => ( binunionT_lem
% 2.99/1.52 => ( powersetT_lem
% 2.99/1.52 => ( setminusT_lem
% 2.99/1.52 => ( complementT_lem
% 2.99/1.52 => ( setextT
% 2.99/1.52 => ( subsetTI
% 2.99/1.52 => ( powersetTI1
% 2.99/1.52 => ( powersetTE1
% 2.99/1.52 => ( complementTI1
% 2.99/1.52 => ( complementTE1
% 2.99/1.52 => ( binintersectTELcontra
% 2.99/1.52 => ( binintersectTERcontra
% 2.99/1.52 => ( contrasubsetT
% 2.99/1.52 => ( contrasubsetT1
% 2.99/1.52 => ( contrasubsetT2
% 2.99/1.52 => ( contrasubsetT3
% 2.99/1.52 => ( doubleComplementI1
% 2.99/1.52 => ( doubleComplementE1
% 2.99/1.52 => ( doubleComplementSub1
% 2.99/1.52 => ( doubleComplementSub2
% 2.99/1.52 => ( doubleComplementEq
% 2.99/1.52 => ( complementTnotintersectT
% 2.99/1.52 => ( complementImpComplementIntersect
% 2.99/1.52 => ( complementSubsetComplementIntersect
% 2.99/1.52 => ( complementInPowersetComplementIntersect
% 2.99/1.52 => ( contraSubsetComplement
% 2.99/1.52 => ( complementTcontraSubset
% 2.99/1.52 => ( binunionTILcontra
% 2.99/1.52 => ( binunionTIRcontra
% 2.99/1.52 => ( inIntersectImpInUnion
% 2.99/1.52 => ( inIntersectImpInUnion2
% 2.99/1.52 => ( inIntersectImpInIntersectUnions
% 2.99/1.52 => ( intersectInPowersetIntersectUnions
% 2.99/1.52 => ( inComplementUnionImpNotIn1
% 2.99/1.52 => ( inComplementUnionImpInComplement1
% 2.99/1.52 => ( binunionTE
% 2.99/1.52 => ( binunionTEcontra
% 2.99/1.52 => ( demorgan2a1
% 2.99/1.52 => ( complementUnionInPowersetComplement
% 2.99/1.52 => ( demorgan2a2
% 2.99/1.52 => ( demorgan1a
% 2.99/1.52 => ( demorgan1b
% 2.99/1.52 => ( demorgan1
% 2.99/1.52 => ( demorgan2a
% 2.99/1.52 => ( demorgan2b2
% 2.99/1.52 => ( demorgan2b
% 2.99/1.52 => ( demorgan2
% 2.99/1.52 => ( woz13rule0
% 2.99/1.52 => ( woz13rule1
% 2.99/1.52 => ( woz13rule2
% 2.99/1.52 => ( woz13rule3
% 2.99/1.52 => ( woz13rule4
% 2.99/1.52 => ( woz1_1
% 2.99/1.52 => ( woz1_2
% 2.99/1.52 => ( woz1_3
% 2.99/1.52 => ( woz1_4
% 2.99/1.52 => ( woz1_5
% 2.99/1.52 => ( breln1all2
% 2.99/1.52 => ( breln1SetBreln1
% 2.99/1.52 => ( choice2fnsingleton
% 2.99/1.52 => ( setOfPairsIsBReln1
% 2.99/1.52 => ( breln1all1
% 2.99/1.52 => ( subbreln1
% 2.99/1.52 => ( eqbreln1
% 2.99/1.52 => ( breln1invprop
% 2.99/1.52 => ( breln1invI
% 2.99/1.52 => ( breln1invE
% 2.99/1.52 => ( breln1compprop
% 2.99/1.52 => ( breln1compI
% 2.99/1.52 => ( breln1compE
% 2.99/1.52 => ( breln1compEex
% 2.99/1.52 => ( breln1unionprop
% 2.99/1.52 => ( breln1unionIL
% 2.99/1.52 => ( breln1unionIR
% 2.99/1.52 => ( breln1unionI
% 2.99/1.52 => ( breln1unionE
% 2.99/1.52 => ( breln1unionEcases
% 2.99/1.52 => ( breln1unionCommutes
% 2.99/1.52 => ( woz2Ex
% 2.99/1.52 => ( woz2W
% 2.99/1.52 => ( woz2A
% 2.99/1.52 => ( woz2B
% 2.99/1.52 => ( image1Ex
% 2.99/1.52 => ( image1Ex1
% 2.99/1.52 => ( image1Equiv
% 2.99/1.52 => ( image1E
% 2.99/1.52 => ( image1I
% 2.99/1.52 => ( injFuncInInjFuncSet
% 2.99/1.52 => ( injFuncSetFuncIn
% 2.99/1.52 => ( injFuncSetFuncInj
% 2.99/1.52 => ( surjFuncSetFuncIn
% 2.99/1.52 => ( surjFuncSetFuncSurj
% 2.99/1.52 => ( leftInvIsSurj
% 2.99/1.52 => ( surjCantorThm
% 2.99/1.52 => ( foundation2
% 2.99/1.52 => ( notinself
% 2.99/1.52 => ( notinself2
% 2.99/1.52 => ! [Xx: $i] :
% 2.99/1.52 ( ( in @ Xx @ omega )
% 2.99/1.52 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.99/1.52 ------------------------
% 2.99/1.55 % (28651)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/18Mi)
% 2.99/1.55 % (28647)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_28453 for (2999ds/27Mi)
% 2.99/1.55 % (28650)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_28453 for (2999ds/275Mi)
% 2.99/1.55 % (28649)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_28453 for (2999ds/2Mi)
% 2.99/1.55 % (28649)Instruction limit reached!
% 2.99/1.55 % (28649)------------------------------
% 2.99/1.55 % (28649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.55 % (28649)Termination reason: Unknown
% 2.99/1.55 % (28649)Termination phase: shuffling
% 2.99/1.55
% 2.99/1.55 % (28649)Memory used [KB]: 1663
% 2.99/1.55 % (28649)Time elapsed: 0.002 s
% 2.99/1.55 % (28649)Instructions burned: 2 (million)
% 2.99/1.55 % (28649)------------------------------
% 2.99/1.55 % (28649)------------------------------
% 2.99/1.55 % (28646)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_28453 for (2999ds/4Mi)
% 2.99/1.56 % (28651)Instruction limit reached!
% 2.99/1.56 % (28651)------------------------------
% 2.99/1.56 % (28651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.56 % (28651)Termination reason: Unknown
% 2.99/1.56 % (28651)Termination phase: shuffling
% 2.99/1.56
% 2.99/1.56 % (28651)Memory used [KB]: 1918
% 2.99/1.56 % (28651)Time elapsed: 0.008 s
% 2.99/1.56 % (28651)Instructions burned: 19 (million)
% 2.99/1.56 % (28651)------------------------------
% 2.99/1.56 % (28651)------------------------------
% 2.99/1.56 % (28646)Instruction limit reached!
% 2.99/1.56 % (28646)------------------------------
% 2.99/1.56 % (28646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.56 % (28646)Termination reason: Unknown
% 2.99/1.56 % (28646)Termination phase: shuffling
% 2.99/1.56
% 2.99/1.56 % (28646)Memory used [KB]: 1663
% 2.99/1.56 % (28646)Time elapsed: 0.003 s
% 2.99/1.56 % (28646)Instructions burned: 4 (million)
% 2.99/1.56 % (28646)------------------------------
% 2.99/1.56 % (28646)------------------------------
% 2.99/1.56 % (28647)Instruction limit reached!
% 2.99/1.56 % (28647)------------------------------
% 2.99/1.56 % (28647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.56 % (28647)Termination reason: Unknown
% 2.99/1.56 % (28647)Termination phase: shuffling
% 2.99/1.56
% 2.99/1.56 % (28647)Memory used [KB]: 2174
% 2.99/1.56 % (28647)Time elapsed: 0.010 s
% 2.99/1.56 % (28647)Instructions burned: 27 (million)
% 2.99/1.56 % (28647)------------------------------
% 2.99/1.56 % (28647)------------------------------
% 2.99/1.56 % (28645)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/183Mi)
% 2.99/1.56 % (28648)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/2Mi)
% 2.99/1.57 % (28648)Instruction limit reached!
% 2.99/1.57 % (28648)------------------------------
% 2.99/1.57 % (28648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.57 % (28648)Termination reason: Unknown
% 2.99/1.57 % (28648)Termination phase: shuffling
% 2.99/1.57
% 2.99/1.57 % (28648)Memory used [KB]: 1663
% 2.99/1.57 % (28648)Time elapsed: 0.003 s
% 2.99/1.57 % (28648)Instructions burned: 2 (million)
% 2.99/1.57 % (28648)------------------------------
% 2.99/1.57 % (28648)------------------------------
% 2.99/1.57 % (28654)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_28453 for (2999ds/15Mi)
% 2.99/1.57 % (28652)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/3Mi)
% 2.99/1.57 % (28652)Instruction limit reached!
% 2.99/1.57 % (28652)------------------------------
% 2.99/1.57 % (28652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.57 % (28652)Termination reason: Unknown
% 2.99/1.57 % (28652)Termination phase: shuffling
% 2.99/1.57
% 2.99/1.57 % (28652)Memory used [KB]: 1663
% 2.99/1.57 % (28652)Time elapsed: 0.004 s
% 2.99/1.57 % (28652)Instructions burned: 5 (million)
% 2.99/1.57 % (28652)------------------------------
% 2.99/1.57 % (28652)------------------------------
% 2.99/1.57 % (28654)Instruction limit reached!
% 2.99/1.57 % (28654)------------------------------
% 2.99/1.57 % (28654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.57 % (28654)Termination reason: Unknown
% 2.99/1.57 % (28654)Termination phase: shuffling
% 2.99/1.57
% 2.99/1.57 % (28654)Memory used [KB]: 1918
% 2.99/1.57 % (28654)Time elapsed: 0.007 s
% 2.99/1.57 % (28654)Instructions burned: 17 (million)
% 2.99/1.57 % (28654)------------------------------
% 2.99/1.57 % (28654)------------------------------
% 2.99/1.57 % (28655)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_28453 for (2999ds/3Mi)
% 2.99/1.57 % (28653)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_28453 for (2999ds/37Mi)
% 2.99/1.57 % (28655)Instruction limit reached!
% 2.99/1.57 % (28655)------------------------------
% 2.99/1.57 % (28655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.57 % (28655)Termination reason: Unknown
% 2.99/1.57 % (28655)Termination phase: shuffling
% 2.99/1.57
% 2.99/1.57 % (28655)Memory used [KB]: 1663
% 2.99/1.57 % (28655)Time elapsed: 0.003 s
% 2.99/1.57 % (28655)Instructions burned: 6 (million)
% 2.99/1.57 % (28655)------------------------------
% 2.99/1.57 % (28655)------------------------------
% 2.99/1.58 % (28658)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/16Mi)
% 2.99/1.58 % (28656)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_28453 for (2999ds/1041Mi)
% 2.99/1.59 % (28657)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/7Mi)
% 2.99/1.59 % (28658)Instruction limit reached!
% 2.99/1.59 % (28658)------------------------------
% 2.99/1.59 % (28658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.59 % (28658)Termination reason: Unknown
% 2.99/1.59 % (28658)Termination phase: shuffling
% 2.99/1.59
% 2.99/1.59 % (28658)Memory used [KB]: 1918
% 2.99/1.59 % (28658)Time elapsed: 0.006 s
% 2.99/1.59 % (28658)Instructions burned: 16 (million)
% 2.99/1.59 % (28658)------------------------------
% 2.99/1.59 % (28658)------------------------------
% 2.99/1.59 % (28659)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_28453 for (2999ds/3Mi)
% 2.99/1.59 % (28659)Instruction limit reached!
% 2.99/1.59 % (28659)------------------------------
% 2.99/1.59 % (28659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.59 % (28659)Termination reason: Unknown
% 2.99/1.59 % (28659)Termination phase: shuffling
% 2.99/1.59
% 2.99/1.59 % (28659)Memory used [KB]: 1663
% 2.99/1.59 % (28659)Time elapsed: 0.003 s
% 2.99/1.59 % (28659)Instructions burned: 5 (million)
% 2.99/1.59 % (28659)------------------------------
% 2.99/1.59 % (28659)------------------------------
% 2.99/1.59 % (28657)Instruction limit reached!
% 2.99/1.59 % (28657)------------------------------
% 2.99/1.59 % (28657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.59 % (28657)Termination reason: Unknown
% 2.99/1.59 % (28657)Termination phase: shuffling
% 2.99/1.59
% 2.99/1.59 % (28657)Memory used [KB]: 1663
% 2.99/1.59 % (28657)Time elapsed: 0.005 s
% 2.99/1.59 % (28657)Instructions burned: 7 (million)
% 2.99/1.59 % (28657)------------------------------
% 2.99/1.59 % (28657)------------------------------
% 2.99/1.59 % (28653)Instruction limit reached!
% 2.99/1.59 % (28653)------------------------------
% 2.99/1.59 % (28653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.59 % (28653)Termination reason: Unknown
% 2.99/1.59 % (28653)Termination phase: shuffling
% 2.99/1.59
% 2.99/1.59 % (28653)Memory used [KB]: 2302
% 2.99/1.59 % (28653)Time elapsed: 0.020 s
% 2.99/1.59 % (28653)Instructions burned: 37 (million)
% 2.99/1.59 % (28653)------------------------------
% 2.99/1.59 % (28653)------------------------------
% 3.03/1.60 % (28660)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_28453 for (2999ds/3Mi)
% 3.03/1.60 % (28660)Instruction limit reached!
% 3.03/1.60 % (28660)------------------------------
% 3.03/1.60 % (28660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.60 % (28660)Termination reason: Unknown
% 3.03/1.60 % (28660)Termination phase: shuffling
% 3.03/1.60
% 3.03/1.60 % (28660)Memory used [KB]: 1663
% 3.03/1.60 % (28660)Time elapsed: 0.003 s
% 3.03/1.60 % (28660)Instructions burned: 4 (million)
% 3.03/1.60 % (28660)------------------------------
% 3.03/1.60 % (28660)------------------------------
% 3.03/1.60 % (28662)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/3Mi)
% 3.03/1.60 % (28663)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/4Mi)
% 3.03/1.61 % (28662)Instruction limit reached!
% 3.03/1.61 % (28662)------------------------------
% 3.03/1.61 % (28662)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.61 % (28662)Termination reason: Unknown
% 3.03/1.61 % (28662)Termination phase: shuffling
% 3.03/1.61
% 3.03/1.61 % (28662)Memory used [KB]: 1663
% 3.03/1.61 % (28662)Time elapsed: 0.004 s
% 3.03/1.61 % (28662)Instructions burned: 5 (million)
% 3.03/1.61 % (28662)------------------------------
% 3.03/1.61 % (28662)------------------------------
% 3.03/1.61 % (28661)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/7Mi)
% 3.03/1.61 % (28663)Instruction limit reached!
% 3.03/1.61 % (28663)------------------------------
% 3.03/1.61 % (28663)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.61 % (28663)Termination reason: Unknown
% 3.03/1.61 % (28663)Termination phase: shuffling
% 3.03/1.61
% 3.03/1.61 % (28663)Memory used [KB]: 1663
% 3.03/1.61 % (28663)Time elapsed: 0.004 s
% 3.03/1.61 % (28663)Instructions burned: 5 (million)
% 3.03/1.61 % (28663)------------------------------
% 3.03/1.61 % (28663)------------------------------
% 3.03/1.61 % (28664)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/18Mi)
% 3.03/1.61 % (28661)Instruction limit reached!
% 3.03/1.61 % (28661)------------------------------
% 3.03/1.61 % (28661)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.61 % (28661)Termination reason: Unknown
% 3.03/1.61 % (28661)Termination phase: shuffling
% 3.03/1.61
% 3.03/1.61 % (28661)Memory used [KB]: 1791
% 3.03/1.61 % (28661)Time elapsed: 0.005 s
% 3.03/1.61 % (28661)Instructions burned: 9 (million)
% 3.03/1.61 % (28661)------------------------------
% 3.03/1.61 % (28661)------------------------------
% 3.03/1.61 % (28645)Instruction limit reached!
% 3.03/1.61 % (28645)------------------------------
% 3.03/1.61 % (28645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.61 % (28645)Termination reason: Unknown
% 3.03/1.61 % (28645)Termination phase: Property scanning
% 3.03/1.61
% 3.03/1.61 % (28645)Memory used [KB]: 3070
% 3.03/1.61 % (28645)Time elapsed: 0.049 s
% 3.03/1.61 % (28645)Instructions burned: 184 (million)
% 3.03/1.61 % (28645)------------------------------
% 3.03/1.61 % (28645)------------------------------
% 3.03/1.61 % (28664)Instruction limit reached!
% 3.03/1.61 % (28664)------------------------------
% 3.03/1.61 % (28664)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.61 % (28664)Termination reason: Unknown
% 3.03/1.61 % (28664)Termination phase: shuffling
% 3.03/1.61
% 3.03/1.61 % (28664)Memory used [KB]: 1918
% 3.03/1.61 % (28664)Time elapsed: 0.007 s
% 3.03/1.61 % (28664)Instructions burned: 21 (million)
% 3.03/1.61 % (28664)------------------------------
% 3.03/1.61 % (28664)------------------------------
% 3.03/1.62 % (28665)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/710Mi)
% 3.03/1.62 % (28668)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_28453 for (2999ds/21Mi)
% 3.03/1.62 % (28667)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_28453 for (2999ds/902Mi)
% 3.03/1.62 % (28666)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/6Mi)
% 3.03/1.62 % (28669)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/5Mi)
% 3.03/1.62 % (28666)Instruction limit reached!
% 3.03/1.62 % (28666)------------------------------
% 3.03/1.62 % (28666)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.62 % (28666)Termination reason: Unknown
% 3.03/1.62 % (28666)Termination phase: shuffling
% 3.03/1.62
% 3.03/1.62 % (28666)Memory used [KB]: 1663
% 3.03/1.62 % (28666)Time elapsed: 0.005 s
% 3.03/1.62 % (28666)Instructions burned: 6 (million)
% 3.03/1.62 % (28666)------------------------------
% 3.03/1.62 % (28666)------------------------------
% 3.03/1.62 % (28669)Instruction limit reached!
% 3.03/1.62 % (28669)------------------------------
% 3.03/1.62 % (28669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.62 % (28669)Termination reason: Unknown
% 3.03/1.62 % (28669)Termination phase: shuffling
% 3.03/1.62
% 3.03/1.62 % (28669)Memory used [KB]: 1663
% 3.03/1.62 % (28669)Time elapsed: 0.003 s
% 3.03/1.62 % (28669)Instructions burned: 6 (million)
% 3.03/1.62 % (28669)------------------------------
% 3.03/1.62 % (28669)------------------------------
% 3.03/1.62 % (28668)Instruction limit reached!
% 3.03/1.62 % (28668)------------------------------
% 3.03/1.62 % (28668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.62 % (28668)Termination reason: Unknown
% 3.03/1.62 % (28668)Termination phase: shuffling
% 3.03/1.62
% 3.03/1.62 % (28668)Memory used [KB]: 1918
% 3.03/1.62 % (28668)Time elapsed: 0.007 s
% 3.03/1.62 % (28668)Instructions burned: 21 (million)
% 3.03/1.62 % (28668)------------------------------
% 3.03/1.62 % (28668)------------------------------
% 3.03/1.63 % (28650)Instruction limit reached!
% 3.03/1.63 % (28650)------------------------------
% 3.03/1.63 % (28650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.63 % (28650)Termination reason: Unknown
% 3.03/1.63 % (28650)Termination phase: Property scanning
% 3.03/1.63
% 3.03/1.63 % (28650)Memory used [KB]: 6012
% 3.03/1.63 % (28650)Time elapsed: 0.081 s
% 3.03/1.63 % (28650)Instructions burned: 276 (million)
% 3.03/1.63 % (28650)------------------------------
% 3.03/1.63 % (28650)------------------------------
% 3.03/1.63 % (28671)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_28453 for (2999ds/377Mi)
% 3.03/1.63 % (28672)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_28453 for (2999ds/779Mi)
% 3.03/1.64 % (28670)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_28453 for (2999ds/6Mi)
% 3.03/1.64 % (28673)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_28453 for (2998ds/19Mi)
% 3.03/1.64 % (28670)Instruction limit reached!
% 3.03/1.64 % (28670)------------------------------
% 3.03/1.64 % (28670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.64 % (28670)Termination reason: Unknown
% 3.03/1.64 % (28670)Termination phase: shuffling
% 3.03/1.64
% 3.03/1.64 % (28670)Memory used [KB]: 1663
% 3.03/1.64 % (28670)Time elapsed: 0.005 s
% 3.03/1.64 % (28670)Instructions burned: 8 (million)
% 3.03/1.64 % (28670)------------------------------
% 3.03/1.64 % (28670)------------------------------
% 3.03/1.65 % (28673)Instruction limit reached!
% 3.03/1.65 % (28673)------------------------------
% 3.03/1.65 % (28673)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.65 % (28673)Termination reason: Unknown
% 3.03/1.65 % (28673)Termination phase: shuffling
% 3.03/1.65
% 3.03/1.65 % (28673)Memory used [KB]: 1918
% 3.03/1.65 % (28673)Time elapsed: 0.007 s
% 3.03/1.65 % (28673)Instructions burned: 22 (million)
% 3.03/1.65 % (28673)------------------------------
% 3.03/1.65 % (28673)------------------------------
% 3.03/1.66 % (28675)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_28453 for (2998ds/17Mi)
% 3.03/1.66 % (28675)Instruction limit reached!
% 3.03/1.66 % (28675)------------------------------
% 3.03/1.66 % (28675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.66 % (28675)Termination reason: Unknown
% 3.03/1.66 % (28675)Termination phase: shuffling
% 3.03/1.66
% 3.03/1.66 % (28675)Memory used [KB]: 1918
% 3.03/1.66 % (28675)Time elapsed: 0.007 s
% 3.03/1.66 % (28675)Instructions burned: 19 (million)
% 3.03/1.66 % (28675)------------------------------
% 3.03/1.66 % (28675)------------------------------
% 3.03/1.66 % (28674)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_28453 for (2998ds/879Mi)
% 3.03/1.67 % (28676)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_28453 for (2998ds/3Mi)
% 3.03/1.67 % (28676)Instruction limit reached!
% 3.03/1.67 % (28676)------------------------------
% 3.03/1.67 % (28676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.67 % (28676)Termination reason: Unknown
% 3.03/1.67 % (28676)Termination phase: shuffling
% 3.03/1.67
% 3.03/1.67 % (28676)Memory used [KB]: 1663
% 3.03/1.67 % (28676)Time elapsed: 0.003 s
% 3.03/1.67 % (28676)Instructions burned: 5 (million)
% 3.03/1.67 % (28676)------------------------------
% 3.03/1.67 % (28676)------------------------------
% 3.03/1.68 % (28677)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_28453 for (2998ds/30Mi)
% 3.03/1.69 % (28677)Instruction limit reached!
% 3.03/1.69 % (28677)------------------------------
% 3.03/1.69 % (28677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.69 % (28677)Termination reason: Unknown
% 3.03/1.69 % (28677)Termination phase: shuffling
% 3.03/1.69
% 3.03/1.69 % (28677)Memory used [KB]: 2302
% 3.03/1.69 % (28677)Time elapsed: 0.010 s
% 3.03/1.69 % (28677)Instructions burned: 31 (million)
% 3.03/1.69 % (28677)------------------------------
% 3.03/1.69 % (28677)------------------------------
% 3.03/1.70 % (28678)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_28453 for (2998ds/127Mi)
% 3.03/1.73 % (28671)Instruction limit reached!
% 3.03/1.73 % (28671)------------------------------
% 3.03/1.73 % (28671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.73 % (28671)Termination reason: Unknown
% 3.03/1.73 % (28671)Termination phase: Saturation
% 3.03/1.73
% 3.03/1.73 % (28671)Memory used [KB]: 9466
% 3.03/1.73 % (28671)Time elapsed: 0.101 s
% 3.03/1.73 % (28671)Instructions burned: 379 (million)
% 3.03/1.73 % (28671)------------------------------
% 3.03/1.73 % (28671)------------------------------
% 3.03/1.73 % (28678)Instruction limit reached!
% 3.03/1.73 % (28678)------------------------------
% 3.03/1.73 % (28678)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.73 % (28678)Termination reason: Unknown
% 3.03/1.73 % (28678)Termination phase: Property scanning
% 3.03/1.73
% 3.03/1.73 % (28678)Memory used [KB]: 2942
% 3.03/1.73 % (28678)Time elapsed: 0.035 s
% 3.03/1.73 % (28678)Instructions burned: 129 (million)
% 3.03/1.73 % (28678)------------------------------
% 3.03/1.73 % (28678)------------------------------
% 3.03/1.75 % (28683)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_28453 for (2997ds/3Mi)
% 3.03/1.75 % (28683)Instruction limit reached!
% 3.03/1.75 % (28683)------------------------------
% 3.03/1.75 % (28683)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.75 % (28683)Termination reason: Unknown
% 3.03/1.75 % (28683)Termination phase: shuffling
% 3.03/1.75
% 3.03/1.75 % (28683)Memory used [KB]: 1663
% 3.03/1.75 % (28683)Time elapsed: 0.002 s
% 3.03/1.75 % (28683)Instructions burned: 3 (million)
% 3.03/1.75 % (28683)------------------------------
% 3.03/1.75 % (28683)------------------------------
% 3.03/1.75 % (28681)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_28453 for (2997ds/100Mi)
% 3.03/1.76 % (28686)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_28453 for (2997ds/20Mi)
% 3.03/1.76 % (28686)Instruction limit reached!
% 3.03/1.76 % (28686)------------------------------
% 3.03/1.76 % (28686)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.76 % (28686)Termination reason: Unknown
% 3.03/1.76 % (28686)Termination phase: shuffling
% 3.03/1.76
% 3.03/1.76 % (28686)Memory used [KB]: 2046
% 3.03/1.76 % (28686)Time elapsed: 0.008 s
% 3.03/1.76 % (28686)Instructions burned: 21 (million)
% 3.03/1.76 % (28686)------------------------------
% 3.03/1.76 % (28686)------------------------------
% 3.03/1.77 % (28687)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_28453 for (2997ds/86Mi)
% 3.03/1.79 % (28681)Instruction limit reached!
% 3.03/1.79 % (28681)------------------------------
% 3.03/1.79 % (28681)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.79 % (28681)Termination reason: Unknown
% 3.03/1.79 % (28681)Termination phase: Preprocessing 1
% 3.03/1.79
% 3.03/1.79 % (28681)Memory used [KB]: 2814
% 3.03/1.79 % (28681)Time elapsed: 0.041 s
% 3.03/1.79 % (28681)Instructions burned: 100 (million)
% 3.03/1.79 % (28681)------------------------------
% 3.03/1.79 % (28681)------------------------------
% 3.03/1.79 % (28687)Instruction limit reached!
% 3.03/1.79 % (28687)------------------------------
% 3.03/1.79 % (28687)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.79 % (28687)Termination reason: Unknown
% 3.03/1.79 % (28687)Termination phase: Property scanning
% 3.03/1.79
% 3.03/1.79 % (28687)Memory used [KB]: 2814
% 3.03/1.79 % (28687)Time elapsed: 0.024 s
% 3.03/1.79 % (28687)Instructions burned: 89 (million)
% 3.03/1.79 % (28687)------------------------------
% 3.03/1.79 % (28687)------------------------------
% 3.03/1.80 % (28688)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_28453 for (2997ds/107Mi)
% 3.03/1.80 % (28689)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_28453 for (2997ds/39Mi)
% 3.03/1.82 % (28689)Instruction limit reached!
% 3.03/1.82 % (28689)------------------------------
% 3.03/1.82 % (28689)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.82 % (28689)Termination reason: Unknown
% 3.03/1.82 % (28689)Termination phase: shuffling
% 3.03/1.82
% 3.03/1.82 % (28689)Memory used [KB]: 2430
% 3.03/1.82 % (28689)Time elapsed: 0.012 s
% 3.03/1.82 % (28689)Instructions burned: 40 (million)
% 3.03/1.82 % (28689)------------------------------
% 3.03/1.82 % (28689)------------------------------
% 3.03/1.82 % (28692)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_28453 for (2997ds/448Mi)
% 3.03/1.83 % (28688)Instruction limit reached!
% 3.03/1.83 % (28688)------------------------------
% 3.03/1.83 % (28688)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.83 % (28688)Termination reason: Unknown
% 3.03/1.83 % (28688)Termination phase: SInE selection
% 3.03/1.83
% 3.03/1.83 % (28688)Memory used [KB]: 2814
% 3.03/1.83 % (28688)Time elapsed: 0.029 s
% 3.03/1.83 % (28688)Instructions burned: 107 (million)
% 3.03/1.83 % (28688)------------------------------
% 3.03/1.83 % (28688)------------------------------
% 3.03/1.83 % (28672)Instruction limit reached!
% 3.03/1.83 % (28672)------------------------------
% 3.03/1.83 % (28672)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.83 % (28672)Termination reason: Unknown
% 3.03/1.83 % (28672)Termination phase: Saturation
% 3.03/1.83
% 3.03/1.83 % (28672)Memory used [KB]: 7931
% 3.03/1.83 % (28672)Time elapsed: 0.201 s
% 3.03/1.83 % (28672)Instructions burned: 781 (million)
% 3.03/1.83 % (28672)------------------------------
% 3.03/1.83 % (28672)------------------------------
% 3.03/1.84 % (28693)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_28453 for (2997ds/46Mi)
% 3.03/1.85 % (28701)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_28453 for (2996ds/98Mi)
% 3.03/1.85 % (28693)Instruction limit reached!
% 3.03/1.85 % (28693)------------------------------
% 3.03/1.85 % (28693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.85 % (28693)Termination reason: Unknown
% 3.03/1.85 % (28693)Termination phase: shuffling
% 3.03/1.85
% 3.03/1.85 % (28693)Memory used [KB]: 2558
% 3.03/1.85 % (28693)Time elapsed: 0.016 s
% 3.03/1.85 % (28693)Instructions burned: 46 (million)
% 3.03/1.85 % (28693)------------------------------
% 3.03/1.85 % (28693)------------------------------
% 3.03/1.87 % (28665)Instruction limit reached!
% 3.03/1.87 % (28665)------------------------------
% 3.03/1.87 % (28665)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.87 % (28665)Termination reason: Unknown
% 3.03/1.87 % (28665)Termination phase: Saturation
% 3.03/1.87
% 3.03/1.87 % (28665)Memory used [KB]: 11897
% 3.03/1.87 % (28665)Time elapsed: 0.251 s
% 3.03/1.87 % (28665)Instructions burned: 710 (million)
% 3.03/1.87 % (28665)------------------------------
% 3.03/1.87 % (28665)------------------------------
% 3.03/1.87 % (28705)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_28453 for (2996ds/507Mi)
% 3.03/1.88 % (28709)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_28453 for (2996ds/149Mi)
% 3.03/1.89 % (28701)Instruction limit reached!
% 3.03/1.89 % (28701)------------------------------
% 3.03/1.89 % (28701)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.89 % (28701)Termination reason: Unknown
% 3.03/1.89 % (28701)Termination phase: Property scanning
% 3.03/1.89
% 3.03/1.89 % (28701)Memory used [KB]: 2814
% 3.03/1.89 % (28701)Time elapsed: 0.040 s
% 3.03/1.89 % (28701)Instructions burned: 101 (million)
% 3.03/1.89 % (28701)------------------------------
% 3.03/1.89 % (28701)------------------------------
% 3.03/1.90 % (28716)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_28453 for (2996ds/39Mi)
% 3.03/1.91 % (28716)Instruction limit reached!
% 3.03/1.91 % (28716)------------------------------
% 3.03/1.91 % (28716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.91 % (28716)Termination reason: Unknown
% 3.03/1.91 % (28716)Termination phase: shuffling
% 3.03/1.91
% 3.03/1.91 % (28716)Memory used [KB]: 2430
% 3.03/1.91 % (28716)Time elapsed: 0.013 s
% 3.03/1.91 % (28716)Instructions burned: 40 (million)
% 3.03/1.91 % (28716)------------------------------
% 3.03/1.91 % (28716)------------------------------
% 3.03/1.92 % (28656)Instruction limit reached!
% 3.03/1.92 % (28656)------------------------------
% 3.03/1.92 % (28656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.92 % (28656)Termination reason: Unknown
% 3.03/1.92 % (28656)Termination phase: Saturation
% 3.03/1.92
% 3.03/1.92 % (28656)Memory used [KB]: 18549
% 3.03/1.92 % (28656)Time elapsed: 0.342 s
% 3.03/1.92 % (28656)Instructions burned: 1043 (million)
% 3.03/1.92 % (28656)------------------------------
% 3.03/1.92 % (28656)------------------------------
% 3.03/1.93 % (28722)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_28453 for (2996ds/6Mi)
% 3.03/1.93 % (28722)Instruction limit reached!
% 3.03/1.93 % (28722)------------------------------
% 3.03/1.93 % (28722)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.93 % (28722)Termination reason: Unknown
% 3.03/1.93 % (28722)Termination phase: shuffling
% 3.03/1.93
% 3.03/1.93 % (28722)Memory used [KB]: 1663
% 3.03/1.93 % (28722)Time elapsed: 0.005 s
% 3.03/1.93 % (28722)Instructions burned: 7 (million)
% 3.03/1.93 % (28722)------------------------------
% 3.03/1.93 % (28722)------------------------------
% 3.03/1.94 % (28725)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_28453 for (2996ds/1087Mi)
% 3.03/1.94 % (28709)Instruction limit reached!
% 3.03/1.94 % (28709)------------------------------
% 3.03/1.94 % (28709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.94 % (28709)Termination reason: Unknown
% 3.03/1.94 % (28709)Termination phase: Function definition elimination
% 3.03/1.94
% 3.03/1.94 % (28709)Memory used [KB]: 2942
% 3.03/1.94 % (28709)Time elapsed: 0.066 s
% 3.03/1.94 % (28709)Instructions burned: 151 (million)
% 3.03/1.94 % (28709)------------------------------
% 3.03/1.94 % (28709)------------------------------
% 3.03/1.94 % (28726)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_28453 for (2996ds/47Mi)
% 3.03/1.96 % (28730)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_28453 for (2995ds/44Mi)
% 3.03/1.97 % (28667)Instruction limit reached!
% 3.03/1.97 % (28667)------------------------------
% 3.03/1.97 % (28667)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.97 % (28667)Termination reason: Unknown
% 3.03/1.97 % (28667)Termination phase: Saturation
% 3.03/1.97
% 3.03/1.97 % (28667)Memory used [KB]: 13048
% 3.03/1.97 % (28667)Time elapsed: 0.349 s
% 3.03/1.97 % (28667)Instructions burned: 902 (million)
% 3.03/1.97 % (28667)------------------------------
% 3.03/1.97 % (28667)------------------------------
% 3.03/1.97 % (28726)Instruction limit reached!
% 3.03/1.97 % (28726)------------------------------
% 3.03/1.97 % (28726)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.97 % (28726)Termination reason: Unknown
% 3.03/1.97 % (28726)Termination phase: shuffling
% 3.03/1.97
% 3.03/1.97 % (28726)Memory used [KB]: 2558
% 3.03/1.97 % (28726)Time elapsed: 0.024 s
% 3.03/1.97 % (28726)Instructions burned: 49 (million)
% 3.03/1.97 % (28726)------------------------------
% 3.03/1.97 % (28726)------------------------------
% 3.03/1.98 % (28732)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_28453 for (2995ds/432Mi)
% 3.03/1.98 % (28730)Instruction limit reached!
% 3.03/1.98 % (28730)------------------------------
% 3.03/1.98 % (28730)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.98 % (28730)Termination reason: Unknown
% 3.03/1.98 % (28730)Termination phase: shuffling
% 3.03/1.98
% 3.03/1.98 % (28730)Memory used [KB]: 2430
% 3.03/1.98 % (28730)Time elapsed: 0.022 s
% 3.03/1.98 % (28730)Instructions burned: 44 (million)
% 3.03/1.98 % (28730)------------------------------
% 3.03/1.98 % (28730)------------------------------
% 3.03/1.98 % (28733)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_28453 for (2995ds/62Mi)
% 3.03/1.99 % (28674)Instruction limit reached!
% 3.03/1.99 % (28674)------------------------------
% 3.03/1.99 % (28674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/1.99 % (28674)Termination reason: Unknown
% 3.03/1.99 % (28674)Termination phase: Saturation
% 3.03/1.99
% 3.03/1.99 % (28674)Memory used [KB]: 10746
% 3.03/1.99 % (28674)Time elapsed: 0.335 s
% 3.03/1.99 % (28674)Instructions burned: 880 (million)
% 3.03/1.99 % (28674)------------------------------
% 3.03/1.99 % (28674)------------------------------
% 3.03/2.00 % (28692)Instruction limit reached!
% 3.03/2.00 % (28692)------------------------------
% 3.03/2.00 % (28692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/2.00 % (28692)Termination reason: Unknown
% 3.03/2.00 % (28692)Termination phase: Saturation
% 3.03/2.00
% 3.03/2.00 % (28692)Memory used [KB]: 7803
% 3.03/2.00 % (28692)Time elapsed: 0.174 s
% 3.03/2.00 % (28692)Instructions burned: 448 (million)
% 3.03/2.00 % (28692)------------------------------
% 3.03/2.00 % (28692)------------------------------
% 3.03/2.00 % (28735)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_28453 for (2995ds/31Mi)
% 3.03/2.01 % (28733)Instruction limit reached!
% 3.03/2.01 % (28733)------------------------------
% 3.03/2.01 % (28733)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.03/2.01 % (28733)Termination reason: Unknown
% 3.03/2.01 % (28733)Termination phase: shuffling
% 3.03/2.01
% 3.03/2.01 % (28733)Memory used [KB]: 2814
% 3.03/2.01 % (28733)Time elapsed: 0.030 s
% 3.03/2.01 % (28733)Instructions burned: 63 (million)
% 3.03/2.01 % (28733)------------------------------
% 3.03/2.01 % (28733)------------------------------
% 6.26/2.01 % (28735)Instruction limit reached!
% 6.26/2.01 % (28735)------------------------------
% 6.26/2.01 % (28735)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.26/2.01 % (28735)Termination reason: Unknown
% 6.26/2.01 % (28735)Termination phase: shuffling
% 6.26/2.01
% 6.26/2.01 % (28735)Memory used [KB]: 2302
% 6.26/2.01 % (28735)Time elapsed: 0.016 s
% 6.26/2.01 % (28735)Instructions burned: 32 (million)
% 6.26/2.01 % (28735)------------------------------
% 6.26/2.01 % (28735)------------------------------
% 6.26/2.01 % (28736)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_28453 for (2995ds/96Mi)
% 6.26/2.01 % (28738)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_28453 for (2995ds/453Mi)
% 6.26/2.02 % (28739)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_28453 for (2995ds/71Mi)
% 6.26/2.03 % (28740)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_28453 for (2995ds/83Mi)
% 6.26/2.04 % (28739)Instruction limit reached!
% 6.26/2.04 % (28739)------------------------------
% 6.26/2.04 % (28739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.26/2.04 % (28739)Termination reason: Unknown
% 6.26/2.04 % (28739)Termination phase: Property scanning
% 6.26/2.04
% 6.26/2.04 % (28739)Memory used [KB]: 2814
% 6.26/2.04 % (28739)Time elapsed: 0.021 s
% 6.26/2.04 % (28739)Instructions burned: 72 (million)
% 6.26/2.04 % (28739)------------------------------
% 6.26/2.04 % (28739)------------------------------
% 6.26/2.05 % (28742)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_28453 for (2994ds/11Mi)
% 6.26/2.06 % (28742)Instruction limit reached!
% 6.26/2.06 % (28742)------------------------------
% 6.26/2.06 % (28742)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.26/2.06 % (28742)Termination reason: Unknown
% 6.26/2.06 % (28742)Termination phase: shuffling
% 6.26/2.06
% 6.26/2.06 % (28742)Memory used [KB]: 1791
% 6.26/2.06 % (28742)Time elapsed: 0.005 s
% 6.26/2.06 % (28742)Instructions burned: 12 (million)
% 6.26/2.06 % (28742)------------------------------
% 6.26/2.06 % (28742)------------------------------
% 6.26/2.06 % (28736)Instruction limit reached!
% 6.26/2.06 % (28736)------------------------------
% 6.26/2.06 % (28736)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.26/2.06 % (28736)Termination reason: Unknown
% 6.26/2.06 % (28736)Termination phase: Property scanning
% 6.26/2.06
% 6.26/2.06 % (28736)Memory used [KB]: 2814
% 6.26/2.06 % (28736)Time elapsed: 0.045 s
% 6.26/2.06 % (28736)Instructions burned: 96 (million)
% 6.26/2.06 % (28736)------------------------------
% 6.26/2.06 % (28736)------------------------------
% 6.26/2.07 % (28744)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_28453 for (2994ds/41Mi)
% 6.26/2.07 % (28740)Instruction limit reached!
% 6.26/2.07 % (28740)------------------------------
% 6.26/2.07 % (28740)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.26/2.07 % (28740)Termination reason: Unknown
% 6.26/2.07 % (28740)Termination phase: Property scanning
% 6.26/2.07
% 6.26/2.07 % (28740)Memory used [KB]: 2814
% 6.26/2.07 % (28740)Time elapsed: 0.040 s
% 6.26/2.07 % (28740)Instructions burned: 84 (million)
% 6.26/2.07 % (28740)------------------------------
% 6.26/2.07 % (28740)------------------------------
% 6.26/2.07 % (28745)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28453 for (2994ds/6Mi)
% 6.26/2.08 % (28745)Instruction limit reached!
% 6.26/2.08 % (28745)------------------------------
% 6.26/2.08 % (28745)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.26/2.08 % (28744)Instruction limit reached!
% 6.26/2.08 % (28744)------------------------------
% 6.26/2.08 % (28744)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.26/2.08 % (28744)Termination reason: Unknown
% 6.26/2.08 % (28744)Termination phase: shuffling
% 6.26/2.08
% 6.26/2.08 % (28744)Memory used [KB]: 2430
% 6.26/2.08 % (28744)Time elapsed: 0.013 s
% 6.26/2.08 % (28744)Instructions burned: 43 (million)
% 6.26/2.08 % (28744)------------------------------
% 6.26/2.08 % (28744)------------------------------
% 6.26/2.08 % (28745)Termination reason: Unknown
% 6.26/2.08 % (28745)Termination phase: shuffling
% 6.26/2.08
% 6.26/2.08 % (28745)Memory used [KB]: 1663
% 6.26/2.08 % (28745)Time elapsed: 0.005 s
% 6.26/2.08 % (28745)Instructions burned: 7 (million)
% 6.26/2.08 % (28745)------------------------------
% 6.26/2.08 % (28745)------------------------------
% 6.26/2.09 % (28746)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_28453 for (2994ds/150Mi)
% 6.26/2.09 % (28748)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_28453 for (2994ds/109Mi)
% 6.26/2.09 % (28747)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_28453 for (2994ds/343Mi)
% 6.26/2.10 % (28705)Instruction limit reached!
% 6.26/2.10 % (28705)------------------------------
% 6.26/2.10 % (28705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.26/2.10 % (28705)Termination reason: Unknown
% 6.26/2.10 % (28705)Termination phase: Saturation
% 6.26/2.10
% 6.26/2.10 % (28705)Memory used [KB]: 8315
% 6.26/2.10 % (28705)Time elapsed: 0.236 s
% 6.26/2.10 % (28705)Instructions burned: 507 (million)
% 6.26/2.10 % (28705)------------------------------
% 6.26/2.10 % (28705)------------------------------
% 6.77/2.12 % (28749)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_28453 for (2994ds/253Mi)
% 6.77/2.13 % (28748)Instruction limit reached!
% 6.77/2.13 % (28748)------------------------------
% 6.77/2.13 % (28748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.77/2.13 % (28748)Termination reason: Unknown
% 6.77/2.13 % (28748)Termination phase: SInE selection
% 6.77/2.13
% 6.77/2.13 % (28748)Memory used [KB]: 2814
% 6.77/2.13 % (28748)Time elapsed: 0.036 s
% 6.77/2.13 % (28748)Instructions burned: 112 (million)
% 6.77/2.13 % (28748)------------------------------
% 6.77/2.13 % (28748)------------------------------
% 6.77/2.14 % (28752)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_28453 for (2993ds/140Mi)
% 6.77/2.15 % (28732)Instruction limit reached!
% 6.77/2.15 % (28732)------------------------------
% 6.77/2.15 % (28732)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.77/2.15 % (28732)Termination reason: Unknown
% 6.77/2.15 % (28732)Termination phase: Saturation
% 6.77/2.15
% 6.77/2.15 % (28732)Memory used [KB]: 3326
% 6.77/2.15 % (28732)Time elapsed: 0.176 s
% 6.77/2.15 % (28732)Instructions burned: 433 (million)
% 6.77/2.15 % (28732)------------------------------
% 6.77/2.15 % (28732)------------------------------
% 6.77/2.16 % (28746)Instruction limit reached!
% 6.77/2.16 % (28746)------------------------------
% 6.77/2.16 % (28746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.77/2.16 % (28746)Termination reason: Unknown
% 6.77/2.16 % (28746)Termination phase: Preprocessing 3
% 6.77/2.16
% 6.77/2.16 % (28746)Memory used [KB]: 5373
% 6.77/2.16 % (28746)Time elapsed: 0.074 s
% 6.77/2.16 % (28746)Instructions burned: 150 (million)
% 6.77/2.16 % (28746)------------------------------
% 6.77/2.16 % (28746)------------------------------
% 6.77/2.17 % (28754)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_28453 for (2993ds/614Mi)
% 6.77/2.17 % (28755)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_28453 for (2993ds/202Mi)
% 6.77/2.18 % (28752)Instruction limit reached!
% 6.77/2.18 % (28752)------------------------------
% 6.77/2.18 % (28752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.77/2.18 % (28752)Termination reason: Unknown
% 6.77/2.18 % (28752)Termination phase: Property scanning
% 6.77/2.18
% 6.77/2.18 % (28752)Memory used [KB]: 2942
% 6.77/2.18 % (28752)Time elapsed: 0.038 s
% 6.77/2.18 % (28752)Instructions burned: 140 (million)
% 6.77/2.18 % (28752)------------------------------
% 6.77/2.18 % (28752)------------------------------
% 6.77/2.18 % (28757)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_28453 for (2993ds/339Mi)
% 6.77/2.19 % (28725)First to succeed.
% 6.77/2.19 % (28738)Instruction limit reached!
% 6.77/2.19 % (28738)------------------------------
% 6.77/2.19 % (28738)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.77/2.19 % (28738)Termination reason: Unknown
% 6.77/2.19 % (28738)Termination phase: Saturation
% 6.77/2.19
% 6.77/2.19 % (28738)Memory used [KB]: 7803
% 6.77/2.19 % (28738)Time elapsed: 0.175 s
% 6.77/2.19 % (28738)Instructions burned: 456 (million)
% 6.77/2.19 % (28738)------------------------------
% 6.77/2.19 % (28738)------------------------------
% 6.77/2.19 % (28725)Refutation found. Thanks to Tanya!
% 6.77/2.19 % SZS status Theorem for DTF2THF_28453
% 6.77/2.19 % SZS output start Proof for DTF2THF_28453
% 6.77/2.19 thf(func_def_0, type, in: $i > $i > $o).
% 6.77/2.19 thf(func_def_1, type, exu: ($i > $o) > $o).
% 6.77/2.19 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 6.77/2.19 thf(func_def_8, type, powerset: $i > $i).
% 6.77/2.19 thf(func_def_10, type, setunion: $i > $i).
% 6.77/2.19 thf(func_def_19, type, descr: ($i > $o) > $i).
% 6.77/2.19 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_26, type, prop2set: $o > $i).
% 6.77/2.19 thf(func_def_36, type, nonempty: $i > $o).
% 6.77/2.19 thf(func_def_69, type, set2prop: $i > $o).
% 6.77/2.19 thf(func_def_88, type, subset: $i > $i > $o).
% 6.77/2.19 thf(func_def_89, type, disjoint: $i > $i > $o).
% 6.77/2.19 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 6.77/2.19 thf(func_def_114, type, binunion: $i > $i > $i).
% 6.77/2.19 thf(func_def_122, type, binintersect: $i > $i > $i).
% 6.77/2.19 thf(func_def_135, type, regular: $i > $o).
% 6.77/2.19 thf(func_def_136, type, setminus: $i > $i > $i).
% 6.77/2.19 thf(func_def_147, type, symdiff: $i > $i > $i).
% 6.77/2.19 thf(func_def_153, type, iskpair: $i > $o).
% 6.77/2.19 thf(func_def_158, type, kpair: $i > $i > $i).
% 6.77/2.19 thf(func_def_160, type, cartprod: $i > $i > $i).
% 6.77/2.19 thf(func_def_177, type, singleton: $i > $o).
% 6.77/2.19 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_184, type, atmost1p: $i > $o).
% 6.77/2.19 thf(func_def_185, type, atleast2p: $i > $o).
% 6.77/2.19 thf(func_def_186, type, atmost2p: $i > $o).
% 6.77/2.19 thf(func_def_187, type, upairsetp: $i > $o).
% 6.77/2.19 thf(func_def_191, type, kfst: $i > $i).
% 6.77/2.19 thf(func_def_203, type, ksnd: $i > $i).
% 6.77/2.19 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 6.77/2.19 thf(func_def_222, type, func: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_223, type, funcSet: $i > $i > $i).
% 6.77/2.19 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 6.77/2.19 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 6.77/2.19 thf(func_def_323, type, breln1: $i > $i > $o).
% 6.77/2.19 thf(func_def_325, type, breln1Set: $i > $i).
% 6.77/2.19 thf(func_def_327, type, transitive: $i > $i > $o).
% 6.77/2.19 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 6.77/2.19 thf(func_def_329, type, reflexive: $i > $i > $o).
% 6.77/2.19 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 6.77/2.19 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 6.77/2.19 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 6.77/2.19 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 6.77/2.19 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 6.77/2.19 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 6.77/2.19 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 6.77/2.19 thf(func_def_378, type, omegaS: $i > $i).
% 6.77/2.19 thf(func_def_393, type, sP0: $i > $i > $o).
% 6.77/2.19 thf(func_def_395, type, sP2: $i > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_396, type, sP3: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_397, type, sP4: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_398, type, sP5: ($i > $o) > $i > $i > $o).
% 6.77/2.19 thf(func_def_399, type, sP6: $i > ($i > $i > $o) > $i > $o).
% 6.77/2.19 thf(func_def_400, type, sP7: $i > ($i > $i > $o) > $o).
% 6.77/2.19 thf(func_def_403, type, sP10: $i > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_404, type, sP11: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_406, type, sP13: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_407, type, sP14: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_408, type, sP15: $i > $i > $o).
% 6.77/2.19 thf(func_def_409, type, sP16: $i > $o).
% 6.77/2.19 thf(func_def_410, type, sP17: $i > $i > $o).
% 6.77/2.19 thf(func_def_411, type, sP18: $i > $i > $o).
% 6.77/2.19 thf(func_def_412, type, sP19: $i > $i > $o).
% 6.77/2.19 thf(func_def_413, type, sP20: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_414, type, sP21: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_415, type, sP22: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_417, type, sP24: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_419, type, sP26: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_420, type, sP27: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_422, type, sP29: $i > ($i > $o) > $i > $o).
% 6.77/2.19 thf(func_def_423, type, sP30: $o > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_424, type, sP31: $i > $i > $i > $o > $o).
% 6.77/2.19 thf(func_def_425, type, sP32: $i > $i > $o > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_426, type, sP33: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_428, type, sP35: $i > $i > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_431, type, sP38: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_434, type, sP41: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_436, type, sP43: $i > $i > ($i > $i) > $o).
% 6.77/2.19 thf(func_def_438, type, sP45: ($i > $o) > $o).
% 6.77/2.19 thf(func_def_439, type, sP46: ($i > $i > $o) > $i > $i > $o).
% 6.77/2.19 thf(func_def_440, type, sP47: ($i > $i > $o) > $i > $i > $o).
% 6.77/2.19 thf(func_def_442, type, sP49: $i > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_443, type, sP50: $i > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_444, type, sP51: $i > $i > ($i > $i) > $o).
% 6.77/2.19 thf(func_def_446, type, sP53: $i > ($i > $o) > ($i > $o) > $i > $o).
% 6.77/2.19 thf(func_def_448, type, sP55: $i > $i > $o).
% 6.77/2.19 thf(func_def_451, type, sP58: $i > $i > $o > $o).
% 6.77/2.19 thf(func_def_452, type, sP59: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_453, type, sP60: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_454, type, sP61: ($i > $o) > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_455, type, sP62: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_456, type, sP63: $i > ($i > $o) > $i > $o).
% 6.77/2.19 thf(func_def_457, type, sP64: $i > $i > ($i > $i) > $o).
% 6.77/2.19 thf(func_def_460, type, sP67: ($i > $o) > $i > $o).
% 6.77/2.19 thf(func_def_461, type, sP68: $i > $i > $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_462, type, sP69: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_464, type, sP71: ($i > $o) > $o).
% 6.77/2.19 thf(func_def_465, type, sP72: $i > $i > $o > $o).
% 6.77/2.19 thf(func_def_466, type, sP73: ($i > $o) > $o).
% 6.77/2.19 thf(func_def_467, type, sP74: $o > $i > $i > $o).
% 6.77/2.19 thf(func_def_468, type, sP75: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_469, type, sP76: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_470, type, sP77: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_471, type, sP78: ($i > $o) > $i > $o).
% 6.77/2.19 thf(func_def_472, type, sP79: $i > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_473, type, sP80: $i > $o).
% 6.77/2.19 thf(func_def_475, type, sP82: ($i > $o) > $o).
% 6.77/2.19 thf(func_def_476, type, sP83: $o > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_477, type, sP84: $i > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_478, type, sP85: $i > $o).
% 6.77/2.19 thf(func_def_479, type, sP86: $i > $o).
% 6.77/2.19 thf(func_def_480, type, sP87: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_481, type, sP88: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_482, type, sP89: $i > ($i > $o) > $o).
% 6.77/2.19 thf(func_def_484, type, sP91: $i > ($i > $i) > $i > $o).
% 6.77/2.19 thf(func_def_486, type, sP93: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_487, type, sP94: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_488, type, sP95: $i > $i > $o > $o).
% 6.77/2.19 thf(func_def_490, type, sP97: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_494, type, sP101: ($i > $i) > $i > $i > $o).
% 6.77/2.19 thf(func_def_496, type, sP103: $i > $i > $o).
% 6.77/2.19 thf(func_def_498, type, sP105: ($i > $o) > $o).
% 6.77/2.19 thf(func_def_514, type, sK121: $i > $i > $i).
% 6.77/2.19 thf(func_def_517, type, sK124: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_518, type, sK125: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_519, type, sK126: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_520, type, sK127: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_528, type, sK135: $i > $i).
% 6.77/2.19 thf(func_def_538, type, sK145: ($i > $o) > $i > $i > $i).
% 6.77/2.19 thf(func_def_539, type, sK146: ($i > $o) > $i > $i > $i).
% 6.77/2.19 thf(func_def_540, type, sK147: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_543, type, sK150: $i > $o).
% 6.77/2.19 thf(func_def_548, type, sK155: $i > $i > $o).
% 6.77/2.19 thf(func_def_550, type, sK157: $i > ($i > $i > $o) > $i).
% 6.77/2.19 thf(func_def_551, type, sK158: $i > $i > ($i > $i > $o) > $i).
% 6.77/2.19 thf(func_def_552, type, sK159: $i > ($i > $i > $o) > $i).
% 6.77/2.19 thf(func_def_553, type, sK160: $i > ($i > $i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_576, type, sK183: $i > $i).
% 6.77/2.19 thf(func_def_577, type, sK184: ($i > $i) > $i > $i > $i).
% 6.77/2.19 thf(func_def_600, type, sK207: $i > $i).
% 6.77/2.19 thf(func_def_601, type, sK208: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_602, type, sK209: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_620, type, sK227: $i > $i > $o).
% 6.77/2.19 thf(func_def_623, type, sK230: $i > $o).
% 6.77/2.19 thf(func_def_624, type, sK231: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_625, type, sK232: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_634, type, sK241: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_635, type, sK242: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_639, type, sK246: $i > $i > $i).
% 6.77/2.19 thf(func_def_640, type, sK247: $i > $i > $i).
% 6.77/2.19 thf(func_def_641, type, sK248: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_642, type, sK249: $i > $i > $i).
% 6.77/2.19 thf(func_def_643, type, sK250: $i > $i > $i).
% 6.77/2.19 thf(func_def_644, type, sK251: $i > $i > $i).
% 6.77/2.19 thf(func_def_645, type, sK252: $i > $i > $i).
% 6.77/2.19 thf(func_def_646, type, sK253: $i > $i > $i).
% 6.77/2.19 thf(func_def_647, type, sK254: $i > $i).
% 6.77/2.19 thf(func_def_648, type, sK255: $i > $i).
% 6.77/2.19 thf(func_def_649, type, sK256: $i > $i).
% 6.77/2.19 thf(func_def_650, type, sK257: $i > $i).
% 6.77/2.19 thf(func_def_651, type, sK258: $i > $i > $i).
% 6.77/2.19 thf(func_def_652, type, sK259: $i > $i > $i).
% 6.77/2.19 thf(func_def_653, type, sK260: $i > $i > $i).
% 6.77/2.19 thf(func_def_654, type, sK261: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_655, type, sK262: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_657, type, sK264: $i > $i).
% 6.77/2.19 thf(func_def_663, type, sK270: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_664, type, sK271: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_665, type, sK272: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_666, type, sK273: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_691, type, sK298: $i > $i > $o).
% 6.77/2.19 thf(func_def_695, type, sK302: $i > $o).
% 6.77/2.19 thf(func_def_696, type, sK303: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_697, type, sK304: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_723, type, sK330: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_724, type, sK331: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_725, type, sK332: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_726, type, sK333: $i > $o).
% 6.77/2.19 thf(func_def_730, type, sK337: $i > $i).
% 6.77/2.19 thf(func_def_731, type, sK338: ($i > $i) > $i > $i > $i).
% 6.77/2.19 thf(func_def_739, type, sK346: $i > ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_741, type, sK348: $i > $o).
% 6.77/2.19 thf(func_def_750, type, sK357: $i > $i > $o).
% 6.77/2.19 thf(func_def_760, type, sK367: $i > $i > $o > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_772, type, sK379: $i > $o).
% 6.77/2.19 thf(func_def_773, type, sK380: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_774, type, sK381: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_775, type, sK382: $i > $i > $i).
% 6.77/2.19 thf(func_def_782, type, sK389: $i > $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_796, type, sK403: $i > $i).
% 6.77/2.19 thf(func_def_798, type, sK405: $i > ($i > $i) > $i > $i).
% 6.77/2.19 thf(func_def_799, type, sK406: $i > $o).
% 6.77/2.19 thf(func_def_806, type, sK413: $i > $o).
% 6.77/2.19 thf(func_def_807, type, sK414: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_808, type, sK415: $i > $o).
% 6.77/2.19 thf(func_def_810, type, sK417: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_811, type, sK418: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_812, type, sK419: $i > $o).
% 6.77/2.19 thf(func_def_816, type, sK423: $i > $o).
% 6.77/2.19 thf(func_def_817, type, sK424: $i > $o).
% 6.77/2.19 thf(func_def_818, type, sK425: ($i > $o) > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_819, type, sK426: ($i > $o) > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_823, type, sK430: $i > $o).
% 6.77/2.19 thf(func_def_824, type, sK431: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_825, type, sK432: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_827, type, sK434: $i > $i).
% 6.77/2.19 thf(func_def_829, type, sK436: $i > ($i > $i) > $i > $i).
% 6.77/2.19 thf(func_def_830, type, sK437: $i > $i > ($i > $i) > $i).
% 6.77/2.19 thf(func_def_840, type, sK447: ($i > $o) > $i).
% 6.77/2.19 thf(func_def_841, type, sK448: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_842, type, sK449: $i > $o).
% 6.77/2.19 thf(func_def_883, type, sK490: $i > $o).
% 6.77/2.19 thf(func_def_890, type, sK497: $i > ($i > $i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_891, type, sK498: ($i > $i > $o) > $i > $i > $i).
% 6.77/2.19 thf(func_def_892, type, sK499: ($i > $i > $o) > $i > $i > $i).
% 6.77/2.19 thf(func_def_893, type, sK500: ($i > $i > $o) > $i > $i > $i).
% 6.77/2.19 thf(func_def_896, type, sK503: $i > $i > $o).
% 6.77/2.19 thf(func_def_907, type, sK514: $i > $i > $i).
% 6.77/2.19 thf(func_def_917, type, sK524: $i > $i > $o).
% 6.77/2.19 thf(func_def_921, type, sK528: $i > $o).
% 6.77/2.19 thf(func_def_924, type, sK531: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_925, type, sK532: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_926, type, sK533: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_927, type, sK534: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_934, type, sK541: $i > $i > $o).
% 6.77/2.19 thf(func_def_939, type, sK546: $i > $i > $o).
% 6.77/2.19 thf(func_def_948, type, sK555: $i > $i).
% 6.77/2.19 thf(func_def_949, type, sK556: ($i > $i) > $i > $i > $i).
% 6.77/2.19 thf(func_def_950, type, sK557: $i > $i > ($i > $i) > $i).
% 6.77/2.19 thf(func_def_954, type, sK561: $i > ($i > $o) > ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_955, type, sK562: $i > ($i > $o) > ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_958, type, sK565: $i > $o).
% 6.77/2.19 thf(func_def_959, type, sK566: $i > $o).
% 6.77/2.19 thf(func_def_962, type, sK569: $i > $i).
% 6.77/2.19 thf(func_def_985, type, sK592: $i > $i > $i).
% 6.77/2.19 thf(func_def_988, type, sK595: $i > $i > $i).
% 6.77/2.19 thf(func_def_990, type, sK597: $i > $o).
% 6.77/2.19 thf(func_def_1000, type, sK607: ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1001, type, sK608: $i > $o).
% 6.77/2.19 thf(func_def_1006, type, sK613: $i > $o).
% 6.77/2.19 thf(func_def_1011, type, sK618: $i > $i > $o).
% 6.77/2.19 thf(func_def_1025, type, sK632: $i > $o).
% 6.77/2.19 thf(func_def_1031, type, sK638: $i > $i).
% 6.77/2.19 thf(func_def_1034, type, sK641: $i > $i > ($i > $i) > $i).
% 6.77/2.19 thf(func_def_1042, type, sK649: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1043, type, sK650: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_1049, type, sK656: $i > $o).
% 6.77/2.19 thf(func_def_1055, type, sK662: $i > $i).
% 6.77/2.19 thf(func_def_1056, type, sK663: ($i > $o) > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1057, type, sK664: ($i > $o) > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1058, type, sK665: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1062, type, sK669: $i > $o).
% 6.77/2.19 thf(func_def_1077, type, sK684: $i > $i > $o).
% 6.77/2.19 thf(func_def_1083, type, sK690: $i > ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_1084, type, sK691: $i > ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_1085, type, sK692: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1088, type, sK695: $i > $o).
% 6.77/2.19 thf(func_def_1090, type, sK697: $i > $i).
% 6.77/2.19 thf(func_def_1111, type, sK718: $i > $i).
% 6.77/2.19 thf(func_def_1112, type, sK719: ($i > $i) > $i > $i > $i).
% 6.77/2.19 thf(func_def_1113, type, sK720: $i > $i > ($i > $i) > $i).
% 6.77/2.19 thf(func_def_1114, type, sK721: $i > $i > $i > ($i > $i) > $i).
% 6.77/2.19 thf(func_def_1163, type, sK770: $i > $i).
% 6.77/2.19 thf(func_def_1169, type, sK776: $i > $i > $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1170, type, sK777: $i > $i > $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1171, type, sK778: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_1175, type, sK782: $i > $o).
% 6.77/2.19 thf(func_def_1176, type, sK783: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1177, type, sK784: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1214, type, sK821: $i > $i > $i).
% 6.77/2.19 thf(func_def_1216, type, sK823: $i > $o).
% 6.77/2.19 thf(func_def_1218, type, sK825: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1219, type, sK826: ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1220, type, sK827: $i > $o).
% 6.77/2.19 thf(func_def_1229, type, sK836: $i > $i > $o > $i).
% 6.77/2.19 thf(func_def_1239, type, sK846: ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1240, type, sK847: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1241, type, sK848: $i > $o).
% 6.77/2.19 thf(func_def_1260, type, sK867: $i > $o).
% 6.77/2.19 thf(func_def_1268, type, sK875: $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_1275, type, sK882: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1276, type, sK883: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1287, type, sK894: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_1288, type, sK895: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_1289, type, sK896: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1290, type, sK897: $i > $o).
% 6.77/2.19 thf(func_def_1304, type, sK911: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1318, type, sK925: $i > $i > $i).
% 6.77/2.19 thf(func_def_1319, type, sK926: $i > $i).
% 6.77/2.19 thf(func_def_1339, type, sK946: $i > $i).
% 6.77/2.19 thf(func_def_1340, type, sK947: ($i > $i) > $i > $i > $i).
% 6.77/2.19 thf(func_def_1341, type, sK948: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1342, type, sK949: ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1343, type, sK950: $i > $o).
% 6.77/2.19 thf(func_def_1357, type, sK964: $i > $i > $i > $i > $o).
% 6.77/2.19 thf(func_def_1377, type, sK984: $i > $i).
% 6.77/2.19 thf(func_def_1378, type, sK985: $i > $i).
% 6.77/2.19 thf(func_def_1397, type, sK1004: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1413, type, sK1020: $i > $i).
% 6.77/2.19 thf(func_def_1437, type, sK1044: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1438, type, sK1045: $i > $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1452, type, sK1059: $i > $o).
% 6.77/2.19 thf(func_def_1453, type, sK1060: ($i > $o) > $i > $i).
% 6.77/2.19 thf(func_def_1454, type, sK1061: $i > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1459, type, sK1066: $i > $o).
% 6.77/2.19 thf(func_def_1473, type, sK1080: $i > $i).
% 6.77/2.19 thf(func_def_1476, type, sK1083: $i > ($i > $i) > $i > $i).
% 6.77/2.19 thf(func_def_1484, type, sK1091: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1485, type, sK1092: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1495, type, sK1102: $i > $o).
% 6.77/2.19 thf(func_def_1508, type, sK1115: $i > $i > $i > $i).
% 6.77/2.19 thf(func_def_1518, type, sK1125: $i > $i > $i).
% 6.77/2.19 thf(func_def_1521, type, sK1128: $i > $i > $i).
% 6.77/2.19 thf(func_def_1533, type, sK1140: $i > $o).
% 6.77/2.19 thf(func_def_1534, type, sK1141: $i > $o).
% 6.77/2.19 thf(func_def_1535, type, sK1142: ($i > $o) > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1536, type, sK1143: ($i > $o) > ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1548, type, sK1155: $i > $i).
% 6.77/2.19 thf(func_def_1549, type, sK1156: $i > $i).
% 6.77/2.19 thf(func_def_1550, type, sK1157: ($i > $i) > $i > $i).
% 6.77/2.19 thf(func_def_1551, type, sK1158: ($i > $i) > $i > $i > $i).
% 6.77/2.19 thf(func_def_1558, type, sK1165: $i > $i > $i).
% 6.77/2.19 thf(func_def_1559, type, sK1166: $i > $i > $i).
% 6.77/2.19 thf(func_def_1560, type, sK1167: $i > $o).
% 6.77/2.19 thf(func_def_1561, type, sK1168: ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1562, type, sK1169: ($i > $o) > $i).
% 6.77/2.19 thf(func_def_1563, type, sK1170: $i > $o).
% 6.77/2.19 thf(func_def_1567, type, ph1174: !>[X0: $tType]:(X0)).
% 6.77/2.19 thf(f8483,plain,(
% 6.77/2.19 $false),
% 6.77/2.19 inference(subsumption_resolution,[status(thm)],[f8477,f5192])).
% 6.77/2.19 thf(f5192,plain,(
% 6.77/2.19 ($true = (in @ sK1162 @ omega))),
% 6.77/2.19 inference(cnf_transformation,[status(thm)],[f3438])).
% 6.77/2.19 thf(f3438,plain,(
% 6.77/2.19 (lamp = $true) & (vacuousDall = $true) & (exu__Cong = $true) & (omegaIndAx = $true) & (cartprodfstpairEq = $true) & (dsetconstr__Cong = $true) & (binunionEcases = $true) & (emptyset__Cong = $true) & (demorgan2a2 = $true) & (setextsub = $true) & (setukpairinjL2 = $true) & (setoftrueEq = $true) & (descr__Cong = $true) & (setminusLsub = $true) & (quantDeMorgan2 = $true) & (complementImpComplementIntersect = $true) & (ex1I2 = $true) & (setminusERneg = $true) & (complementT_lem = $true) & (setminusT_lem = $true) & (breln1all2 = $true) & (omega0Ax = $true) & (doubleComplementSub2 = $true) & (cartprodpairin = $true) & (intersectInPowersetIntersectUnions = $true) & (ubforcartprodlem3 = $true) & (symdiffI2 = $true) & (woz1_2 = $true) & (setunionE = $true) & (setunionsingleton1 = $true) & (ap2apEq2 = $true) & (cartprodsndin = $true) & (bs114d = $true) & (exuI3 = $true) & (dpsetconstrERa = $true) & (symdiffIneg1 = $true) & (demorgan1b = $true) & (theprop = $true) & (binintersectSubset2 = $true) & (image1I = $true) & (setukpairIL = $true) & (ubforcartprodlem2 = $true) & (exuEu = $true) & (upairequniteq = $true) & (setadjoinOr = $true) & (surjCantorThm = $true) & (binintersectEL = $true) & (funcGraphProp4 = $true) & (breln1all1 = $true) & (emptysetAx = $true) & (iftrueProp2 = $true) & (emptyE1 = $true) & (notdexE = $true) & (ex1I = $true) & (symdiffE = $true) & (demorgan2a = $true) & (kpairsurjEq = $true) & (cartprodsndpairEq = $true) & (notdallE = $true) & (kpairiskpair = $true) & (powerset__Cong = $true) & (woz2A = $true) & (powersetI = $true) & (setadjoinSub = $true) & (setOfPairsIsBReln = $true) & (nonemptyImpWitness = $true) & (image1Ex1 = $true) & (eqbreln1 = $true) & (demorgan2a1 = $true) & (exuE3e = $true) & (cartprodpairmemEL = $true) & (setminusI = $true) & (binintersectSubset4 = $true) & (singletoninpowerset = $true) & (binunionT_lem = $true) & (cartprodpairmemER = $true) & (woz2W = $true) & (powersetAx = $true) & (setukpairinjR11 = $true) & (setadjoinIL = $true) & (powersetTE1 = $true) & (upairset2IR = $true) & (setextT = $true) & (woz1_3 = $true) & (contraSubsetComplement = $true) & (setunionI = $true) & (inComplementUnionImpInComplement1 = $true) & (secondinupair = $true) & (setminusIRneg = $true) & (funcext = $true) & (complementTcontraSubset = $true) & (uniqinunit = $true) & (choice2fnsingleton = $true) & (symdiffI1 = $true) & (woz13rule0 = $true) & (breln1unionE = $true) & (image1Equiv = $true) & (setunion__Cong = $true) & (image1E = $true) & (iffalse = $true) & (binintersectER = $true) & (singletonsswitch = $true) & (binintersectTERcontra = $true) & (iffalseProp1 = $true) & (ubforcartprodlem1 = $true) & (breln1SetBreln1 = $true) & (woz13rule4 = $true) & (inIntersectImpInUnion = $true) & (powersetE1 = $true) & (binunionRsub = $true) & (brelnall2 = $true) & (notinself2 = $true) & (doubleComplementI1 = $true) & (emptysetsubset = $true) & (upairsubunion = $true) & (injFuncSetFuncInj = $true) & (complementTI1 = $true) & (setminusSubset2 = $true) & (binunionTILcontra = $true) & (setext = $true) & (binunionIR = $true) & (setukpairinjR12 = $true) & (breln1compE = $true) & (setukpairinjL = $true) & (setukpairIR = $true) & (setukpairinjL1 = $true) & (subsetI2 = $true) & (setunionsingleton = $true) & (foundation2 = $true) & (funcinfuncset = $true) & (binunionE = $true) & (ex1E2 = $true) & (iftrueProp1 = $true) & (powersetI1 = $true) & (funcImageSingleton = $true) & (cartprodmempair = $true) & (wellorderingAx = $true) & (notequalI1 = $true) & (demorgan1a = $true) & (replAx = $true) & (dsetconstrI = $true) & (foundationAx = $true) & (kfstsingleton = $true) & (lam2p = $true) & (demorgan1 = $true) & (quantDeMorgan3 = $true) & (subsetE = $true) & (inIntersectImpInUnion2 = $true) & (complementTE1 = $true) & (iftrue = $true) & (woz13rule3 = $true) & (injFuncSetFuncIn = $true) & (breln1unionIL = $true) & (contrasubsetT3 = $true) & (setukpairinjR2 = $true) & (subset2powerset = $true) & (sepSubset = $true) & (complementUnionInPowersetComplement = $true) & (eqimpsubset1 = $true) & (doubleComplementE1 = $true) & (subsetI1 = $true) & (woz2B = $true) & (powersetT_lem = $true) & (setukpairinjR1 = $true) & (woz1_5 = $true) & (beta2 = $true) & (woz2Ex = $true) & (apProp = $true) & (cartprodmempair1 = $true) & (subsetE2 = $true) & (omega__Cong = $true) & (subbreln1 = $true) & (prop2setE = $true) & (omegaSAx = $true) & (powersetTI1 = $true) & (binintersectSubset1 = $true) & (image1Ex = $true) & (nonemptyI1 = $true) & (notinself = $true) & (doubleComplementSub1 = $true) & (binunionLsub = $true) & (app = $true) & (nonemptyE1 = $true) & (emptyinPowerset = $true) & (upairsetIL = $true) & (eta1 = $true) & (setOfPairsIsBReln1 = $true) & (notinemptyset = $true) & (setminusER = $true) & (emptyinunitempty = $true) & (quantDeMorgan4 = $true) & (complementSubsetComplementIntersect = $true) & (symdiffIneg2 = $true) & (setminusSubset1 = $true) & (funcGraphProp2 = $true) & (subbreln = $true) & (doubleComplementEq = $true) & (inIntersectImpInIntersectUnions = $true) & (surjFuncSetFuncIn = $true) & (exuE3u = $true) & (dpsetconstrI = $true) & (demorgan2b = $true) & (breln1invE = $true) & (subsetTI = $true) & (cartprodpairsurjEq = $true) & (emptysetE = $true) & (binunionTIRcontra = $true) & (($true = (in @ sK1162 @ omega)) & ($true != (in @ (omegaS @ sK1162) @ omega))) & (setadjoinAx = $true) & (breln1unionprop = $true) & (dsetconstrER = $true) & (upairset2E = $true) & (prop2setI = $true) & (setadjoinSub2 = $true) & (demorgan2b2 = $true) & (theeq = $true) & (setminusEL = $true) & (upairsetE = $true) & (woz1_4 = $true) & (binintersectLsub = $true) & (emptyI = $true) & (complementTnotintersectT = $true) & (complementInPowersetComplementIntersect = $true) & (funcGraphProp3 = $true) & (powersetE = $true) & (setadjoinIR = $true) & (emptysetimpfalse = $true) & (exuE2 = $true) & (setbeta = $true) & (contrasubsetT1 = $true) & (setadjoin__Cong = $true) & (woz13rule1 = $true) & (binunionTE = $true) & (breln1unionIR = $true) & (subPowSU = $true) & (binintersectTELcontra = $true) & (setminusELneg = $true) & (binintersectSubset5 = $true) & (quantDeMorgan1 = $true) & (exuI2 = $true) & (eqimpsubset2 = $true) & (disjointsetsI1 = $true) & (injFuncInInjFuncSet = $true) & (beta1 = $true) & (binintersectRsub = $true) & (setukpairinjR = $true) & (brelnall1 = $true) & (exuE1 = $true) & (exuI1 = $true) & (setadjoinE = $true) & (kpairp = $true) & (surjFuncSetFuncSurj = $true) & (setunionAx = $true) & (breln1compprop = $true) & (iftrueorfalse = $true) & (emptyInPowerset = $true) & (dpsetconstrEL1 = $true) & (inCongP = $true) & (inComplementUnionImpNotIn1 = $true) & (iffalseProp2 = $true) & (inPowerset = $true) & (sepInPowerset = $true) & (breln1invprop = $true) & (setminusILneg = $true) & (eta2 = $true) & (ap2apEq1 = $true) & (lamProp = $true) & (powersetsubset = $true) & (leftInvIsSurj = $true) & (dpsetconstrSub = $true) & (subsetRefl = $true) & (nonemptyI = $true) & (setunionE2 = $true) & (lam2lamEq = $true) & (funcext2 = $true) & (subsetTrans = $true) & (singletonprop = $true) & (ex1E1 = $true) & (singletonsuniq = $true) & (ksndpairEq = $true) & (breln1invI = $true) & (demorgan2 = $true) & (woz13rule2 = $true) & (binintersectI = $true) & (singletoninpowunion = $true) & (kfstpairEq = $true) & (notequalI2 = $true) & (funcextLem = $true) & (ifp = $true) & (cartprodfstin = $true) & (cartprodmempaircEq = $true) & (binintersectSubset3 = $true) & (prop2set2propI = $true) & (funcGraphProp1 = $true) & (descrp = $true) & (contrasubsetT = $true) & (ifSingleton = $true) & (binintersectT_lem = $true) & (ksndsingleton = $true) & (breln1unionI = $true) & (breln1unionEcases = $true) & (eqinunit = $true) & (notsubsetI = $true) & (setextAx = $true) & (contrasubsetT2 = $true) & (binunionTEcontra = $true) & (upairinpowunion = $true) & (infuncsetfunc = $true) & (breln1compEex = $true) & (dpsetconstrEL2 = $true) & (notinsingleton = $true) & (ap2p = $true) & (binunionIL = $true) & (eqbreln = $true) & (singletonsubset = $true) & (in__Cong = $true) & (woz1_1 = $true) & (setunionsingleton2 = $true) & (noeltsimpempty = $true) & (dpsetconstrER = $true) & (upairsetIR = $true) & (breln1unionCommutes = $true) & (breln1compI = $true) & (dsetconstrEL = $true) & (subsetemptysetimpeq = $true)),
% 6.77/2.19 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1162])],[f1130,f3437])).
% 6.77/2.19 thf(f3437,plain,(
% 6.77/2.19 ? [X0 : $i] : (((in @ X0 @ omega) = $true) & ((in @ (omegaS @ X0) @ omega) != $true)) => (($true = (in @ sK1162 @ omega)) & ($true != (in @ (omegaS @ sK1162) @ omega)))),
% 6.77/2.19 introduced(definition,[],[choice_axiom])).
% 6.77/2.19 thf(f1130,plain,(
% 6.77/2.19 (lamp = $true) & (vacuousDall = $true) & (exu__Cong = $true) & (omegaIndAx = $true) & (cartprodfstpairEq = $true) & (dsetconstr__Cong = $true) & (binunionEcases = $true) & (emptyset__Cong = $true) & (demorgan2a2 = $true) & (setextsub = $true) & (setukpairinjL2 = $true) & (setoftrueEq = $true) & (descr__Cong = $true) & (setminusLsub = $true) & (quantDeMorgan2 = $true) & (complementImpComplementIntersect = $true) & (ex1I2 = $true) & (setminusERneg = $true) & (complementT_lem = $true) & (setminusT_lem = $true) & (breln1all2 = $true) & (omega0Ax = $true) & (doubleComplementSub2 = $true) & (cartprodpairin = $true) & (intersectInPowersetIntersectUnions = $true) & (ubforcartprodlem3 = $true) & (symdiffI2 = $true) & (woz1_2 = $true) & (setunionE = $true) & (setunionsingleton1 = $true) & (ap2apEq2 = $true) & (cartprodsndin = $true) & (bs114d = $true) & (exuI3 = $true) & (dpsetconstrERa = $true) & (symdiffIneg1 = $true) & (demorgan1b = $true) & (theprop = $true) & (binintersectSubset2 = $true) & (image1I = $true) & (setukpairIL = $true) & (ubforcartprodlem2 = $true) & (exuEu = $true) & (upairequniteq = $true) & (setadjoinOr = $true) & (surjCantorThm = $true) & (binintersectEL = $true) & (funcGraphProp4 = $true) & (breln1all1 = $true) & (emptysetAx = $true) & (iftrueProp2 = $true) & (emptyE1 = $true) & (notdexE = $true) & (ex1I = $true) & (symdiffE = $true) & (demorgan2a = $true) & (kpairsurjEq = $true) & (cartprodsndpairEq = $true) & (notdallE = $true) & (kpairiskpair = $true) & (powerset__Cong = $true) & (woz2A = $true) & (powersetI = $true) & (setadjoinSub = $true) & (setOfPairsIsBReln = $true) & (nonemptyImpWitness = $true) & (image1Ex1 = $true) & (eqbreln1 = $true) & (demorgan2a1 = $true) & (exuE3e = $true) & (cartprodpairmemEL = $true) & (setminusI = $true) & (binintersectSubset4 = $true) & (singletoninpowerset = $true) & (binunionT_lem = $true) & (cartprodpairmemER = $true) & (woz2W = $true) & (powersetAx = $true) & (setukpairinjR11 = $true) & (setadjoinIL = $true) & (powersetTE1 = $true) & (upairset2IR = $true) & (setextT = $true) & (woz1_3 = $true) & (contraSubsetComplement = $true) & (setunionI = $true) & (inComplementUnionImpInComplement1 = $true) & (secondinupair = $true) & (setminusIRneg = $true) & (funcext = $true) & (complementTcontraSubset = $true) & (uniqinunit = $true) & (choice2fnsingleton = $true) & (symdiffI1 = $true) & (woz13rule0 = $true) & (breln1unionE = $true) & (image1Equiv = $true) & (setunion__Cong = $true) & (image1E = $true) & (iffalse = $true) & (binintersectER = $true) & (singletonsswitch = $true) & (binintersectTERcontra = $true) & (iffalseProp1 = $true) & (ubforcartprodlem1 = $true) & (breln1SetBreln1 = $true) & (woz13rule4 = $true) & (inIntersectImpInUnion = $true) & (powersetE1 = $true) & (binunionRsub = $true) & (brelnall2 = $true) & (notinself2 = $true) & (doubleComplementI1 = $true) & (emptysetsubset = $true) & (upairsubunion = $true) & (injFuncSetFuncInj = $true) & (complementTI1 = $true) & (setminusSubset2 = $true) & (binunionTILcontra = $true) & (setext = $true) & (binunionIR = $true) & (setukpairinjR12 = $true) & (breln1compE = $true) & (setukpairinjL = $true) & (setukpairIR = $true) & (setukpairinjL1 = $true) & (subsetI2 = $true) & (setunionsingleton = $true) & (foundation2 = $true) & (funcinfuncset = $true) & (binunionE = $true) & (ex1E2 = $true) & (iftrueProp1 = $true) & (powersetI1 = $true) & (funcImageSingleton = $true) & (cartprodmempair = $true) & (wellorderingAx = $true) & (notequalI1 = $true) & (demorgan1a = $true) & (replAx = $true) & (dsetconstrI = $true) & (foundationAx = $true) & (kfstsingleton = $true) & (lam2p = $true) & (demorgan1 = $true) & (quantDeMorgan3 = $true) & (subsetE = $true) & (inIntersectImpInUnion2 = $true) & (complementTE1 = $true) & (iftrue = $true) & (woz13rule3 = $true) & (injFuncSetFuncIn = $true) & (breln1unionIL = $true) & (contrasubsetT3 = $true) & (setukpairinjR2 = $true) & (subset2powerset = $true) & (sepSubset = $true) & (complementUnionInPowersetComplement = $true) & (eqimpsubset1 = $true) & (doubleComplementE1 = $true) & (subsetI1 = $true) & (woz2B = $true) & (powersetT_lem = $true) & (setukpairinjR1 = $true) & (woz1_5 = $true) & (beta2 = $true) & (woz2Ex = $true) & (apProp = $true) & (cartprodmempair1 = $true) & (subsetE2 = $true) & (omega__Cong = $true) & (subbreln1 = $true) & (prop2setE = $true) & (omegaSAx = $true) & (powersetTI1 = $true) & (binintersectSubset1 = $true) & (image1Ex = $true) & (nonemptyI1 = $true) & (notinself = $true) & (doubleComplementSub1 = $true) & (binunionLsub = $true) & (app = $true) & (nonemptyE1 = $true) & (emptyinPowerset = $true) & (upairsetIL = $true) & (eta1 = $true) & (setOfPairsIsBReln1 = $true) & (notinemptyset = $true) & (setminusER = $true) & (emptyinunitempty = $true) & (quantDeMorgan4 = $true) & (complementSubsetComplementIntersect = $true) & (symdiffIneg2 = $true) & (setminusSubset1 = $true) & (funcGraphProp2 = $true) & (subbreln = $true) & (doubleComplementEq = $true) & (inIntersectImpInIntersectUnions = $true) & (surjFuncSetFuncIn = $true) & (exuE3u = $true) & (dpsetconstrI = $true) & (demorgan2b = $true) & (breln1invE = $true) & (subsetTI = $true) & (cartprodpairsurjEq = $true) & (emptysetE = $true) & (binunionTIRcontra = $true) & ? [X0 : $i] : (((in @ X0 @ omega) = $true) & ((in @ (omegaS @ X0) @ omega) != $true)) & (setadjoinAx = $true) & (breln1unionprop = $true) & (dsetconstrER = $true) & (upairset2E = $true) & (prop2setI = $true) & (setadjoinSub2 = $true) & (demorgan2b2 = $true) & (theeq = $true) & (setminusEL = $true) & (upairsetE = $true) & (woz1_4 = $true) & (binintersectLsub = $true) & (emptyI = $true) & (complementTnotintersectT = $true) & (complementInPowersetComplementIntersect = $true) & (funcGraphProp3 = $true) & (powersetE = $true) & (setadjoinIR = $true) & (emptysetimpfalse = $true) & (exuE2 = $true) & (setbeta = $true) & (contrasubsetT1 = $true) & (setadjoin__Cong = $true) & (woz13rule1 = $true) & (binunionTE = $true) & (breln1unionIR = $true) & (subPowSU = $true) & (binintersectTELcontra = $true) & (setminusELneg = $true) & (binintersectSubset5 = $true) & (quantDeMorgan1 = $true) & (exuI2 = $true) & (eqimpsubset2 = $true) & (disjointsetsI1 = $true) & (injFuncInInjFuncSet = $true) & (beta1 = $true) & (binintersectRsub = $true) & (setukpairinjR = $true) & (brelnall1 = $true) & (exuE1 = $true) & (exuI1 = $true) & (setadjoinE = $true) & (kpairp = $true) & (surjFuncSetFuncSurj = $true) & (setunionAx = $true) & (breln1compprop = $true) & (iftrueorfalse = $true) & (emptyInPowerset = $true) & (dpsetconstrEL1 = $true) & (inCongP = $true) & (inComplementUnionImpNotIn1 = $true) & (iffalseProp2 = $true) & (inPowerset = $true) & (sepInPowerset = $true) & (breln1invprop = $true) & (setminusILneg = $true) & (eta2 = $true) & (ap2apEq1 = $true) & (lamProp = $true) & (powersetsubset = $true) & (leftInvIsSurj = $true) & (dpsetconstrSub = $true) & (subsetRefl = $true) & (nonemptyI = $true) & (setunionE2 = $true) & (lam2lamEq = $true) & (funcext2 = $true) & (subsetTrans = $true) & (singletonprop = $true) & (ex1E1 = $true) & (singletonsuniq = $true) & (ksndpairEq = $true) & (breln1invI = $true) & (demorgan2 = $true) & (woz13rule2 = $true) & (binintersectI = $true) & (singletoninpowunion = $true) & (kfstpairEq = $true) & (notequalI2 = $true) & (funcextLem = $true) & (ifp = $true) & (cartprodfstin = $true) & (cartprodmempaircEq = $true) & (binintersectSubset3 = $true) & (prop2set2propI = $true) & (funcGraphProp1 = $true) & (descrp = $true) & (contrasubsetT = $true) & (ifSingleton = $true) & (binintersectT_lem = $true) & (ksndsingleton = $true) & (breln1unionI = $true) & (breln1unionEcases = $true) & (eqinunit = $true) & (notsubsetI = $true) & (setextAx = $true) & (contrasubsetT2 = $true) & (binunionTEcontra = $true) & (upairinpowunion = $true) & (infuncsetfunc = $true) & (breln1compEex = $true) & (dpsetconstrEL2 = $true) & (notinsingleton = $true) & (ap2p = $true) & (binunionIL = $true) & (eqbreln = $true) & (singletonsubset = $true) & (in__Cong = $true) & (woz1_1 = $true) & (setunionsingleton2 = $true) & (noeltsimpempty = $true) & (dpsetconstrER = $true) & (upairsetIR = $true) & (breln1unionCommutes = $true) & (breln1compI = $true) & (dsetconstrEL = $true) & (subsetemptysetimpeq = $true)),
% 6.77/2.19 inference(flattening,[status(thm)],[f1129])).
% 6.77/2.19 thf(f1129,plain,(
% 6.77/2.19 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (((in @ X0 @ omega) = $true) & ((in @ (omegaS @ X0) @ omega) != $true)) & (notinself2 = $true)) & (notinself = $true)) & (foundation2 = $true)) & (surjCantorThm = $true)) & (leftInvIsSurj = $true)) & (surjFuncSetFuncSurj = $true)) & (surjFuncSetFuncIn = $true)) & (injFuncSetFuncInj = $true)) & (injFuncSetFuncIn = $true)) & (injFuncInInjFuncSet = $true)) & (image1I = $true)) & (image1E = $true)) & (image1Equiv = $true)) & (image1Ex1 = $true)) & (image1Ex = $true)) & (woz2B = $true)) & (woz2A = $true)) & (woz2W = $true)) & (woz2Ex = $true)) & (breln1unionCommutes = $true)) & (breln1unionEcases = $true)) & (breln1unionE = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 6.77/2.19 inference(ennf_transformation,[status(thm)],[f812])).
% 6.77/2.19 thf(f812,plain,(
% 6.77/2.19 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ((breln1unionE = $true) => ((breln1unionEcases = $true) => ((breln1unionCommutes = $true) => ((woz2Ex = $true) => ((woz2W = $true) => ((woz2A = $true) => ((woz2B = $true) => ((image1Ex = $true) => ((image1Ex1 = $true) => ((image1Equiv = $true) => ((image1E = $true) => ((image1I = $true) => ((injFuncInInjFuncSet = $true) => ((injFuncSetFuncIn = $true) => ((injFuncSetFuncInj = $true) => ((surjFuncSetFuncIn = $true) => ((surjFuncSetFuncSurj = $true) => ((leftInvIsSurj = $true) => ((surjCantorThm = $true) => ((foundation2 = $true) => ((notinself = $true) => ((notinself2 = $true) => ! [X0 : $i] : (((in @ X0 @ omega) = $true) => ((in @ (omegaS @ X0) @ omega) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.77/2.19 inference(fool_elimination,[status(thm)],[f811])).
% 6.77/2.19 thf(f811,plain,(
% 6.77/2.19 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => ! [X0 : $i] : ((in @ X0 @ omega) => (in @ (omegaS @ X0) @ omega))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.77/2.19 inference(rectify,[status(thm)],[f329])).
% 6.77/2.19 thf(f329,negated_conjecture,(
% 6.77/2.19 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => ! [X1 : $i] : ((in @ X1 @ omega) => (in @ (omegaS @ X1) @ omega))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.77/2.19 inference(negated_conjecture,[status(cth)],[f328])).
% 6.77/2.19 thf(f328,conjecture,(
% 6.77/2.19 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => ! [X1 : $i] : ((in @ X1 @ omega) => (in @ (omegaS @ X1) @ omega)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.77/2.19 file('/export/starexec/sandbox/tmp/tmp.KvMWiWE9BC/DTF2THF_28453.p',omegaSp)).
% 6.77/2.19 thf(f8477,plain,(
% 6.77/2.19 ($true != (in @ sK1162 @ omega))),
% 6.77/2.19 inference(trivial_inequality_removal,[status(thm)],[f8471])).
% 6.77/2.19 thf(f8471,plain,(
% 6.77/2.19 ($true != $true) | ($true != (in @ sK1162 @ omega))),
% 6.77/2.19 inference(superposition,[status(thm)],[f7003,f6780])).
% 6.77/2.19 thf(f6780,plain,(
% 6.77/2.19 ( ! [X1 : $i] : (((in @ (setadjoin @ X1 @ X1) @ omega) = $true) | ((in @ X1 @ omega) != $true)) )),
% 6.77/2.19 inference(trivial_inequality_removal,[status(thm)],[f6187])).
% 6.77/2.19 thf(f6187,plain,(
% 6.77/2.19 ( ! [X1 : $i] : (((in @ (setadjoin @ X1 @ X1) @ omega) = $true) | ($true != $true) | ((in @ X1 @ omega) != $true)) )),
% 6.77/2.19 inference(definition_unfolding,[status(thm)],[f4525,f5226])).
% 6.77/2.19 thf(f5226,plain,(
% 6.77/2.19 (omegaSAx = $true)),
% 6.77/2.19 inference(cnf_transformation,[status(thm)],[f3438])).
% 6.77/2.19 thf(f4525,plain,(
% 6.77/2.19 ( ! [X1 : $i] : (((in @ X1 @ omega) != $true) | ((in @ (setadjoin @ X1 @ X1) @ omega) = $true) | (omegaSAx != $true)) )),
% 6.77/2.19 inference(cnf_transformation,[status(thm)],[f2820])).
% 6.77/2.19 thf(f2820,plain,(
% 6.77/2.19 ((omegaSAx = $true) | (($true = (in @ sK808 @ omega)) & ($true != (in @ (setadjoin @ sK808 @ sK808) @ omega)))) & (! [X1 : $i] : (((in @ X1 @ omega) != $true) | ((in @ (setadjoin @ X1 @ X1) @ omega) = $true)) | (omegaSAx != $true))),
% 6.77/2.19 inference(skolemisation,[status(esa),new_symbols(skolem,[sK808])],[f2818,f2819])).
% 6.77/2.19 thf(f2819,plain,(
% 6.77/2.19 ? [X0 : $i] : (((in @ X0 @ omega) = $true) & ($true != (in @ (setadjoin @ X0 @ X0) @ omega))) => (($true = (in @ sK808 @ omega)) & ($true != (in @ (setadjoin @ sK808 @ sK808) @ omega)))),
% 6.77/2.19 introduced(definition,[],[choice_axiom])).
% 6.77/2.19 thf(f2818,plain,(
% 6.77/2.19 ((omegaSAx = $true) | ? [X0 : $i] : (((in @ X0 @ omega) = $true) & ($true != (in @ (setadjoin @ X0 @ X0) @ omega)))) & (! [X1 : $i] : (((in @ X1 @ omega) != $true) | ((in @ (setadjoin @ X1 @ X1) @ omega) = $true)) | (omegaSAx != $true))),
% 6.77/2.19 inference(rectify,[status(thm)],[f2817])).
% 6.77/2.19 thf(f2817,plain,(
% 6.77/2.19 ((omegaSAx = $true) | ? [X0 : $i] : (((in @ X0 @ omega) = $true) & ($true != (in @ (setadjoin @ X0 @ X0) @ omega)))) & (! [X0 : $i] : (((in @ X0 @ omega) != $true) | ($true = (in @ (setadjoin @ X0 @ X0) @ omega))) | (omegaSAx != $true))),
% 6.77/2.19 inference(nnf_transformation,[status(thm)],[f1314])).
% 6.77/2.19 thf(f1314,plain,(
% 6.77/2.19 (omegaSAx = $true) <=> ! [X0 : $i] : (((in @ X0 @ omega) != $true) | ($true = (in @ (setadjoin @ X0 @ X0) @ omega)))),
% 6.77/2.19 inference(ennf_transformation,[status(thm)],[f788])).
% 6.77/2.19 thf(f788,plain,(
% 6.77/2.19 (omegaSAx = $true) <=> ! [X0 : $i] : (((in @ X0 @ omega) = $true) => ($true = (in @ (setadjoin @ X0 @ X0) @ omega)))),
% 6.77/2.19 inference(fool_elimination,[status(thm)],[f787])).
% 6.77/2.19 thf(f787,plain,(
% 6.77/2.19 (! [X0 : $i] : ((in @ X0 @ omega) => (in @ (setadjoin @ X0 @ X0) @ omega)) = omegaSAx)),
% 6.77/2.19 inference(rectify,[status(thm)],[f8])).
% 6.77/2.19 thf(f8,axiom,(
% 6.77/2.19 (! [X1 : $i] : ((in @ X1 @ omega) => (in @ (setadjoin @ X1 @ X1) @ omega)) = omegaSAx)),
% 6.77/2.19 file('/export/starexec/sandbox/tmp/tmp.KvMWiWE9BC/DTF2THF_28453.p',omegaSAx)).
% 6.77/2.19 thf(f7003,plain,(
% 6.77/2.19 ($true != (in @ (setadjoin @ sK1162 @ sK1162) @ omega))),
% 6.77/2.19 inference(beta_eta_normalization,[status(thm)],[f6602])).
% 6.77/2.19 thf(f6602,plain,(
% 6.77/2.19 ($true != (in @ ((^[Y0 : $i]: (setadjoin @ Y0 @ Y0)) @ sK1162) @ omega))),
% 6.77/2.19 inference(definition_unfolding,[status(thm)],[f5191,f3935])).
% 6.77/2.19 thf(f3935,plain,(
% 6.77/2.19 (omegaS = (^[Y0 : $i]: (setadjoin @ Y0 @ Y0)))),
% 6.77/2.19 inference(cnf_transformation,[status(thm)],[f800])).
% 6.77/2.19 thf(f800,plain,(
% 6.77/2.19 (omegaS = (^[Y0 : $i]: (setadjoin @ Y0 @ Y0)))),
% 6.77/2.19 inference(fool_elimination,[status(thm)],[f799])).
% 6.77/2.19 thf(f799,plain,(
% 6.77/2.19 (omegaS = (^[X0 : $i] : (setadjoin @ X0 @ X0)))),
% 6.77/2.19 inference(rectify,[status(thm)],[f327])).
% 6.77/2.19 thf(f327,axiom,(
% 6.77/2.19 (omegaS = (^[X1 : $i] : (setadjoin @ X1 @ X1)))),
% 6.77/2.19 file('/export/starexec/sandbox/tmp/tmp.KvMWiWE9BC/DTF2THF_28453.p',omegaS)).
% 6.77/2.19 thf(f5191,plain,(
% 6.77/2.19 ($true != (in @ (omegaS @ sK1162) @ omega))),
% 6.77/2.19 inference(cnf_transformation,[status(thm)],[f3438])).
% 6.77/2.19 % SZS output end Proof for DTF2THF_28453
% 6.77/2.19 % (28725)------------------------------
% 6.77/2.19 % (28725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.77/2.19 % (28725)Termination reason: Refutation
% 6.77/2.19
% 6.77/2.19 % (28725)Memory used [KB]: 12025
% 6.77/2.19 % (28725)Time elapsed: 0.252 s
% 6.77/2.19 % (28725)Instructions burned: 527 (million)
% 6.77/2.19 % (28725)------------------------------
% 6.77/2.19 % (28725)------------------------------
% 6.77/2.19 % (28644)Success in time 0.665 s
%------------------------------------------------------------------------------