%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU821^1 : 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 : n011.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 09:37:29 AM UTC 2026
% Result : Theorem 3.09s 1.81s
% Output : Refutation 3.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 5
% Syntax : Number of formulae : 24 ( 6 unt; 0 typ; 0 def)
% Number of atoms : 6250 (1737 equ; 0 cnn)
% Maximal formula atoms : 340 ( 260 avg)
% Number of connectives : 2848 ( 24 ~; 8 |;1026 &; 90 @)
% ( 2 <=>;1698 =>; 0 <=; 0 <~>)
% Maximal formula depth : 346 ( 116 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 592 ( 589 usr; 342 con; 0-6 aty)
% Number of variables : 31 ( 0 ^; 20 !; 11 ?; 31 :)
% 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_382,type,
peano3: $i > $i > $i > $o ).
thf(func_def_383,type,
peano4: $i > $i > $i > $o ).
thf(func_def_384,type,
peano5: $i > $i > $i > $o ).
thf(func_def_386,type,
peano: $i > $i > $i > $o ).
thf(func_def_387,type,
transitiveset: $i > $o ).
thf(func_def_392,type,
stricttotalorderedByIn: $i > $o ).
thf(func_def_393,type,
wellorderedByIn: $i > $o ).
thf(func_def_394,type,
ordinal: $i > $o ).
thf(func_def_395,type,
limitOrdinal: $i > $o ).
thf(func_def_400,type,
succOrdinal: $i > $o ).
thf(func_def_421,type,
sP4: $i > $i > $o ).
thf(func_def_422,type,
sP5: $i > $o ).
thf(func_def_423,type,
sP6: $i > $i > $o ).
thf(func_def_424,type,
sP7: $i > $i > $o ).
thf(func_def_425,type,
sP8: $i > $i > $i > $o ).
thf(func_def_428,type,
sP11: $i > $i > $i > $o > $o ).
thf(func_def_430,type,
sP13: $i > $i > $i > $i > $o ).
thf(func_def_434,type,
sP17: $i > $i > $i > $o ).
thf(func_def_439,type,
sK22: $i > $i > $i ).
thf(func_def_456,type,
sK39: $i > $o ).
thf(func_def_471,type,
sK54: $i > $i > $i > $i > $i > $i ).
thf(func_def_473,type,
sK56: $i > $i ).
thf(func_def_474,type,
sK57: $i > $i ).
thf(func_def_475,type,
sK58: $i > $i ).
thf(func_def_476,type,
sK59: ( $i > $i ) > $i > $i ).
thf(func_def_477,type,
sK60: $i > ( $i > $i ) > $i > $i ).
thf(func_def_478,type,
sK61: ( $i > $o ) > $i > $i ).
thf(func_def_480,type,
sK63: $i > $o ).
thf(func_def_499,type,
sK82: $i > $i ).
thf(func_def_501,type,
sK84: ( $i > $o ) > $i ).
thf(func_def_502,type,
sK85: $i > $o ).
thf(func_def_508,type,
sK91: $i > $i > $o ).
thf(func_def_512,type,
sK95: $i > $i > $o ).
thf(func_def_515,type,
sK98: $i > $i > $i > $i ).
thf(func_def_538,type,
sK121: $i > $o ).
thf(func_def_539,type,
sK122: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_540,type,
sK123: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_544,type,
sK127: $i > $o ).
thf(func_def_569,type,
sK152: $i > $o ).
thf(func_def_570,type,
sK153: ( $i > $i ) > $i > $i > $i ).
thf(func_def_573,type,
sK156: $i > $i ).
thf(func_def_581,type,
sK164: $i > $i > $i > $i > $i ).
thf(func_def_600,type,
sK183: $i > $o ).
thf(func_def_614,type,
sK197: $i > $i ).
thf(func_def_617,type,
sK200: $i > $o ).
thf(func_def_620,type,
sK203: ( $i > $o ) > $i ).
thf(func_def_621,type,
sK204: $i > $o ).
thf(func_def_622,type,
sK205: $i > $i ).
thf(func_def_646,type,
sK229: $i > $o ).
thf(func_def_652,type,
sK235: $i > $o ).
thf(func_def_671,type,
sK254: $i > $o ).
thf(func_def_672,type,
sK255: $i > $o ).
thf(func_def_673,type,
sK256: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_674,type,
sK257: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_693,type,
sK276: $o > $i > $i > $i > $i > $i > $i ).
thf(func_def_697,type,
sK280: $i > $o ).
thf(func_def_700,type,
sK283: $i > ( $i > $o ) > $i ).
thf(func_def_701,type,
sK284: $i > ( $i > $o ) > $i ).
thf(func_def_702,type,
sK285: $i > $i ).
thf(func_def_708,type,
sK291: $i > $o ).
thf(func_def_714,type,
sK297: $i > $i > $i ).
thf(func_def_715,type,
sK298: $i > $i ).
thf(func_def_718,type,
sK301: $i > $i > ( $i > $i ) > $i ).
thf(func_def_730,type,
sK313: $i > $i > $o ).
thf(func_def_749,type,
sK332: $i > ( $i > $o ) > $i ).
thf(func_def_750,type,
sK333: $i > $o ).
thf(func_def_752,type,
sK335: $i > $i > $i ).
thf(func_def_775,type,
sK358: $i > $i > $i > $i > $i ).
thf(func_def_776,type,
sK359: $i > $i > $i > $i > $i ).
thf(func_def_781,type,
sK364: $i > $o ).
thf(func_def_784,type,
sK367: $i > ( $i > $o ) > $i ).
thf(func_def_814,type,
sK397: $i > $i > $i > $i ).
thf(func_def_818,type,
sK401: $i > ( $i > $o ) > $i ).
thf(func_def_819,type,
sK402: $i > $o ).
thf(func_def_826,type,
sK409: $i > $i > $i ).
thf(func_def_827,type,
sK410: $i > $i > $i ).
thf(func_def_828,type,
sK411: $i > $i > $i ).
thf(func_def_829,type,
sK412: $i > $i > $i ).
thf(func_def_830,type,
sK413: $i > $i > $i ).
thf(func_def_831,type,
sK414: $i > $i > $i > $i ).
thf(func_def_832,type,
sK415: $i > $i ).
thf(func_def_833,type,
sK416: $i > $i ).
thf(func_def_834,type,
sK417: $i > $i ).
thf(func_def_835,type,
sK418: $i > $i ).
thf(func_def_836,type,
sK419: $i > $i > $i ).
thf(func_def_837,type,
sK420: $i > $i > $i > $i ).
thf(func_def_838,type,
sK421: $i > $i > $i > $i ).
thf(func_def_839,type,
sK422: $i > $i > $i ).
thf(func_def_840,type,
sK423: $i > $i > $i ).
thf(func_def_841,type,
sK424: $i > $i ).
thf(func_def_843,type,
sK426: $i > $i ).
thf(func_def_844,type,
sK427: $i > $i ).
thf(func_def_852,type,
sK435: $i > ( $i > $o ) > $i ).
thf(func_def_853,type,
sK436: $i > $o ).
thf(func_def_860,type,
sK443: $i > $i > $o ).
thf(func_def_867,type,
sK450: $i > ( $i > $i ) > $i > $i ).
thf(func_def_869,type,
sK452: $i > $i ).
thf(func_def_873,type,
sK456: $i > $i > $i ).
thf(func_def_888,type,
sK471: $i > $i > $i ).
thf(func_def_893,type,
sK476: $i > $i > ( $i > $i ) > $i ).
thf(func_def_894,type,
sK477: $i > $i ).
thf(func_def_898,type,
sK481: $i > $i > ( $i > $i ) > $i ).
thf(func_def_899,type,
sK482: $i > $i ).
thf(func_def_910,type,
sK493: $i > $i > $i > $i ).
thf(func_def_911,type,
sK494: $i > $i > $i > $i ).
thf(func_def_912,type,
sK495: $i > $i > $i > $i ).
thf(func_def_913,type,
sK496: $i > $i > $i > $i ).
thf(func_def_947,type,
sK530: $i > $i ).
thf(func_def_951,type,
sK534: ( $i > $o ) > $i > $i > $i ).
thf(func_def_952,type,
sK535: ( $i > $o ) > $i > $i > $i ).
thf(func_def_955,type,
sK538: $i > $o ).
thf(func_def_958,type,
sK541: $i > $i > $o ).
thf(func_def_959,type,
sK542: $i > $i ).
thf(func_def_960,type,
sK543: $i > $i ).
thf(func_def_961,type,
sK544: ( $i > $i > $o ) > $i > $i ).
thf(func_def_962,type,
sK545: ( $i > $i > $o ) > $i > $i ).
thf(func_def_963,type,
sK546: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_965,type,
sK548: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_966,type,
sK549: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_970,type,
sK553: $i > $o ).
thf(func_def_974,type,
sK557: $i > $i ).
thf(func_def_990,type,
sK573: ( $i > $o ) > $i > $i > $i ).
thf(func_def_991,type,
sK574: ( $i > $o ) > $i > $i > $i ).
thf(func_def_994,type,
sK577: $i > $o ).
thf(func_def_997,type,
sK580: $i > $i ).
thf(func_def_1015,type,
sK598: $i > $i > $o ).
thf(func_def_1021,type,
sK604: $i > $i > $o ).
thf(func_def_1024,type,
sK607: $i > $i ).
thf(func_def_1026,type,
sK609: $i > $i ).
thf(func_def_1037,type,
sK620: $o > $i > $i > $i ).
thf(func_def_1048,type,
sK631: $i > $i ).
thf(func_def_1051,type,
sK634: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1052,type,
sK635: $i > $o ).
thf(func_def_1053,type,
sK636: $i > $i ).
thf(func_def_1054,type,
sK637: ( $i > $o ) > $i ).
thf(func_def_1068,type,
sK651: $i > $i > $i > $i ).
thf(func_def_1069,type,
sK652: $i > $i > $i > $i ).
thf(func_def_1080,type,
sK663: $i > $o ).
thf(func_def_1088,type,
sK671: $i > $i > $o ).
thf(func_def_1093,type,
sK676: $i > $o ).
thf(func_def_1094,type,
sK677: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1118,type,
sK701: $i > $i ).
thf(func_def_1125,type,
sK708: $i > $i > $i ).
thf(func_def_1141,type,
sK724: $i > $i > $o ).
thf(func_def_1164,type,
sK747: $i > ( $i > $o ) > $i ).
thf(func_def_1165,type,
sK748: $i > $o ).
thf(func_def_1177,type,
sK760: $i > $i ).
thf(func_def_1183,type,
sK766: $i > $i > $i ).
thf(func_def_1184,type,
sK767: $i > $i > $i ).
thf(func_def_1187,type,
sK770: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1189,type,
sK772: $i > $i ).
thf(func_def_1203,type,
sK786: ( $i > $o ) > $i > $i ).
thf(func_def_1205,type,
sK788: $i > $o ).
thf(func_def_1216,type,
sK799: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1218,type,
sK801: $i > $i > $o ).
thf(func_def_1222,type,
sK805: $i > $i ).
thf(func_def_1228,type,
sK811: $i > $i > $i > $i > $i ).
thf(func_def_1242,type,
sK825: $i > $o ).
thf(func_def_1264,type,
sK847: $i > $o ).
thf(func_def_1319,type,
sK902: $i > $o ).
thf(func_def_1322,type,
sK905: $i > ( $i > $o ) > $i ).
thf(func_def_1329,type,
sK912: $i > $i ).
thf(func_def_1333,type,
sK916: $i > $i > $i > $i ).
thf(func_def_1334,type,
sK917: $i > $i > $i > $i ).
thf(func_def_1342,type,
sK925: $i > $o ).
thf(func_def_1349,type,
sK932: $i > $i ).
thf(func_def_1352,type,
sK935: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1359,type,
sK942: $i > $i ).
thf(func_def_1378,type,
sK961: $i > $i > $i > $i > $i ).
thf(func_def_1379,type,
sK962: $i > $o ).
thf(func_def_1380,type,
sK963: $i > $o ).
thf(func_def_1381,type,
sK964: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1382,type,
sK965: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1396,type,
sK979: $i > $i > $i > $i > $i ).
thf(func_def_1397,type,
sK980: $i > $i > $i > $i > $i ).
thf(func_def_1398,type,
sK981: $i > $i > $i > $i > $i ).
thf(func_def_1399,type,
sK982: $i > $i > $i > $i > $i ).
thf(func_def_1414,type,
sK997: $i > $o ).
thf(func_def_1416,type,
sK999: ( $i > $o ) > $i > $i ).
thf(func_def_1417,type,
sK1000: ( $i > $o ) > $i > $i ).
thf(func_def_1451,type,
sK1034: $i > $i > $i > $i ).
thf(func_def_1452,type,
sK1035: $i > $i > $i > $i ).
thf(func_def_1471,type,
sK1054: $i > $i > $i ).
thf(func_def_1474,type,
sK1057: $i > $o ).
thf(func_def_1475,type,
sK1058: $i > $o ).
thf(func_def_1476,type,
sK1059: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1477,type,
sK1060: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1490,type,
sK1073: $i > $o ).
thf(func_def_1493,type,
sK1076: ( $i > $o ) > $i > $i ).
thf(func_def_1495,type,
sK1078: $i > $o ).
thf(func_def_1500,type,
sK1083: $i > $i ).
thf(func_def_1502,type,
sK1085: $i > ( $i > $i ) > $i > $i > $i ).
thf(func_def_1503,type,
sK1086: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1515,type,
sK1098: $i > $i > $i ).
thf(func_def_1525,type,
sK1108: ( $i > $o ) > $i ).
thf(func_def_1526,type,
sK1109: ( $i > $o ) > $i ).
thf(func_def_1527,type,
sK1110: $i > $o ).
thf(f7295,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f4827,f6763]) ).
thf(f6763,plain,
! [X1: $i] :
( $true
!= ( in @ X1 @ X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f5100]) ).
thf(f5100,plain,
! [X1: $i] :
( ( $true != $true )
| ( $true
!= ( in @ X1 @ X1 ) ) ),
inference(definition_unfolding,[status(thm)],[f3237,f5021]) ).
thf(f5021,plain,
notinself = $true,
inference(cnf_transformation,[status(thm)],[f3222]) ).
thf(f3222,plain,
( ( breln1all1 = $true )
& ( quantDeMorgan2 = $true )
& ( subsetI2 = $true )
& ( lam2p = $true )
& ( woz13rule2 = $true )
& ( eqinunit = $true )
& ( setminusILneg = $true )
& ( setadjoinIR = $true )
& ( complementTI1 = $true )
& ( binunionTEcontra = $true )
& ( inIntersectImpInUnion = $true )
& ( inIntersectImpInUnion2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( setOfPairsIsBReln = $true )
& ( setminusI = $true )
& ( dpsetconstrEL2 = $true )
& ( funcGraphProp1 = $true )
& ( app = $true )
& ( breln1unionCommutes = $true )
& ( inPowerset = $true )
& ( nonemptyImpWitness = $true )
& ( setukpairinjR12 = $true )
& ( breln1all2 = $true )
& ( setbeta = $true )
& ( demorgan2a = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( powersetAx = $true )
& ( sepSubset = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( exuE3u = $true )
& ( subsetE = $true )
& ( subset2powerset = $true )
& ( exuEu = $true )
& ( breln1invprop = $true )
& ( demorgan1b = $true )
& ( powersetTI1 = $true )
& ( iffalse = $true )
& ( ubforcartprodlem2 = $true )
& ( binintersectI = $true )
& ( ubforcartprodlem1 = $true )
& ( powersetE = $true )
& ( singletonsuniq = $true )
& ( replAx = $true )
& ( setadjoinE = $true )
& ( foundationAx = $true )
& ( binunionLsub = $true )
& ( ap2apEq1 = $true )
& ( image1I = $true )
& ( emptyinPowerset = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( upairset2E = $true )
& ( demorgan2a2 = $true )
& ( ordinalMinLem1 = $true )
& ( emptysetAx = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( nonemptyI = $true )
& ( woz2B = $true )
& ( symdiffE = $true )
& ( cartprodmempaircEq = $true )
& ( kpairp = $true )
& ( notinself = $true )
& ( emptyE1 = $true )
& ( emptysetE = $true )
& ( setminusIRneg = $true )
& ( cartprodpairsurjEq = $true )
& ( woz13rule1 = $true )
& ( binunionTILcontra = $true )
& ( exu__Cong = $true )
& ( nonemptyE1 = $true )
& ( upairset2IR = $true )
& ( symdiffI2 = $true )
& ( setadjoinIL = $true )
& ( contrasubsetT3 = $true )
& ( transitivesetOp2 = $true )
& ( binunionIR = $true )
& ( funcGraphProp3 = $true )
& ( emptyI = $true )
& ( cartprodmempair = $true )
& ( setextAx = $true )
& ( injFuncInInjFuncSet = $true )
& ( doubleComplementSub1 = $true )
& ( setminusELneg = $true )
& ( iftrueProp1 = $true )
& ( notequalI1 = $true )
& ( setminusEL = $true )
& ( doubleComplementE1 = $true )
& ( transitivesetOp1 = $true )
& ( woz2A = $true )
& ( brelnall2 = $true )
& ( breln1compE = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( setunionTransitive = $true )
& ( woz13rule0 = $true )
& ( dpsetconstrERa = $true )
& ( setunionsingleton1 = $true )
& ( contrasubsetT2 = $true )
& ( notdallE = $true )
& ( dpsetconstrI = $true )
& ( quantDeMorgan4 = $true )
& ( setextsub = $true )
& ( sepInPowerset = $true )
& ( subsetI1 = $true )
& ( setminusSubset2 = $true )
& ( breln1compEex = $true )
& ( breln1unionIL = $true )
& ( theeq = $true )
& ( eqimpsubset2 = $true )
& ( descrp = $true )
& ( binintersectEL = $true )
& ( setunionE = $true )
& ( ifp = $true )
& ( woz1_5 = $true )
& ( binintersectSubset3 = $true )
& ( surjFuncSetFuncIn = $true )
& ( cartprodsndin = $true )
& ( lamp = $true )
& ( exuI1 = $true )
& ( cartprodpairin = $true )
& ( powerset__Cong = $true )
& ( singletonsswitch = $true )
& ( funcGraphProp2 = $true )
& ( eqbreln = $true )
& ( binintersectRsub = $true )
& ( eta2 = $true )
& ( ap2apEq2 = $true )
& ( kfstsingleton = $true )
& ( demorgan1 = $true )
& ( upairsetE = $true )
& ( setminusT_lem = $true )
& ( exuE3e = $true )
& ( prop2setI = $true )
& ( cartprodpairmemER = $true )
& ( subsetE2 = $true )
& ( image1Equiv = $true )
& ( setukpairinjL = $true )
& ( setextT = $true )
& ( dsetconstrER = $true )
& ( binintersectSubset5 = $true )
& ( omega__Cong = $true )
& ( ex1E1 = $true )
& ( woz13rule4 = $true )
& ( binunionT_lem = $true )
& ( eta1 = $true )
& ( setunionsingleton2 = $true )
& ( ifSingleton = $true )
& ( complementTcontraSubset = $true )
& ( ordinalTransSet1 = $true )
& ( funcext2 = $true )
& ( setadjoinAx = $true )
& ( setukpairinjR11 = $true )
& ( emptysetOrdinal = $true )
& ( setminusSubset1 = $true )
& ( setminusERneg = $true )
& ( dpsetconstrSub = $true )
& ( disjointsetsI1 = $true )
& ( binunionIL = $true )
& ( doubleComplementSub2 = $true )
& ( prop2setE = $true )
& ( peanoSinj = $true )
& ( apProp = $true )
& ( ex1I2 = $true )
& ( cartprodsndpairEq = $true )
& ( woz1_4 = $true )
& ( subbreln = $true )
& ( subPowSU = $true )
& ( inCongP = $true )
& ( breln1invE = $true )
& ( binunionEcases = $true )
& ( binunionE = $true )
& ( setadjoinSub = $true )
& ( binintTransitive = $true )
& ( contraSubsetComplement = $true )
& ( infuncsetfunc = $true )
& ( funcextLem = $true )
& ( powersetsubset = $true )
& ( ubforcartprodlem3 = $true )
& ( quantDeMorgan1 = $true )
& ( complementTnotintersectT = $true )
& ( ex1I = $true )
& ( ap2p = $true )
& ( bs114d = $true )
& ( injFuncSetFuncIn = $true )
& ( setukpairinjL2 = $true )
& ( powersetE1 = $true )
& ( image1Ex1 = $true )
& ( setukpairinjR = $true )
& ( cartprodfstpairEq = $true )
& ( image1E = $true )
& ( breln1invI = $true )
& ( setOfPairsIsBReln1 = $true )
& ( notinemptyset = $true )
& ( omegaSp = $true )
& ( breln1SetBreln1 = $true )
& ( doubleComplementEq = $true )
& ( cartprodpairmemEL = $true )
& ( woz2Ex = $true )
& ( kfstpairEq = $true )
& ( secondinupair = $true )
& ( emptyset__Cong = $true )
& ( setext = $true )
& ( theprop = $true )
& ( surjCantorThm = $true )
& ( setadjoinSub2 = $true )
& ( upairsetIR = $true )
& ( subsetRefl = $true )
& ( subsetTrans = $true )
& ( breln1unionIR = $true )
& ( binunionTE = $true )
& ( upairinpowunion = $true )
& ( binintersectSubset4 = $true )
& ( setminusER = $true )
& ( contrasubsetT1 = $true )
& ( vacuousDall = $true )
& ( kpairsurjEq = $true )
& ( funcImageSingleton = $true )
& ( symdiffIneg2 = $true )
& ( upairequniteq = $true )
& ( demorgan2b = $true )
& ( exuE1 = $true )
& ( omegaSclos = $true )
& ( binintersectTELcontra = $true )
& ( iftrueProp2 = $true )
& ( complementT_lem = $true )
& ( exuI3 = $true )
& ( lamProp = $true )
& ( breln1unionprop = $true )
& ( doubleComplementI1 = $true )
& ( setukpairinjR1 = $true )
& ( omegaIndAx = $true )
& ( setukpairIL = $true )
& ( ksndpairEq = $true )
& ( complementTE1 = $true )
& ( powersetTE1 = $true )
& ( beta1 = $true )
& ( woz1_2 = $true )
& ( lam2lamEq = $true )
& ( demorgan2 = $true )
& ( omegaSAx = $true )
& ( dsetconstr__Cong = $true )
& ( iffalseProp1 = $true )
& ( funcinfuncset = $true )
& ( notsubsetI = $true )
& ( notinsingleton = $true )
& ( upairsetIL = $true )
& ( breln1unionEcases = $true )
& ( ex1E2 = $true )
& ( woz2W = $true )
& ( prop2set2propI = $true )
& ( woz1_1 = $true )
& ( singletoninpowerset = $true )
& ( binunionRsub = $true )
& ( binintersectER = $true )
& ( dsetconstrEL = $true )
& ( ( in @ sK1113 @ sK1112 )
= $true )
& ( ( in @ sK1113 @ sK1113 )
= $true )
& ( ( ordinal @ sK1112 )
= $true )
& ( omega0Ax = $true )
& ( breln1compI = $true )
& ( quantDeMorgan3 = $true )
& ( breln1unionI = $true )
& ( setunionsingleton = $true )
& ( dpsetconstrEL1 = $true )
& ( symdiffI1 = $true )
& ( ordinalTransSet = $true )
& ( symdiffIneg1 = $true )
& ( setunionOrdinalLem1 = $true )
& ( binintersectTERcontra = $true )
& ( powersetI1 = $true )
& ( subbreln1 = $true )
& ( iffalseProp2 = $true )
& ( in__Cong = $true )
& ( notequalI2 = $true )
& ( eqbreln1 = $true )
& ( wellorderingAx = $true )
& ( peano0notS = $true )
& ( subsetTI = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setadjoin__Cong = $true )
& ( eqimpsubset1 = $true )
& ( exuE2 = $true )
& ( funcext = $true )
& ( singletoninpowunion = $true )
& ( demorgan2b2 = $true )
& ( singletonsubset = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectSubset1 = $true )
& ( binintersectLsub = $true )
& ( setunionE2 = $true )
& ( brelnall1 = $true )
& ( nonemptyI1 = $true )
& ( ksndsingleton = $true )
& ( demorgan1a = $true )
& ( surjFuncSetFuncSurj = $true )
& ( beta2 = $true )
& ( setunionAx = $true )
& ( setminusLsub = $true )
& ( foundation2 = $true )
& ( descr__Cong = $true )
& ( image1Ex = $true )
& ( binintersectT_lem = $true )
& ( choice2fnsingleton = $true )
& ( contrasubsetT = $true )
& ( powersetT_lem = $true )
& ( emptyinunitempty = $true )
& ( setunionI = $true )
& ( emptysetsubset = $true )
& ( complementImpComplementIntersect = $true )
& ( notinself2 = $true )
& ( setukpairIR = $true )
& ( woz13rule3 = $true )
& ( exuI2 = $true )
& ( setukpairinjR2 = $true )
& ( demorgan2a1 = $true )
& ( cartprodfstin = $true )
& ( cartprodmempair1 = $true )
& ( notdexE = $true )
& ( setukpairinjL1 = $true )
& ( funcGraphProp4 = $true )
& ( leftInvIsSurj = $true )
& ( breln1compprop = $true )
& ( iftrue = $true )
& ( binintersectSubset2 = $true )
& ( woz1_3 = $true )
& ( emptysetimpfalse = $true )
& ( upairsubunion = $true )
& ( powersetI = $true )
& ( breln1unionE = $true )
& ( binunionTIRcontra = $true )
& ( dsetconstrI = $true )
& ( kpairiskpair = $true )
& ( setunion__Cong = $true )
& ( injFuncSetFuncInj = $true )
& ( dpsetconstrER = $true )
& ( iftrueorfalse = $true )
& ( noeltsimpempty = $true )
& ( singletonprop = $true )
& ( uniqinunit = $true )
& ( emptyInPowerset = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1112,sK1113])],[f1087,f3221,f3220]) ).
thf(f3220,plain,
( ? [X0: $i] :
( ? [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ( $true
= ( in @ X1 @ X1 ) ) )
& ( ( ordinal @ X0 )
= $true ) )
=> ( ? [X1: $i] :
( ( $true
= ( in @ X1 @ sK1112 ) )
& ( $true
= ( in @ X1 @ X1 ) ) )
& ( ( ordinal @ sK1112 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f3221,plain,
( ? [X1: $i] :
( ( $true
= ( in @ X1 @ sK1112 ) )
& ( $true
= ( in @ X1 @ X1 ) ) )
=> ( ( ( in @ sK1113 @ sK1112 )
= $true )
& ( ( in @ sK1113 @ sK1113 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1087,plain,
( ( breln1all1 = $true )
& ( quantDeMorgan2 = $true )
& ( subsetI2 = $true )
& ( lam2p = $true )
& ( woz13rule2 = $true )
& ( eqinunit = $true )
& ( setminusILneg = $true )
& ( setadjoinIR = $true )
& ( complementTI1 = $true )
& ( binunionTEcontra = $true )
& ( inIntersectImpInUnion = $true )
& ( inIntersectImpInUnion2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( setOfPairsIsBReln = $true )
& ( setminusI = $true )
& ( dpsetconstrEL2 = $true )
& ( funcGraphProp1 = $true )
& ( app = $true )
& ( breln1unionCommutes = $true )
& ( inPowerset = $true )
& ( nonemptyImpWitness = $true )
& ( setukpairinjR12 = $true )
& ( breln1all2 = $true )
& ( setbeta = $true )
& ( demorgan2a = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( powersetAx = $true )
& ( sepSubset = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( exuE3u = $true )
& ( subsetE = $true )
& ( subset2powerset = $true )
& ( exuEu = $true )
& ( breln1invprop = $true )
& ( demorgan1b = $true )
& ( powersetTI1 = $true )
& ( iffalse = $true )
& ( ubforcartprodlem2 = $true )
& ( binintersectI = $true )
& ( ubforcartprodlem1 = $true )
& ( powersetE = $true )
& ( singletonsuniq = $true )
& ( replAx = $true )
& ( setadjoinE = $true )
& ( foundationAx = $true )
& ( binunionLsub = $true )
& ( ap2apEq1 = $true )
& ( image1I = $true )
& ( emptyinPowerset = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( upairset2E = $true )
& ( demorgan2a2 = $true )
& ( ordinalMinLem1 = $true )
& ( emptysetAx = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( nonemptyI = $true )
& ( woz2B = $true )
& ( symdiffE = $true )
& ( cartprodmempaircEq = $true )
& ( kpairp = $true )
& ( notinself = $true )
& ( emptyE1 = $true )
& ( emptysetE = $true )
& ( setminusIRneg = $true )
& ( cartprodpairsurjEq = $true )
& ( woz13rule1 = $true )
& ( binunionTILcontra = $true )
& ( exu__Cong = $true )
& ( nonemptyE1 = $true )
& ( upairset2IR = $true )
& ( symdiffI2 = $true )
& ( setadjoinIL = $true )
& ( contrasubsetT3 = $true )
& ( transitivesetOp2 = $true )
& ( binunionIR = $true )
& ( funcGraphProp3 = $true )
& ( emptyI = $true )
& ( cartprodmempair = $true )
& ( setextAx = $true )
& ( injFuncInInjFuncSet = $true )
& ( doubleComplementSub1 = $true )
& ( setminusELneg = $true )
& ( iftrueProp1 = $true )
& ( notequalI1 = $true )
& ( setminusEL = $true )
& ( doubleComplementE1 = $true )
& ( transitivesetOp1 = $true )
& ( woz2A = $true )
& ( brelnall2 = $true )
& ( breln1compE = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( setunionTransitive = $true )
& ( woz13rule0 = $true )
& ( dpsetconstrERa = $true )
& ( setunionsingleton1 = $true )
& ( contrasubsetT2 = $true )
& ( notdallE = $true )
& ( dpsetconstrI = $true )
& ( quantDeMorgan4 = $true )
& ( setextsub = $true )
& ( sepInPowerset = $true )
& ( subsetI1 = $true )
& ( setminusSubset2 = $true )
& ( breln1compEex = $true )
& ( breln1unionIL = $true )
& ( theeq = $true )
& ( eqimpsubset2 = $true )
& ( descrp = $true )
& ( binintersectEL = $true )
& ( setunionE = $true )
& ( ifp = $true )
& ( woz1_5 = $true )
& ( binintersectSubset3 = $true )
& ( surjFuncSetFuncIn = $true )
& ( cartprodsndin = $true )
& ( lamp = $true )
& ( exuI1 = $true )
& ( cartprodpairin = $true )
& ( powerset__Cong = $true )
& ( singletonsswitch = $true )
& ( funcGraphProp2 = $true )
& ( eqbreln = $true )
& ( binintersectRsub = $true )
& ( eta2 = $true )
& ( ap2apEq2 = $true )
& ( kfstsingleton = $true )
& ( demorgan1 = $true )
& ( upairsetE = $true )
& ( setminusT_lem = $true )
& ( exuE3e = $true )
& ( prop2setI = $true )
& ( cartprodpairmemER = $true )
& ( subsetE2 = $true )
& ( image1Equiv = $true )
& ( setukpairinjL = $true )
& ( setextT = $true )
& ( dsetconstrER = $true )
& ( binintersectSubset5 = $true )
& ( omega__Cong = $true )
& ( ex1E1 = $true )
& ( woz13rule4 = $true )
& ( binunionT_lem = $true )
& ( eta1 = $true )
& ( setunionsingleton2 = $true )
& ( ifSingleton = $true )
& ( complementTcontraSubset = $true )
& ( ordinalTransSet1 = $true )
& ( funcext2 = $true )
& ( setadjoinAx = $true )
& ( setukpairinjR11 = $true )
& ( emptysetOrdinal = $true )
& ( setminusSubset1 = $true )
& ( setminusERneg = $true )
& ( dpsetconstrSub = $true )
& ( disjointsetsI1 = $true )
& ( binunionIL = $true )
& ( doubleComplementSub2 = $true )
& ( prop2setE = $true )
& ( peanoSinj = $true )
& ( apProp = $true )
& ( ex1I2 = $true )
& ( cartprodsndpairEq = $true )
& ( woz1_4 = $true )
& ( subbreln = $true )
& ( subPowSU = $true )
& ( inCongP = $true )
& ( breln1invE = $true )
& ( binunionEcases = $true )
& ( binunionE = $true )
& ( setadjoinSub = $true )
& ( binintTransitive = $true )
& ( contraSubsetComplement = $true )
& ( infuncsetfunc = $true )
& ( funcextLem = $true )
& ( powersetsubset = $true )
& ( ubforcartprodlem3 = $true )
& ( quantDeMorgan1 = $true )
& ( complementTnotintersectT = $true )
& ( ex1I = $true )
& ( ap2p = $true )
& ( bs114d = $true )
& ( injFuncSetFuncIn = $true )
& ( setukpairinjL2 = $true )
& ( powersetE1 = $true )
& ( image1Ex1 = $true )
& ( setukpairinjR = $true )
& ( cartprodfstpairEq = $true )
& ( image1E = $true )
& ( breln1invI = $true )
& ( setOfPairsIsBReln1 = $true )
& ( notinemptyset = $true )
& ( omegaSp = $true )
& ( breln1SetBreln1 = $true )
& ( doubleComplementEq = $true )
& ( cartprodpairmemEL = $true )
& ( woz2Ex = $true )
& ( kfstpairEq = $true )
& ( secondinupair = $true )
& ( emptyset__Cong = $true )
& ( setext = $true )
& ( theprop = $true )
& ( surjCantorThm = $true )
& ( setadjoinSub2 = $true )
& ( upairsetIR = $true )
& ( subsetRefl = $true )
& ( subsetTrans = $true )
& ( breln1unionIR = $true )
& ( binunionTE = $true )
& ( upairinpowunion = $true )
& ( binintersectSubset4 = $true )
& ( setminusER = $true )
& ( contrasubsetT1 = $true )
& ( vacuousDall = $true )
& ( kpairsurjEq = $true )
& ( funcImageSingleton = $true )
& ( symdiffIneg2 = $true )
& ( upairequniteq = $true )
& ( demorgan2b = $true )
& ( exuE1 = $true )
& ( omegaSclos = $true )
& ( binintersectTELcontra = $true )
& ( iftrueProp2 = $true )
& ( complementT_lem = $true )
& ( exuI3 = $true )
& ( lamProp = $true )
& ( breln1unionprop = $true )
& ( doubleComplementI1 = $true )
& ( setukpairinjR1 = $true )
& ( omegaIndAx = $true )
& ( setukpairIL = $true )
& ( ksndpairEq = $true )
& ( complementTE1 = $true )
& ( powersetTE1 = $true )
& ( beta1 = $true )
& ( woz1_2 = $true )
& ( lam2lamEq = $true )
& ( demorgan2 = $true )
& ( omegaSAx = $true )
& ( dsetconstr__Cong = $true )
& ( iffalseProp1 = $true )
& ( funcinfuncset = $true )
& ( notsubsetI = $true )
& ( notinsingleton = $true )
& ( upairsetIL = $true )
& ( breln1unionEcases = $true )
& ( ex1E2 = $true )
& ( woz2W = $true )
& ( prop2set2propI = $true )
& ( woz1_1 = $true )
& ( singletoninpowerset = $true )
& ( binunionRsub = $true )
& ( binintersectER = $true )
& ( dsetconstrEL = $true )
& ? [X0: $i] :
( ? [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ( $true
= ( in @ X1 @ X1 ) ) )
& ( ( ordinal @ X0 )
= $true ) )
& ( omega0Ax = $true )
& ( breln1compI = $true )
& ( quantDeMorgan3 = $true )
& ( breln1unionI = $true )
& ( setunionsingleton = $true )
& ( dpsetconstrEL1 = $true )
& ( symdiffI1 = $true )
& ( ordinalTransSet = $true )
& ( symdiffIneg1 = $true )
& ( setunionOrdinalLem1 = $true )
& ( binintersectTERcontra = $true )
& ( powersetI1 = $true )
& ( subbreln1 = $true )
& ( iffalseProp2 = $true )
& ( in__Cong = $true )
& ( notequalI2 = $true )
& ( eqbreln1 = $true )
& ( wellorderingAx = $true )
& ( peano0notS = $true )
& ( subsetTI = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setadjoin__Cong = $true )
& ( eqimpsubset1 = $true )
& ( exuE2 = $true )
& ( funcext = $true )
& ( singletoninpowunion = $true )
& ( demorgan2b2 = $true )
& ( singletonsubset = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectSubset1 = $true )
& ( binintersectLsub = $true )
& ( setunionE2 = $true )
& ( brelnall1 = $true )
& ( nonemptyI1 = $true )
& ( ksndsingleton = $true )
& ( demorgan1a = $true )
& ( surjFuncSetFuncSurj = $true )
& ( beta2 = $true )
& ( setunionAx = $true )
& ( setminusLsub = $true )
& ( foundation2 = $true )
& ( descr__Cong = $true )
& ( image1Ex = $true )
& ( binintersectT_lem = $true )
& ( choice2fnsingleton = $true )
& ( contrasubsetT = $true )
& ( powersetT_lem = $true )
& ( emptyinunitempty = $true )
& ( setunionI = $true )
& ( emptysetsubset = $true )
& ( complementImpComplementIntersect = $true )
& ( notinself2 = $true )
& ( setukpairIR = $true )
& ( woz13rule3 = $true )
& ( exuI2 = $true )
& ( setukpairinjR2 = $true )
& ( demorgan2a1 = $true )
& ( cartprodfstin = $true )
& ( cartprodmempair1 = $true )
& ( notdexE = $true )
& ( setukpairinjL1 = $true )
& ( funcGraphProp4 = $true )
& ( leftInvIsSurj = $true )
& ( breln1compprop = $true )
& ( iftrue = $true )
& ( binintersectSubset2 = $true )
& ( woz1_3 = $true )
& ( emptysetimpfalse = $true )
& ( upairsubunion = $true )
& ( powersetI = $true )
& ( breln1unionE = $true )
& ( binunionTIRcontra = $true )
& ( dsetconstrI = $true )
& ( kpairiskpair = $true )
& ( setunion__Cong = $true )
& ( injFuncSetFuncInj = $true )
& ( dpsetconstrER = $true )
& ( iftrueorfalse = $true )
& ( noeltsimpempty = $true )
& ( singletonprop = $true )
& ( uniqinunit = $true )
& ( emptyInPowerset = $true ) ),
inference(flattening,[status(thm)],[f1086]) ).
thf(f1086,plain,
( ? [X0: $i] :
( ? [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ( $true
= ( in @ X1 @ X1 ) ) )
& ( ( ordinal @ X0 )
= $true ) )
& ( emptysetOrdinal = $true )
& ( setunionOrdinalLem1 = $true )
& ( ordinalTransSet1 = $true )
& ( ordinalTransSet = $true )
& ( ordinalMinLem1 = $true )
& ( setunionTransitive = $true )
& ( transitivesetOp2 = $true )
& ( binintTransitive = $true )
& ( transitivesetOp1 = $true )
& ( peanoSinj = $true )
& ( peano0notS = $true )
& ( omegaSclos = $true )
& ( omegaSp = $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)],[f1021]) ).
thf(f1021,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 )
=> ( ( omegaSp = $true )
=> ( ( omegaSclos = $true )
=> ( ( peano0notS = $true )
=> ( ( peanoSinj = $true )
=> ( ( transitivesetOp1 = $true )
=> ( ( binintTransitive = $true )
=> ( ( transitivesetOp2 = $true )
=> ( ( setunionTransitive = $true )
=> ( ( ordinalMinLem1 = $true )
=> ( ( ordinalTransSet = $true )
=> ( ( ordinalTransSet1 = $true )
=> ( ( setunionOrdinalLem1 = $true )
=> ( ( emptysetOrdinal = $true )
=> ! [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
=> ! [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
=> ( $true
!= ( in @ X1 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f935]) ).
thf(f935,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 )
=> ( ( omegaSp = $true )
=> ( ( omegaSclos = $true )
=> ( ( peano0notS = $true )
=> ( ( peanoSinj = $true )
=> ( ( transitivesetOp1 = $true )
=> ( ( binintTransitive = $true )
=> ( ( transitivesetOp2 = $true )
=> ( ( setunionTransitive = $true )
=> ( ( ordinalMinLem1 = $true )
=> ( ( ordinalTransSet = $true )
=> ( ( ordinalTransSet1 = $true )
=> ( ( setunionOrdinalLem1 = $true )
=> ( ( emptysetOrdinal = $true )
=> ! [X0: $i] :
( ( ( ordinal @ X0 )
= $true )
=> ! [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
=> ( $true
!= ( in @ X1 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f934]) ).
thf(f934,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
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ( binintTransitive
=> ( transitivesetOp2
=> ( setunionTransitive
=> ( ordinalMinLem1
=> ( ordinalTransSet
=> ( ordinalTransSet1
=> ( setunionOrdinalLem1
=> ( emptysetOrdinal
=> ! [X0: $i] :
( ( ordinal @ X0 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ~ ( in @ X1 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f343]) ).
thf(f343,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
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ( binintTransitive
=> ( transitivesetOp2
=> ( setunionTransitive
=> ( ordinalMinLem1
=> ( ordinalTransSet
=> ( ordinalTransSet1
=> ( setunionOrdinalLem1
=> ( emptysetOrdinal
=> ! [X11: $i] :
( ( ordinal @ X11 )
=> ! [X3: $i] :
( ( in @ X3 @ X11 )
=> ~ ( in @ X3 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f342]) ).
thf(f342,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
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ( binintTransitive
=> ( transitivesetOp2
=> ( setunionTransitive
=> ( ordinalMinLem1
=> ( ordinalTransSet
=> ( ordinalTransSet1
=> ( setunionOrdinalLem1
=> ( emptysetOrdinal
=> ! [X11: $i] :
( ( ordinal @ X11 )
=> ! [X3: $i] :
( ( in @ X3 @ X11 )
=> ~ ( in @ X3 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.zpLud4Dut7/DTF2THF_5045.p',ordinalIrrefl) ).
thf(f3237,plain,
! [X1: $i] :
( ( $true
!= ( in @ X1 @ X1 ) )
| ( notinself != $true ) ),
inference(cnf_transformation,[status(thm)],[f1539]) ).
thf(f1539,plain,
( ( ( notinself = $true )
| ( $true
= ( in @ sK27 @ sK27 ) ) )
& ( ! [X1: $i] :
( $true
!= ( in @ X1 @ X1 ) )
| ( notinself != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK27])],[f1537,f1538]) ).
thf(f1538,plain,
( ? [X0: $i] :
( ( in @ X0 @ X0 )
= $true )
=> ( $true
= ( in @ sK27 @ sK27 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1537,plain,
( ( ( notinself = $true )
| ? [X0: $i] :
( ( in @ X0 @ X0 )
= $true ) )
& ( ! [X1: $i] :
( $true
!= ( in @ X1 @ X1 ) )
| ( notinself != $true ) ) ),
inference(rectify,[status(thm)],[f1536]) ).
thf(f1536,plain,
( ( ( notinself = $true )
| ? [X0: $i] :
( ( in @ X0 @ X0 )
= $true ) )
& ( ! [X0: $i] :
( ( in @ X0 @ X0 )
!= $true )
| ( notinself != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1078]) ).
thf(f1078,plain,
( ( notinself = $true )
<=> ! [X0: $i] :
( ( in @ X0 @ X0 )
!= $true ) ),
inference(flattening,[status(thm)],[f1014]) ).
thf(f1014,plain,
( ! [X0: $i] :
( ( in @ X0 @ X0 )
!= $true )
<=> ( notinself = $true ) ),
inference(fool_elimination,[status(thm)],[f1013]) ).
thf(f1013,plain,
! [X0: $i] :
( ( ~ ( in @ X0 @ X0 ) )
= notinself ),
inference(rectify,[status(thm)],[f324]) ).
thf(f324,axiom,
! [X3: $i] :
( ( ~ ( in @ X3 @ X3 ) )
= notinself ),
file('/export/starexec/sandbox/tmp/tmp.zpLud4Dut7/DTF2THF_5045.p',notinself) ).
thf(f4827,plain,
( ( in @ sK1113 @ sK1113 )
= $true ),
inference(cnf_transformation,[status(thm)],[f3222]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU821^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n011.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 12:34:47 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
% 0.28/0.47 ---- Original DTF file ---
% 0.28/0.47 thf(spec,logic,$$dhol).
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % File : SEU821^1 : TPTP v9.2.1. Released v3.7.0.
% 0.28/0.47 % Domain : Set Theory
% 0.28/0.47 % Problem : Ordinals
% 0.28/0.47 % Version : Especial.
% 0.28/0.47 % English : (! X:i.ordinal X -> (! A:i.in A X -> ~(in A A)))
% 0.28/0.47
% 0.28/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.28/0.47 % Source : [Bro08]
% 0.28/0.47 % Names : ZFC323g [Bro08]
% 0.28/0.47
% 0.28/0.47 % Status : Theorem
% 0.28/0.47 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.28/0.47 % Syntax : Number of formulae : 743 ( 341 unt; 401 typ; 341 def)
% 0.28/0.47 % Number of atoms : 2110 ( 487 equ; 0 cnn)
% 0.28/0.47 % Maximal formula atoms : 340 ( 6 avg)
% 0.28/0.47 % Number of connectives : 4562 ( 78 ~; 17 |; 60 &;3305 @)
% 0.28/0.47 % ( 17 <=>;1085 =>; 0 <=; 0 <~>)
% 0.28/0.47 % Maximal formula depth : 345 ( 2 avg)
% 0.28/0.47 % Number of types : 2 ( 0 usr)
% 0.28/0.47 % Number of type conns : 195 ( 195 >; 0 *; 0 +; 0 <<)
% 0.28/0.47 % Number of symbols : 404 ( 401 usr; 341 con; 0-4 aty)
% 0.28/0.47 % Number of variables : 1182 ( 77 ^;1060 !; 45 ?;1182 :)
% 0.28/0.47 % SPC : TH0_THM_EQU_NAR
% 0.28/0.47
% 0.28/0.47 % Comments : http://mathgate.info/detsetitem.php?id=518
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 thf(in_type,type,
% 0.28/0.47 in: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(exu_type,type,
% 0.28/0.47 exu: ( $i > $o ) > $o ).
% 0.28/0.47
% 0.28/0.47 thf(exu,definition,
% 0.28/0.47 ( exu
% 0.28/0.47 = ( ^ [Xphi: $i > $o] :
% 0.28/0.47 ? [Xx: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 & ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setextAx_type,type,
% 0.28/0.47 setextAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setextAx,definition,
% 0.28/0.47 ( setextAx
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 <=> ( in @ Xx @ B ) )
% 0.28/0.47 => ( A = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset_type,type,
% 0.28/0.47 emptyset: $i ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetAx_type,type,
% 0.28/0.47 emptysetAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetAx,definition,
% 0.28/0.47 ( emptysetAx
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin_type,type,
% 0.28/0.47 setadjoin: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinAx_type,type,
% 0.28/0.47 setadjoinAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinAx,definition,
% 0.28/0.47 ( setadjoinAx
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.47 <=> ( ( Xy = Xx )
% 0.28/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powerset_type,type,
% 0.28/0.47 powerset: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(powersetAx_type,type,
% 0.28/0.47 powersetAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetAx,definition,
% 0.28/0.47 ( powersetAx
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 <=> ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunion_type,type,
% 0.28/0.47 setunion: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setunionAx_type,type,
% 0.28/0.47 setunionAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionAx,definition,
% 0.28/0.47 ( setunionAx
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 <=> ? [B: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omega_type,type,
% 0.28/0.47 omega: $i ).
% 0.28/0.47
% 0.28/0.47 thf(omega0Ax_type,type,
% 0.28/0.47 omega0Ax: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omega0Ax,definition,
% 0.28/0.47 ( omega0Ax
% 0.28/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omegaSAx_type,type,
% 0.28/0.47 omegaSAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omegaSAx,definition,
% 0.28/0.47 ( omegaSAx
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ omega )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omegaIndAx_type,type,
% 0.28/0.47 omegaIndAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omegaIndAx,definition,
% 0.28/0.47 ( omegaIndAx
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( ( in @ emptyset @ A )
% 0.28/0.47 & ! [Xx: $i] :
% 0.28/0.47 ( ( ( in @ Xx @ omega )
% 0.28/0.47 & ( in @ Xx @ A ) )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ omega )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(replAx_type,type,
% 0.28/0.47 replAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(replAx,definition,
% 0.28/0.47 ( replAx
% 0.28/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.28/0.47 => ? [B: $i] :
% 0.28/0.47 ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 <=> ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(foundationAx_type,type,
% 0.28/0.47 foundationAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(foundationAx,definition,
% 0.28/0.47 ( foundationAx
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.47 => ? [B: $i] :
% 0.28/0.47 ( ( in @ B @ A )
% 0.28/0.47 & ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(wellorderingAx_type,type,
% 0.28/0.47 wellorderingAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(wellorderingAx,definition,
% 0.28/0.47 ( wellorderingAx
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ? [B: $i] :
% 0.28/0.47 ( ! [C: $i] :
% 0.28/0.47 ( ( in @ C @ B )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ C )
% 0.28/0.47 => ( in @ Xx @ A ) ) )
% 0.28/0.47 & ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ( in @ Xx @ A )
% 0.28/0.47 & ( in @ Xy @ A ) )
% 0.28/0.47 => ( ! [C: $i] :
% 0.28/0.47 ( ( in @ C @ B )
% 0.28/0.47 => ( ( in @ Xx @ C )
% 0.28/0.47 <=> ( in @ Xy @ C ) ) )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 & ! [C: $i,D: $i] :
% 0.28/0.47 ( ( ( in @ C @ B )
% 0.28/0.47 & ( in @ D @ B ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ C )
% 0.28/0.47 => ( in @ Xx @ D ) )
% 0.28/0.47 | ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ D )
% 0.28/0.47 => ( in @ Xx @ C ) ) ) )
% 0.28/0.47 & ! [C: $i] :
% 0.28/0.47 ( ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ C )
% 0.28/0.47 => ( in @ Xx @ A ) )
% 0.28/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.28/0.47 => ? [D: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ D @ B )
% 0.28/0.47 & ( in @ Xx @ C )
% 0.28/0.47 & ~ ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ D )
% 0.28/0.47 & ( in @ Xy @ C ) )
% 0.28/0.47 & ! [E: $i] :
% 0.28/0.47 ( ( in @ E @ B )
% 0.28/0.47 => ( ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ E )
% 0.28/0.47 => ( in @ Xy @ D ) )
% 0.28/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(descr_type,type,
% 0.28/0.47 descr: ( $i > $o ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(descrp_type,type,
% 0.28/0.47 descrp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(descrp,definition,
% 0.28/0.47 ( descrp
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ( Xphi
% 0.28/0.47 @ ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr_type,type,
% 0.28/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrI_type,type,
% 0.28/0.47 dsetconstrI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrI,definition,
% 0.28/0.47 ( dsetconstrI
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrEL_type,type,
% 0.28/0.47 dsetconstrEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrEL,definition,
% 0.28/0.47 ( dsetconstrEL
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrER_type,type,
% 0.28/0.47 dsetconstrER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrER,definition,
% 0.28/0.47 ( dsetconstrER
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE1_type,type,
% 0.28/0.47 exuE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE1,definition,
% 0.28/0.47 ( exuE1
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 & ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(prop2set_type,type,
% 0.28/0.47 prop2set: $o > $i ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setE_type,type,
% 0.28/0.47 prop2setE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setE,definition,
% 0.28/0.47 ( prop2setE
% 0.28/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetE_type,type,
% 0.28/0.47 emptysetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetE,definition,
% 0.28/0.47 ( emptysetE
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ emptyset )
% 0.28/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetimpfalse_type,type,
% 0.28/0.47 emptysetimpfalse: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetimpfalse,definition,
% 0.28/0.47 ( emptysetimpfalse
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ emptyset )
% 0.28/0.47 => $false ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notinemptyset_type,type,
% 0.28/0.47 notinemptyset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notinemptyset,definition,
% 0.28/0.47 ( notinemptyset
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3e_type,type,
% 0.28/0.47 exuE3e: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3e,definition,
% 0.28/0.47 ( exuE3e
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setext_type,type,
% 0.28/0.47 setext: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setext,definition,
% 0.28/0.47 ( setext
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) )
% 0.28/0.47 => ( A = B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyI_type,type,
% 0.28/0.47 emptyI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyI,definition,
% 0.28/0.47 ( emptyI
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ A )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(noeltsimpempty_type,type,
% 0.28/0.47 noeltsimpempty: $o ).
% 0.28/0.47
% 0.28/0.47 thf(noeltsimpempty,definition,
% 0.28/0.47 ( noeltsimpempty
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ A )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setbeta_type,type,
% 0.28/0.47 setbeta: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setbeta,definition,
% 0.28/0.47 ( setbeta
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonempty_type,type,
% 0.28/0.47 nonempty: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyE1_type,type,
% 0.28/0.47 nonemptyE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyE1,definition,
% 0.28/0.47 ( nonemptyE1
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( nonempty @ A )
% 0.28/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI_type,type,
% 0.28/0.47 nonemptyI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI,definition,
% 0.28/0.47 ( nonemptyI
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( nonempty
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI1_type,type,
% 0.28/0.47 nonemptyI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI1,definition,
% 0.28/0.47 ( nonemptyI1
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIL_type,type,
% 0.28/0.47 setadjoinIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIL,definition,
% 0.28/0.47 ( setadjoinIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinunitempty_type,type,
% 0.28/0.47 emptyinunitempty: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinunitempty,definition,
% 0.28/0.47 ( emptyinunitempty
% 0.28/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIR_type,type,
% 0.28/0.47 setadjoinIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIR,definition,
% 0.28/0.47 ( setadjoinIR
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinE_type,type,
% 0.28/0.47 setadjoinE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinE,definition,
% 0.28/0.47 ( setadjoinE
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ( ( Xy = Xx )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ Xy @ A )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinOr_type,type,
% 0.28/0.47 setadjoinOr: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinOr,definition,
% 0.28/0.47 ( setadjoinOr
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.47 => ( ( Xy = Xx )
% 0.28/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setoftrueEq_type,type,
% 0.28/0.47 setoftrueEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setoftrueEq,definition,
% 0.28/0.47 ( setoftrueEq
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : $true )
% 0.28/0.47 = A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI_type,type,
% 0.28/0.47 powersetI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI,definition,
% 0.28/0.47 ( powersetI
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) )
% 0.28/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinPowerset_type,type,
% 0.28/0.47 emptyinPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinPowerset,definition,
% 0.28/0.47 ( emptyinPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyInPowerset_type,type,
% 0.28/0.47 emptyInPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyInPowerset,definition,
% 0.28/0.47 ( emptyInPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE_type,type,
% 0.28/0.47 powersetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE,definition,
% 0.28/0.47 ( powersetE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionI_type,type,
% 0.28/0.47 setunionI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionI,definition,
% 0.28/0.47 ( setunionI
% 0.28/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( ( in @ B @ A )
% 0.28/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE_type,type,
% 0.28/0.47 setunionE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE,definition,
% 0.28/0.47 ( setunionE
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ! [B: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( ( in @ B @ A )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subPowSU_type,type,
% 0.28/0.47 subPowSU: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subPowSU,definition,
% 0.28/0.47 ( subPowSU
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE2_type,type,
% 0.28/0.47 exuE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE2,definition,
% 0.28/0.47 ( exuE2
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyImpWitness_type,type,
% 0.28/0.47 nonemptyImpWitness: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyImpWitness,definition,
% 0.28/0.47 ( nonemptyImpWitness
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( nonempty @ A )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & $true ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(uniqinunit_type,type,
% 0.28/0.47 uniqinunit: $o ).
% 0.28/0.47
% 0.28/0.47 thf(uniqinunit,definition,
% 0.28/0.47 ( uniqinunit
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notinsingleton_type,type,
% 0.28/0.47 notinsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notinsingleton,definition,
% 0.28/0.47 ( notinsingleton
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx != Xy )
% 0.28/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqinunit_type,type,
% 0.28/0.47 eqinunit: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqinunit,definition,
% 0.28/0.47 ( eqinunit
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsswitch_type,type,
% 0.28/0.47 singletonsswitch: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsswitch,definition,
% 0.28/0.47 ( singletonsswitch
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetE_type,type,
% 0.28/0.47 upairsetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetE,definition,
% 0.28/0.47 ( upairsetE
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xx )
% 0.28/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIL_type,type,
% 0.28/0.47 upairsetIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIL,definition,
% 0.28/0.47 ( upairsetIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIR_type,type,
% 0.28/0.47 upairsetIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIR,definition,
% 0.28/0.47 ( upairsetIR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyE1_type,type,
% 0.28/0.47 emptyE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyE1,definition,
% 0.28/0.47 ( emptyE1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = emptyset )
% 0.28/0.47 => $false ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(vacuousDall_type,type,
% 0.28/0.47 vacuousDall: $o ).
% 0.28/0.47
% 0.28/0.47 thf(vacuousDall,definition,
% 0.28/0.47 ( vacuousDall
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ emptyset )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan1_type,type,
% 0.28/0.47 quantDeMorgan1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan1,definition,
% 0.28/0.47 ( quantDeMorgan1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan2_type,type,
% 0.28/0.47 quantDeMorgan2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan2,definition,
% 0.28/0.47 ( quantDeMorgan2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) )
% 0.28/0.47 => ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan3_type,type,
% 0.28/0.47 quantDeMorgan3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan3,definition,
% 0.28/0.47 ( quantDeMorgan3
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan4_type,type,
% 0.28/0.47 quantDeMorgan4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan4,definition,
% 0.28/0.47 ( quantDeMorgan4
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) )
% 0.28/0.47 => ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setI_type,type,
% 0.28/0.47 prop2setI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setI,definition,
% 0.28/0.47 ( prop2setI
% 0.28/0.47 = ( ! [Xphi: $o] :
% 0.28/0.47 ( Xphi
% 0.28/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(set2prop_type,type,
% 0.28/0.47 set2prop: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2set2propI_type,type,
% 0.28/0.47 prop2set2propI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2set2propI,definition,
% 0.28/0.47 ( prop2set2propI
% 0.28/0.47 = ( ! [Xphi: $o] :
% 0.28/0.47 ( Xphi
% 0.28/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notdexE_type,type,
% 0.28/0.47 notdexE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notdexE,definition,
% 0.28/0.47 ( notdexE
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notdallE_type,type,
% 0.28/0.47 notdallE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notdallE,definition,
% 0.28/0.47 ( notdallE
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI1_type,type,
% 0.28/0.47 exuI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI1,definition,
% 0.28/0.47 ( exuI1
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 & ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI3_type,type,
% 0.28/0.47 exuI3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI3,definition,
% 0.28/0.47 ( exuI3
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.28/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI2_type,type,
% 0.28/0.47 exuI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI2,definition,
% 0.28/0.47 ( exuI2
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 <=> ( Xy = Xx ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inCongP_type,type,
% 0.28/0.47 inCongP: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inCongP,definition,
% 0.28/0.47 ( inCongP
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong_type,type,
% 0.28/0.47 in__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong,definition,
% 0.28/0.47 ( in__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3u_type,type,
% 0.28/0.47 exuE3u: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3u,definition,
% 0.28/0.47 ( exuE3u
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exu__Cong_type,type,
% 0.28/0.47 exu__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exu__Cong,definition,
% 0.28/0.47 ( exu__Cong
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 <=> ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset__Cong_type,type,
% 0.28/0.47 emptyset__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset__Cong,definition,
% 0.28/0.47 ( emptyset__Cong
% 0.28/0.47 = ( emptyset = emptyset ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin__Cong_type,type,
% 0.28/0.47 setadjoin__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin__Cong,definition,
% 0.28/0.47 ( setadjoin__Cong
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ! [Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( Xz = Xu )
% 0.28/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.28/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powerset__Cong_type,type,
% 0.28/0.47 powerset__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powerset__Cong,definition,
% 0.28/0.47 ( powerset__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( ( powerset @ A )
% 0.28/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunion__Cong_type,type,
% 0.28/0.47 setunion__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunion__Cong,definition,
% 0.28/0.47 ( setunion__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( ( setunion @ A )
% 0.28/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omega__Cong_type,type,
% 0.28/0.47 omega__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omega__Cong,definition,
% 0.28/0.47 ( omega__Cong
% 0.28/0.47 = ( omega = omega ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuEu_type,type,
% 0.28/0.47 exuEu: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuEu,definition,
% 0.28/0.47 ( exuEu
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(descr__Cong_type,type,
% 0.28/0.47 descr__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(descr__Cong,definition,
% 0.28/0.47 ( descr__Cong
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.28/0.47 => ( ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr__Cong_type,type,
% 0.28/0.47 dsetconstr__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr__Cong,definition,
% 0.28/0.47 ( dsetconstr__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.28/0.47 => ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset_type,type,
% 0.28/0.47 subset: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(disjoint_type,type,
% 0.28/0.47 disjoint: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setsmeet_type,type,
% 0.28/0.47 setsmeet: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI1_type,type,
% 0.28/0.47 subsetI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI1,definition,
% 0.28/0.47 ( subsetI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset2_type,type,
% 0.28/0.47 eqimpsubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset2,definition,
% 0.28/0.47 ( eqimpsubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset1_type,type,
% 0.28/0.47 eqimpsubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset1,definition,
% 0.28/0.47 ( eqimpsubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2_type,type,
% 0.28/0.47 subsetI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2,definition,
% 0.28/0.47 ( subsetI2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetsubset_type,type,
% 0.28/0.47 emptysetsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetsubset,definition,
% 0.28/0.47 ( emptysetsubset
% 0.28/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE_type,type,
% 0.28/0.47 subsetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE,definition,
% 0.28/0.47 ( subsetE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE2_type,type,
% 0.28/0.47 subsetE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE2,definition,
% 0.28/0.47 ( subsetE2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notsubsetI_type,type,
% 0.28/0.47 notsubsetI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notsubsetI,definition,
% 0.28/0.47 ( notsubsetI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI1_type,type,
% 0.28/0.47 notequalI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI1,definition,
% 0.28/0.47 ( notequalI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ~ ( subset @ A @ B )
% 0.28/0.47 => ( A != B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI2_type,type,
% 0.28/0.47 notequalI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI2,definition,
% 0.28/0.47 ( notequalI2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( A != B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetRefl_type,type,
% 0.28/0.47 subsetRefl: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetRefl,definition,
% 0.28/0.47 ( subsetRefl
% 0.28/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTrans_type,type,
% 0.28/0.47 subsetTrans: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTrans,definition,
% 0.28/0.47 ( subsetTrans
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( subset @ B @ C )
% 0.28/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub_type,type,
% 0.28/0.47 setadjoinSub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub,definition,
% 0.28/0.47 ( setadjoinSub
% 0.28/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub2_type,type,
% 0.28/0.47 setadjoinSub2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub2,definition,
% 0.28/0.47 ( setadjoinSub2
% 0.28/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset2powerset_type,type,
% 0.28/0.47 subset2powerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subset2powerset,definition,
% 0.28/0.47 ( subset2powerset
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setextsub_type,type,
% 0.28/0.47 setextsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setextsub,definition,
% 0.28/0.47 ( setextsub
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( subset @ B @ A )
% 0.28/0.47 => ( A = B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetemptysetimpeq_type,type,
% 0.28/0.47 subsetemptysetimpeq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetemptysetimpeq,definition,
% 0.28/0.47 ( subsetemptysetimpeq
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( subset @ A @ emptyset )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI1_type,type,
% 0.28/0.47 powersetI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI1,definition,
% 0.28/0.47 ( powersetI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ B @ A )
% 0.28/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE1_type,type,
% 0.28/0.47 powersetE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE1,definition,
% 0.28/0.47 ( powersetE1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inPowerset_type,type,
% 0.28/0.47 inPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inPowerset,definition,
% 0.28/0.47 ( inPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetsubset_type,type,
% 0.28/0.47 powersetsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetsubset,definition,
% 0.28/0.47 ( powersetsubset
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sepInPowerset_type,type,
% 0.28/0.47 sepInPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(sepInPowerset,definition,
% 0.28/0.47 ( sepInPowerset
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( in
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sepSubset_type,type,
% 0.28/0.47 sepSubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(sepSubset,definition,
% 0.28/0.47 ( sepSubset
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( subset
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunion_type,type,
% 0.28/0.47 binunion: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIL_type,type,
% 0.28/0.47 binunionIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIL,definition,
% 0.28/0.47 ( binunionIL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2IR_type,type,
% 0.28/0.47 upairset2IR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2IR,definition,
% 0.28/0.47 ( upairset2IR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIR_type,type,
% 0.28/0.47 binunionIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIR,definition,
% 0.28/0.47 ( binunionIR
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionEcases_type,type,
% 0.28/0.47 binunionEcases: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionEcases,definition,
% 0.28/0.47 ( binunionEcases
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.28/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.47 => ( ( ( in @ Xx @ A )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ Xx @ B )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionE_type,type,
% 0.28/0.47 binunionE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionE,definition,
% 0.28/0.47 ( binunionE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionLsub_type,type,
% 0.28/0.47 binunionLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionLsub,definition,
% 0.28/0.47 ( binunionLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionRsub_type,type,
% 0.28/0.47 binunionRsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionRsub,definition,
% 0.28/0.47 ( binunionRsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersect_type,type,
% 0.28/0.47 binintersect: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectI_type,type,
% 0.28/0.47 binintersectI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectI,definition,
% 0.28/0.47 ( binintersectI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset5_type,type,
% 0.28/0.47 binintersectSubset5: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset5,definition,
% 0.28/0.47 ( binintersectSubset5
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( subset @ C @ A )
% 0.28/0.47 => ( ( subset @ C @ B )
% 0.28/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectEL_type,type,
% 0.28/0.47 binintersectEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectEL,definition,
% 0.28/0.47 ( binintersectEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectLsub_type,type,
% 0.28/0.47 binintersectLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectLsub,definition,
% 0.28/0.47 ( binintersectLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset2_type,type,
% 0.28/0.47 binintersectSubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset2,definition,
% 0.28/0.47 ( binintersectSubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset3_type,type,
% 0.28/0.47 binintersectSubset3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset3,definition,
% 0.28/0.47 ( binintersectSubset3
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( binintersect @ A @ B )
% 0.28/0.47 = B )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectER_type,type,
% 0.28/0.47 binintersectER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectER,definition,
% 0.28/0.47 ( binintersectER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(disjointsetsI1_type,type,
% 0.28/0.47 disjointsetsI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(disjointsetsI1,definition,
% 0.28/0.47 ( disjointsetsI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( in @ Xx @ B ) )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectRsub_type,type,
% 0.28/0.47 binintersectRsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectRsub,definition,
% 0.28/0.47 ( binintersectRsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset4_type,type,
% 0.28/0.47 binintersectSubset4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset4,definition,
% 0.28/0.47 ( binintersectSubset4
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ B @ A )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset1_type,type,
% 0.28/0.47 binintersectSubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset1,definition,
% 0.28/0.47 ( binintersectSubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( binintersect @ A @ B )
% 0.28/0.47 = A )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(bs114d_type,type,
% 0.28/0.47 bs114d: $o ).
% 0.28/0.47
% 0.28/0.47 thf(bs114d,definition,
% 0.28/0.47 ( bs114d
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.28/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(regular_type,type,
% 0.28/0.47 regular: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminus_type,type,
% 0.28/0.47 setminus: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI_type,type,
% 0.28/0.47 setminusI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI,definition,
% 0.28/0.47 ( setminusI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusEL_type,type,
% 0.28/0.47 setminusEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusEL,definition,
% 0.28/0.47 ( setminusEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusER_type,type,
% 0.28/0.47 setminusER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusER,definition,
% 0.28/0.47 ( setminusER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset2_type,type,
% 0.28/0.47 setminusSubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset2,definition,
% 0.28/0.47 ( setminusSubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusERneg_type,type,
% 0.28/0.47 setminusERneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusERneg,definition,
% 0.28/0.47 ( setminusERneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusELneg_type,type,
% 0.28/0.47 setminusELneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusELneg,definition,
% 0.28/0.47 ( setminusELneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusILneg_type,type,
% 0.28/0.47 setminusILneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusILneg,definition,
% 0.28/0.47 ( setminusILneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusIRneg_type,type,
% 0.28/0.47 setminusIRneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusIRneg,definition,
% 0.28/0.47 ( setminusIRneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusLsub_type,type,
% 0.28/0.47 setminusLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusLsub,definition,
% 0.28/0.47 ( setminusLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1_type,type,
% 0.28/0.47 setminusSubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1,definition,
% 0.28/0.47 ( setminusSubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiff_type,type,
% 0.28/0.47 symdiff: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffE_type,type,
% 0.28/0.47 symdiffE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffE,definition,
% 0.28/0.47 ( symdiffE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI1_type,type,
% 0.28/0.47 symdiffI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI1,definition,
% 0.28/0.47 ( symdiffI1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI2_type,type,
% 0.28/0.47 symdiffI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI2,definition,
% 0.28/0.47 ( symdiffI2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg1_type,type,
% 0.28/0.47 symdiffIneg1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg1,definition,
% 0.28/0.47 ( symdiffIneg1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg2_type,type,
% 0.28/0.47 symdiffIneg2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg2,definition,
% 0.28/0.47 ( symdiffIneg2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iskpair_type,type,
% 0.28/0.47 iskpair: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(secondinupair_type,type,
% 0.28/0.47 secondinupair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(secondinupair,definition,
% 0.28/0.47 ( secondinupair
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIL_type,type,
% 0.28/0.47 setukpairIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIL,definition,
% 0.28/0.47 ( setukpairIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIR_type,type,
% 0.28/0.47 setukpairIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIR,definition,
% 0.28/0.47 ( setukpairIR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpairiskpair_type,type,
% 0.28/0.47 kpairiskpair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairiskpair,definition,
% 0.28/0.47 ( kpairiskpair
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpair_type,type,
% 0.28/0.47 kpair: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(kpairp_type,type,
% 0.28/0.47 kpairp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairp,definition,
% 0.28/0.47 ( kpairp
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprod_type,type,
% 0.28/0.47 cartprod: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsubset_type,type,
% 0.28/0.47 singletonsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsubset,definition,
% 0.28/0.47 ( singletonsubset
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowerset_type,type,
% 0.28/0.47 singletoninpowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowerset,definition,
% 0.28/0.47 ( singletoninpowerset
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowunion_type,type,
% 0.28/0.47 singletoninpowunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowunion,definition,
% 0.28/0.47 ( singletoninpowunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2E_type,type,
% 0.28/0.47 upairset2E: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2E,definition,
% 0.28/0.47 ( upairset2E
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xx )
% 0.28/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsubunion_type,type,
% 0.28/0.47 upairsubunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsubunion,definition,
% 0.28/0.47 ( upairsubunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairinpowunion_type,type,
% 0.28/0.47 upairinpowunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairinpowunion,definition,
% 0.28/0.47 ( upairinpowunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem1_type,type,
% 0.28/0.47 ubforcartprodlem1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem1,definition,
% 0.28/0.47 ( ubforcartprodlem1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem2_type,type,
% 0.28/0.47 ubforcartprodlem2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem2,definition,
% 0.28/0.47 ( ubforcartprodlem2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem3_type,type,
% 0.28/0.47 ubforcartprodlem3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem3,definition,
% 0.28/0.47 ( ubforcartprodlem3
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairin_type,type,
% 0.28/0.47 cartprodpairin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairin,definition,
% 0.28/0.47 ( cartprodpairin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair1_type,type,
% 0.28/0.47 cartprodmempair1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair1,definition,
% 0.28/0.47 ( cartprodmempair1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 & ( Xu
% 0.28/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair_type,type,
% 0.28/0.47 cartprodmempair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair,definition,
% 0.28/0.47 ( cartprodmempair
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE2_type,type,
% 0.28/0.47 setunionE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE2,definition,
% 0.28/0.47 ( setunionE2
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 => ? [X: $i] :
% 0.28/0.47 ( ( in @ X @ A )
% 0.28/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton1_type,type,
% 0.28/0.47 setunionsingleton1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton1,definition,
% 0.28/0.47 ( setunionsingleton1
% 0.28/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton2_type,type,
% 0.28/0.47 setunionsingleton2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton2,definition,
% 0.28/0.47 ( setunionsingleton2
% 0.28/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton_type,type,
% 0.28/0.47 setunionsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton,definition,
% 0.28/0.47 ( setunionsingleton
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.28/0.47 = Xx ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singleton_type,type,
% 0.28/0.47 singleton: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonprop_type,type,
% 0.28/0.47 singletonprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonprop,definition,
% 0.28/0.47 ( singletonprop
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) )
% 0.28/0.47 => ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1_type,type,
% 0.28/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E1_type,type,
% 0.28/0.47 ex1E1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E1,definition,
% 0.28/0.47 ( ex1E1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I_type,type,
% 0.28/0.47 ex1I: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I,definition,
% 0.28/0.47 ( ex1I
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xy = Xx ) ) )
% 0.28/0.47 => ( ex1 @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I2_type,type,
% 0.28/0.47 ex1I2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I2,definition,
% 0.28/0.47 ( ex1I2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) )
% 0.28/0.47 => ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsuniq_type,type,
% 0.28/0.47 singletonsuniq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsuniq,definition,
% 0.28/0.47 ( singletonsuniq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.28/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(atmost1p_type,type,
% 0.28/0.47 atmost1p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(atleast2p_type,type,
% 0.28/0.47 atleast2p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(atmost2p_type,type,
% 0.28/0.47 atmost2p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetp_type,type,
% 0.28/0.47 upairsetp: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL1_type,type,
% 0.28/0.47 setukpairinjL1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL1,definition,
% 0.28/0.47 ( setukpairinjL1
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kfstsingleton_type,type,
% 0.28/0.47 kfstsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kfstsingleton,definition,
% 0.28/0.47 ( kfstsingleton
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.28/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(theprop_type,type,
% 0.28/0.47 theprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(theprop,definition,
% 0.28/0.47 ( theprop
% 0.28/0.47 = ( ! [X: $i] :
% 0.28/0.47 ( ( singleton @ X )
% 0.28/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kfst_type,type,
% 0.28/0.47 kfst: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(kfstpairEq_type,type,
% 0.28/0.47 kfstpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kfstpairEq,definition,
% 0.28/0.47 ( kfstpairEq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xx ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstin_type,type,
% 0.28/0.47 cartprodfstin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstin,definition,
% 0.28/0.47 ( cartprodfstin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL2_type,type,
% 0.28/0.47 setukpairinjL2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL2,definition,
% 0.28/0.48 ( setukpairinjL2
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjL_type,type,
% 0.28/0.48 setukpairinjL: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjL,definition,
% 0.28/0.48 ( setukpairinjL
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.28/0.48 = ( kpair @ Xz @ Xu ) )
% 0.28/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR11_type,type,
% 0.28/0.48 setukpairinjR11: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR11,definition,
% 0.28/0.48 ( setukpairinjR11
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( Xx = Xy )
% 0.28/0.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR12_type,type,
% 0.28/0.48 setukpairinjR12: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR12,definition,
% 0.28/0.48 ( setukpairinjR12
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( Xx = Xy )
% 0.28/0.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR1_type,type,
% 0.28/0.48 setukpairinjR1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR1,definition,
% 0.28/0.48 ( setukpairinjR1
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.48 => ( ( Xz = Xu )
% 0.28/0.48 => ( Xy = Xu ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(upairequniteq_type,type,
% 0.28/0.48 upairequniteq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(upairequniteq,definition,
% 0.28/0.48 ( upairequniteq
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.48 = ( setadjoin @ Xz @ emptyset ) )
% 0.28/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR2_type,type,
% 0.28/0.48 setukpairinjR2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR2,definition,
% 0.28/0.48 ( setukpairinjR2
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR_type,type,
% 0.28/0.48 setukpairinjR: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setukpairinjR,definition,
% 0.28/0.48 ( setukpairinjR
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.28/0.48 = ( kpair @ Xz @ Xu ) )
% 0.28/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ksndsingleton_type,type,
% 0.28/0.48 ksndsingleton: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ksndsingleton,definition,
% 0.28/0.48 ( ksndsingleton
% 0.28/0.48 = ( ! [Xu: $i] :
% 0.28/0.48 ( ( iskpair @ Xu )
% 0.28/0.48 => ( singleton
% 0.28/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.28/0.48 @ ^ [Xx: $i] :
% 0.28/0.48 ( Xu
% 0.28/0.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ksnd_type,type,
% 0.28/0.48 ksnd: $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(ksndpairEq_type,type,
% 0.28/0.48 ksndpairEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ksndpairEq,definition,
% 0.28/0.48 ( ksndpairEq
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.28/0.48 = Xy ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(kpairsurjEq_type,type,
% 0.28/0.48 kpairsurjEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(kpairsurjEq,definition,
% 0.28/0.48 ( kpairsurjEq
% 0.28/0.48 = ( ! [Xu: $i] :
% 0.28/0.48 ( ( iskpair @ Xu )
% 0.28/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.28/0.48 = Xu ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodsndin_type,type,
% 0.28/0.48 cartprodsndin: $o ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodsndin,definition,
% 0.28/0.48 ( cartprodsndin
% 0.28/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodpairmemEL_type,type,
% 0.28/0.48 cartprodpairmemEL: $o ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodpairmemEL,definition,
% 0.28/0.48 ( cartprodpairmemEL
% 0.28/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.28/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodpairmemER_type,type,
% 0.28/0.48 cartprodpairmemER: $o ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodpairmemER,definition,
% 0.28/0.48 ( cartprodpairmemER
% 0.28/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.28/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodmempaircEq_type,type,
% 0.28/0.48 cartprodmempaircEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodmempaircEq,definition,
% 0.28/0.48 ( cartprodmempaircEq
% 0.28/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( kpair @ Xx @ Xy )
% 0.28/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodfstpairEq_type,type,
% 0.28/0.48 cartprodfstpairEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodfstpairEq,definition,
% 0.28/0.48 ( cartprodfstpairEq
% 0.28/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.28/0.48 = Xx ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodsndpairEq_type,type,
% 0.28/0.48 cartprodsndpairEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodsndpairEq,definition,
% 0.28/0.48 ( cartprodsndpairEq
% 0.28/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.28/0.48 = Xy ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodpairsurjEq_type,type,
% 0.28/0.48 cartprodpairsurjEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(cartprodpairsurjEq,definition,
% 0.28/0.48 ( cartprodpairsurjEq
% 0.28/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.28/0.48 = Xu ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln_type,type,
% 0.28/0.48 breln: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstr_type,type,
% 0.28/0.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrI_type,type,
% 0.28/0.48 dpsetconstrI: $o ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrI,definition,
% 0.28/0.48 ( dpsetconstrI
% 0.28/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( Xphi @ Xx @ Xy )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.48 @ ( dpsetconstr @ A @ B
% 0.28/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrSub_type,type,
% 0.28/0.48 dpsetconstrSub: $o ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrSub,definition,
% 0.28/0.48 ( dpsetconstrSub
% 0.28/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.48 ( subset
% 0.28/0.48 @ ( dpsetconstr @ A @ B
% 0.28/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.28/0.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setOfPairsIsBReln_type,type,
% 0.28/0.48 setOfPairsIsBReln: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setOfPairsIsBReln,definition,
% 0.28/0.48 ( setOfPairsIsBReln
% 0.28/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.48 ( breln @ A @ B
% 0.28/0.48 @ ( dpsetconstr @ A @ B
% 0.28/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrERa_type,type,
% 0.28/0.48 dpsetconstrERa: $o ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrERa,definition,
% 0.28/0.48 ( dpsetconstrERa
% 0.28/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.48 @ ( dpsetconstr @ A @ B
% 0.28/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrEL1_type,type,
% 0.28/0.48 dpsetconstrEL1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrEL1,definition,
% 0.28/0.48 ( dpsetconstrEL1
% 0.28/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.48 @ ( dpsetconstr @ A @ B
% 0.28/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrEL2_type,type,
% 0.28/0.48 dpsetconstrEL2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrEL2,definition,
% 0.28/0.48 ( dpsetconstrEL2
% 0.28/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.48 @ ( dpsetconstr @ A @ B
% 0.28/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrER_type,type,
% 0.28/0.48 dpsetconstrER: $o ).
% 0.28/0.48
% 0.28/0.48 thf(dpsetconstrER,definition,
% 0.28/0.48 ( dpsetconstrER
% 0.28/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.48 @ ( dpsetconstr @ A @ B
% 0.28/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(func_type,type,
% 0.28/0.48 func: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcSet_type,type,
% 0.28/0.48 funcSet: $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(funcImageSingleton_type,type,
% 0.28/0.48 funcImageSingleton: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcImageSingleton,definition,
% 0.28/0.48 ( funcImageSingleton
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( singleton
% 0.28/0.48 @ ( dsetconstr @ B
% 0.28/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(apProp_type,type,
% 0.28/0.48 apProp: $o ).
% 0.28/0.48
% 0.28/0.48 thf(apProp,definition,
% 0.28/0.48 ( apProp
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in
% 0.28/0.48 @ ( setunion
% 0.28/0.48 @ ( dsetconstr @ B
% 0.28/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.28/0.48 @ B ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ap_type,type,
% 0.28/0.48 ap: $i > $i > $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(app_type,type,
% 0.28/0.48 app: $o ).
% 0.28/0.48
% 0.28/0.48 thf(app,definition,
% 0.28/0.48 ( app
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(infuncsetfunc_type,type,
% 0.28/0.48 infuncsetfunc: $o ).
% 0.28/0.48
% 0.28/0.48 thf(infuncsetfunc,definition,
% 0.28/0.48 ( infuncsetfunc
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ap2p_type,type,
% 0.28/0.48 ap2p: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ap2p,definition,
% 0.28/0.48 ( ap2p
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(funcinfuncset_type,type,
% 0.28/0.48 funcinfuncset: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcinfuncset,definition,
% 0.28/0.48 ( funcinfuncset
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(lamProp_type,type,
% 0.28/0.48 lamProp: $o ).
% 0.28/0.48
% 0.28/0.48 thf(lamProp,definition,
% 0.28/0.48 ( lamProp
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ( func @ A @ B
% 0.28/0.48 @ ( dpsetconstr @ A @ B
% 0.28/0.48 @ ^ [Xx: $i,Xy: $i] :
% 0.28/0.48 ( ( Xf @ Xx )
% 0.28/0.48 = Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(lam_type,type,
% 0.28/0.48 lam: $i > $i > ( $i > $i ) > $i ).
% 0.28/0.48
% 0.28/0.48 thf(lamp_type,type,
% 0.28/0.48 lamp: $o ).
% 0.28/0.48
% 0.28/0.48 thf(lamp,definition,
% 0.28/0.48 ( lamp
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ( func @ A @ B
% 0.28/0.48 @ ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(lam2p_type,type,
% 0.28/0.48 lam2p: $o ).
% 0.28/0.48
% 0.28/0.48 thf(lam2p,definition,
% 0.28/0.48 ( lam2p
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ( in
% 0.28/0.48 @ ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.28/0.48 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(brelnall1_type,type,
% 0.28/0.48 brelnall1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(brelnall1,definition,
% 0.28/0.48 ( brelnall1
% 0.28/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.48 ( ( breln @ A @ B @ R )
% 0.28/0.48 => ! [Xphi: $i > $o] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ R )
% 0.28/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(brelnall2_type,type,
% 0.28/0.48 brelnall2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(brelnall2,definition,
% 0.28/0.48 ( brelnall2
% 0.28/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.48 ( ( breln @ A @ B @ R )
% 0.28/0.48 => ! [Xphi: $i > $o] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ R )
% 0.28/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ex1E2_type,type,
% 0.28/0.48 ex1E2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ex1E2,definition,
% 0.28/0.48 ( ex1E2
% 0.28/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.48 ( ( ex1 @ A
% 0.28/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( Xphi @ Xx )
% 0.28/0.48 => ( ( Xphi @ Xy )
% 0.28/0.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(funcGraphProp1_type,type,
% 0.28/0.48 funcGraphProp1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcGraphProp1,definition,
% 0.28/0.48 ( funcGraphProp1
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(funcGraphProp3_type,type,
% 0.28/0.48 funcGraphProp3: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcGraphProp3,definition,
% 0.28/0.48 ( funcGraphProp3
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(funcGraphProp2_type,type,
% 0.28/0.48 funcGraphProp2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcGraphProp2,definition,
% 0.28/0.48 ( funcGraphProp2
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.28/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(funcextLem_type,type,
% 0.28/0.48 funcextLem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcextLem,definition,
% 0.28/0.48 ( funcextLem
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ! [Xg: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xg )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(funcGraphProp4_type,type,
% 0.28/0.48 funcGraphProp4: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcGraphProp4,definition,
% 0.28/0.48 ( funcGraphProp4
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.28/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(subbreln_type,type,
% 0.28/0.48 subbreln: $o ).
% 0.28/0.48
% 0.28/0.48 thf(subbreln,definition,
% 0.28/0.48 ( subbreln
% 0.28/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.48 ( ( breln @ A @ B @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln @ A @ B @ S )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(eqbreln_type,type,
% 0.28/0.48 eqbreln: $o ).
% 0.28/0.48
% 0.28/0.48 thf(eqbreln,definition,
% 0.28/0.48 ( eqbreln
% 0.28/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.48 ( ( breln @ A @ B @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln @ A @ B @ S )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.28/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(funcext_type,type,
% 0.28/0.48 funcext: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcext,definition,
% 0.28/0.48 ( funcext
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ! [Xg: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xg )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(funcext2_type,type,
% 0.28/0.48 funcext2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(funcext2,definition,
% 0.28/0.48 ( funcext2
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ! [Xg: $i] :
% 0.28/0.48 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ap2apEq1_type,type,
% 0.28/0.48 ap2apEq1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ap2apEq1,definition,
% 0.28/0.48 ( ap2apEq1
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ap2apEq2_type,type,
% 0.28/0.48 ap2apEq2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ap2apEq2,definition,
% 0.28/0.48 ( ap2apEq2
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(beta1_type,type,
% 0.28/0.48 beta1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(beta1,definition,
% 0.28/0.48 ( beta1
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B
% 0.28/0.48 @ ( lam @ A @ B
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.28/0.48 @ Xx )
% 0.28/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(eta1_type,type,
% 0.28/0.48 eta1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(eta1,definition,
% 0.28/0.48 ( eta1
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ( ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.28/0.48 = Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(lam2lamEq_type,type,
% 0.28/0.48 lam2lamEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(lam2lamEq,definition,
% 0.28/0.48 ( lam2lamEq
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ( ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.28/0.48 = ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(beta2_type,type,
% 0.28/0.48 beta2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(beta2,definition,
% 0.28/0.48 ( beta2
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B
% 0.28/0.48 @ ( lam @ A @ B
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.28/0.48 @ Xx )
% 0.28/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(eta2_type,type,
% 0.28/0.48 eta2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(eta2,definition,
% 0.28/0.48 ( eta2
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ( ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.28/0.48 = Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iffalseProp1_type,type,
% 0.28/0.48 iffalseProp1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iffalseProp1,definition,
% 0.28/0.48 ( iffalseProp1
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ~ Xphi
% 0.28/0.48 => ( in @ Xy
% 0.28/0.48 @ ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iffalseProp2_type,type,
% 0.28/0.48 iffalseProp2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iffalseProp2,definition,
% 0.28/0.48 ( iffalseProp2
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ~ Xphi
% 0.28/0.48 => ( ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) )
% 0.28/0.48 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueProp1_type,type,
% 0.28/0.48 iftrueProp1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueProp1,definition,
% 0.28/0.48 ( iftrueProp1
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( Xphi
% 0.28/0.48 => ( in @ Xx
% 0.28/0.48 @ ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueProp2_type,type,
% 0.28/0.48 iftrueProp2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueProp2,definition,
% 0.28/0.48 ( iftrueProp2
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( Xphi
% 0.28/0.48 => ( ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) )
% 0.28/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ifSingleton_type,type,
% 0.28/0.48 ifSingleton: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ifSingleton,definition,
% 0.28/0.48 ( ifSingleton
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( singleton
% 0.28/0.48 @ ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(if_type,type,
% 0.28/0.48 if: $i > $o > $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(ifp_type,type,
% 0.28/0.48 ifp: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ifp,definition,
% 0.28/0.48 ( ifp
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(theeq_type,type,
% 0.28/0.48 theeq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(theeq,definition,
% 0.28/0.48 ( theeq
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ( singleton @ X )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ X )
% 0.28/0.48 => ( ( setunion @ X )
% 0.28/0.48 = Xx ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iftrue_type,type,
% 0.28/0.48 iftrue: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iftrue,definition,
% 0.28/0.48 ( iftrue
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( Xphi
% 0.28/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.28/0.48 = Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iffalse_type,type,
% 0.28/0.48 iffalse: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iffalse,definition,
% 0.28/0.48 ( iffalse
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ~ Xphi
% 0.28/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.28/0.48 = Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueorfalse_type,type,
% 0.28/0.48 iftrueorfalse: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueorfalse,definition,
% 0.28/0.48 ( iftrueorfalse
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectT_lem_type,type,
% 0.28/0.48 binintersectT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectT_lem,definition,
% 0.28/0.48 ( binintersectT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionT_lem_type,type,
% 0.28/0.48 binunionT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionT_lem,definition,
% 0.28/0.48 ( binunionT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(powersetT_lem_type,type,
% 0.28/0.48 powersetT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(powersetT_lem,definition,
% 0.28/0.48 ( powersetT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setminusT_lem_type,type,
% 0.28/0.48 setminusT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setminusT_lem,definition,
% 0.28/0.48 ( setminusT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementT_lem_type,type,
% 0.28/0.48 complementT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementT_lem,definition,
% 0.28/0.48 ( complementT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setextT_type,type,
% 0.28/0.48 setextT: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setextT,definition,
% 0.28/0.48 ( setextT
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ Y ) ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ Y )
% 0.28/0.48 => ( in @ Xx @ X ) ) )
% 0.28/0.48 => ( X = Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(subsetTI_type,type,
% 0.28/0.48 subsetTI: $o ).
% 0.28/0.48
% 0.28/0.48 thf(subsetTI,definition,
% 0.28/0.48 ( subsetTI
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ Y ) ) )
% 0.28/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(powersetTI1_type,type,
% 0.28/0.48 powersetTI1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(powersetTI1,definition,
% 0.28/0.48 ( powersetTI1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ Y ) ) )
% 0.28/0.48 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(powersetTE1_type,type,
% 0.28/0.48 powersetTE1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(powersetTE1,definition,
% 0.28/0.48 ( powersetTE1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ X @ ( powerset @ Y ) )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementTI1_type,type,
% 0.28/0.48 complementTI1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementTI1,definition,
% 0.28/0.48 ( complementTI1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementTE1_type,type,
% 0.28/0.48 complementTE1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementTE1,definition,
% 0.28/0.48 ( complementTE1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectTELcontra_type,type,
% 0.28/0.48 binintersectTELcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectTELcontra,definition,
% 0.28/0.48 ( binintersectTELcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ X )
% 0.28/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectTERcontra_type,type,
% 0.28/0.48 binintersectTERcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectTERcontra,definition,
% 0.28/0.48 ( binintersectTERcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ Y )
% 0.28/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT_type,type,
% 0.28/0.48 contrasubsetT: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT,definition,
% 0.28/0.48 ( contrasubsetT
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ( ( in @ Xx @ Y )
% 0.28/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT1_type,type,
% 0.28/0.48 contrasubsetT1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT1,definition,
% 0.28/0.48 ( contrasubsetT1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( subset @ X @ Y )
% 0.28/0.48 => ( ~ ( in @ Xx @ Y )
% 0.28/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT2_type,type,
% 0.28/0.48 contrasubsetT2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT2,definition,
% 0.28/0.48 ( contrasubsetT2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ Y )
% 0.28/0.48 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT3_type,type,
% 0.28/0.48 contrasubsetT3: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT3,definition,
% 0.28/0.48 ( contrasubsetT3
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.28/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementI1_type,type,
% 0.28/0.48 doubleComplementI1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementI1,definition,
% 0.28/0.48 ( doubleComplementI1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementE1_type,type,
% 0.28/0.48 doubleComplementE1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementE1,definition,
% 0.28/0.48 ( doubleComplementE1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.28/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementSub1_type,type,
% 0.28/0.48 doubleComplementSub1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementSub1,definition,
% 0.28/0.48 ( doubleComplementSub1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementSub2_type,type,
% 0.28/0.48 doubleComplementSub2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementSub2,definition,
% 0.28/0.48 ( doubleComplementSub2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementEq_type,type,
% 0.28/0.48 doubleComplementEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementEq,definition,
% 0.28/0.48 ( doubleComplementEq
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( X
% 0.28/0.48 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementTnotintersectT_type,type,
% 0.28/0.48 complementTnotintersectT: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementTnotintersectT,definition,
% 0.28/0.48 ( complementTnotintersectT
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementImpComplementIntersect_type,type,
% 0.28/0.48 complementImpComplementIntersect: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementImpComplementIntersect,definition,
% 0.28/0.48 ( complementImpComplementIntersect
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementSubsetComplementIntersect_type,type,
% 0.28/0.48 complementSubsetComplementIntersect: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementSubsetComplementIntersect,definition,
% 0.28/0.48 ( complementSubsetComplementIntersect
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementInPowersetComplementIntersect_type,type,
% 0.28/0.48 complementInPowersetComplementIntersect: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementInPowersetComplementIntersect,definition,
% 0.28/0.48 ( complementInPowersetComplementIntersect
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contraSubsetComplement_type,type,
% 0.28/0.48 contraSubsetComplement: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contraSubsetComplement,definition,
% 0.28/0.48 ( contraSubsetComplement
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ Y )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementTcontraSubset_type,type,
% 0.28/0.48 complementTcontraSubset: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementTcontraSubset,definition,
% 0.28/0.48 ( complementTcontraSubset
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTILcontra_type,type,
% 0.28/0.48 binunionTILcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTILcontra,definition,
% 0.28/0.48 ( binunionTILcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTIRcontra_type,type,
% 0.28/0.48 binunionTIRcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTIRcontra,definition,
% 0.28/0.48 ( binunionTIRcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.48 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInUnion_type,type,
% 0.28/0.48 inIntersectImpInUnion: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInUnion,definition,
% 0.28/0.48 ( inIntersectImpInUnion
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInUnion2_type,type,
% 0.28/0.48 inIntersectImpInUnion2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInUnion2,definition,
% 0.28/0.48 ( inIntersectImpInUnion2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInIntersectUnions_type,type,
% 0.28/0.48 inIntersectImpInIntersectUnions: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInIntersectUnions,definition,
% 0.28/0.48 ( inIntersectImpInIntersectUnions
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(intersectInPowersetIntersectUnions_type,type,
% 0.28/0.48 intersectInPowersetIntersectUnions: $o ).
% 0.28/0.48
% 0.28/0.48 thf(intersectInPowersetIntersectUnions,definition,
% 0.28/0.48 ( intersectInPowersetIntersectUnions
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inComplementUnionImpNotIn1_type,type,
% 0.28/0.48 inComplementUnionImpNotIn1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inComplementUnionImpNotIn1,definition,
% 0.28/0.48 ( inComplementUnionImpNotIn1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inComplementUnionImpInComplement1_type,type,
% 0.28/0.48 inComplementUnionImpInComplement1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inComplementUnionImpInComplement1,definition,
% 0.28/0.48 ( inComplementUnionImpInComplement1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTE_type,type,
% 0.28/0.48 binunionTE: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTE,definition,
% 0.28/0.48 ( binunionTE
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.48 => ( ( ( in @ Xx @ X )
% 0.28/0.48 => Xphi )
% 0.28/0.48 => ( ( ( in @ Xx @ Y )
% 0.28/0.48 => Xphi )
% 0.28/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTEcontra_type,type,
% 0.28/0.48 binunionTEcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTEcontra,definition,
% 0.28/0.48 ( binunionTEcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ X )
% 0.28/0.48 => ( ~ ( in @ Xx @ Y )
% 0.28/0.48 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a1_type,type,
% 0.28/0.48 demorgan2a1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a1,definition,
% 0.28/0.48 ( demorgan2a1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementUnionInPowersetComplement_type,type,
% 0.28/0.48 complementUnionInPowersetComplement: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementUnionInPowersetComplement,definition,
% 0.28/0.48 ( complementUnionInPowersetComplement
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a2_type,type,
% 0.28/0.48 demorgan2a2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a2,definition,
% 0.28/0.48 ( demorgan2a2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1a_type,type,
% 0.28/0.48 demorgan1a: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1a,definition,
% 0.28/0.48 ( demorgan1a
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1b_type,type,
% 0.28/0.48 demorgan1b: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1b,definition,
% 0.28/0.48 ( demorgan1b
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1_type,type,
% 0.28/0.48 demorgan1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1,definition,
% 0.28/0.48 ( demorgan1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.28/0.48 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a_type,type,
% 0.28/0.48 demorgan2a: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a,definition,
% 0.28/0.48 ( demorgan2a
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2b2_type,type,
% 0.28/0.48 demorgan2b2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2b2,definition,
% 0.28/0.48 ( demorgan2b2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2b_type,type,
% 0.28/0.48 demorgan2b: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2b,definition,
% 0.28/0.48 ( demorgan2b
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2_type,type,
% 0.28/0.48 demorgan2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2,definition,
% 0.28/0.48 ( demorgan2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.28/0.48 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule0_type,type,
% 0.28/0.48 woz13rule0: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule0,definition,
% 0.28/0.48 ( woz13rule0
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule1_type,type,
% 0.28/0.48 woz13rule1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule1,definition,
% 0.28/0.48 ( woz13rule1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ Z )
% 0.28/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule2_type,type,
% 0.28/0.48 woz13rule2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule2,definition,
% 0.28/0.48 ( woz13rule2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ Y @ Z )
% 0.28/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule3_type,type,
% 0.28/0.48 woz13rule3: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule3,definition,
% 0.28/0.48 ( woz13rule3
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ Y )
% 0.28/0.48 => ( ( subset @ X @ Z )
% 0.28/0.48 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule4_type,type,
% 0.28/0.48 woz13rule4: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule4,definition,
% 0.28/0.48 ( woz13rule4
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ! [W: $i] :
% 0.28/0.48 ( ( in @ W @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ Z )
% 0.28/0.48 => ( ( subset @ Y @ W )
% 0.28/0.48 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_1_type,type,
% 0.28/0.48 woz1_1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_1,definition,
% 0.28/0.48 ( woz1_1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_2_type,type,
% 0.28/0.48 woz1_2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_2,definition,
% 0.28/0.48 ( woz1_2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ! [W: $i] :
% 0.28/0.48 ( ( in @ W @ ( powerset @ A ) )
% 0.28/0.48 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.28/0.48 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_3_type,type,
% 0.28/0.48 woz1_3: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_3,definition,
% 0.28/0.48 ( woz1_3
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_4_type,type,
% 0.28/0.48 woz1_4: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_4,definition,
% 0.28/0.48 ( woz1_4
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_5_type,type,
% 0.28/0.48 woz1_5: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz1_5,definition,
% 0.28/0.48 ( woz1_5
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1_type,type,
% 0.28/0.48 breln1: $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1all2_type,type,
% 0.28/0.48 breln1all2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1all2,definition,
% 0.28/0.48 ( breln1all2
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [Xphi: $i > $o] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ R )
% 0.28/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1Set_type,type,
% 0.28/0.48 breln1Set: $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(breln1SetBreln1_type,type,
% 0.28/0.48 breln1SetBreln1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1SetBreln1,definition,
% 0.28/0.48 ( breln1SetBreln1
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( in @ R @ ( breln1Set @ A ) )
% 0.28/0.48 => ( breln1 @ A @ R ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(transitive_type,type,
% 0.28/0.48 transitive: $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(antisymmetric_type,type,
% 0.28/0.48 antisymmetric: $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(reflexive_type,type,
% 0.28/0.48 reflexive: $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(refltransitive_type,type,
% 0.28/0.48 refltransitive: $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(refllinearorder_type,type,
% 0.28/0.48 refllinearorder: $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(reflwellordering_type,type,
% 0.28/0.48 reflwellordering: $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(choice2fnsingleton_type,type,
% 0.28/0.48 choice2fnsingleton: $o ).
% 0.28/0.48
% 0.28/0.48 thf(choice2fnsingleton,definition,
% 0.28/0.48 ( choice2fnsingleton
% 0.28/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ B )
% 0.28/0.48 & ( Xphi @ Xx @ Xy ) ) )
% 0.28/0.48 => ! [R: $i] :
% 0.28/0.48 ( ( in @ R @ ( breln1Set @ B ) )
% 0.28/0.48 => ( ( reflwellordering @ B @ R )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( singleton
% 0.28/0.48 @ ( dsetconstr @ B
% 0.28/0.48 @ ^ [Xy: $i] :
% 0.28/0.48 ( ( Xphi @ Xx @ Xy )
% 0.28/0.48 & ! [Xz: $i] :
% 0.28/0.48 ( ( in @ Xz @ B )
% 0.28/0.48 => ( ( Xphi @ Xx @ Xz )
% 0.28/0.48 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setOfPairsIsBReln1_type,type,
% 0.28/0.48 setOfPairsIsBReln1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setOfPairsIsBReln1,definition,
% 0.28/0.48 ( setOfPairsIsBReln1
% 0.28/0.48 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.28/0.48 ( breln1 @ A
% 0.28/0.48 @ ( dpsetconstr @ A @ A
% 0.28/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1all1_type,type,
% 0.28/0.48 breln1all1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1all1,definition,
% 0.28/0.48 ( breln1all1
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [Xphi: $i > $o] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ R )
% 0.28/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(subbreln1_type,type,
% 0.28/0.48 subbreln1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(subbreln1,definition,
% 0.28/0.48 ( subbreln1
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(eqbreln1_type,type,
% 0.28/0.48 eqbreln1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(eqbreln1,definition,
% 0.28/0.48 ( eqbreln1
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.28/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1invset_type,type,
% 0.28/0.48 breln1invset: $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(breln1invprop_type,type,
% 0.28/0.48 breln1invprop: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1invprop,definition,
% 0.28/0.48 ( breln1invprop
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1invI_type,type,
% 0.28/0.48 breln1invI: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1invI,definition,
% 0.28/0.48 ( breln1invI
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1invE_type,type,
% 0.28/0.48 breln1invE: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1invE,definition,
% 0.28/0.48 ( breln1invE
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compset_type,type,
% 0.28/0.48 breln1compset: $i > $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compprop_type,type,
% 0.28/0.48 breln1compprop: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compprop,definition,
% 0.28/0.48 ( breln1compprop
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compI_type,type,
% 0.28/0.48 breln1compI: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compI,definition,
% 0.28/0.48 ( breln1compI
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ! [Xz: $i] :
% 0.28/0.48 ( ( in @ Xz @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.28/0.48 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compE_type,type,
% 0.28/0.48 breln1compE: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compE,definition,
% 0.28/0.48 ( breln1compE
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.28/0.48 => ? [Xz: $i] :
% 0.28/0.48 ( ( in @ Xz @ A )
% 0.28/0.48 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.28/0.48 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compEex_type,type,
% 0.28/0.48 breln1compEex: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1compEex,definition,
% 0.28/0.48 ( breln1compEex
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.28/0.48 => ! [Xphi: $o] :
% 0.28/0.48 ( ! [Xz: $i] :
% 0.28/0.48 ( ( in @ Xz @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.28/0.48 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.28/0.48 => Xphi ) ) )
% 0.28/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionprop_type,type,
% 0.28/0.48 breln1unionprop: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionprop,definition,
% 0.28/0.48 ( breln1unionprop
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionIL_type,type,
% 0.28/0.48 breln1unionIL: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionIL,definition,
% 0.28/0.48 ( breln1unionIL
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionIR_type,type,
% 0.28/0.48 breln1unionIR: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionIR,definition,
% 0.28/0.48 ( breln1unionIR
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionI_type,type,
% 0.28/0.48 breln1unionI: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionI,definition,
% 0.28/0.48 ( breln1unionI
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.28/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionE_type,type,
% 0.28/0.48 breln1unionE: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionE,definition,
% 0.28/0.48 ( breln1unionE
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionEcases_type,type,
% 0.28/0.48 breln1unionEcases: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionEcases,definition,
% 0.28/0.48 ( breln1unionEcases
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.28/0.48 => ! [Xphi: $o] :
% 0.28/0.48 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.48 => Xphi )
% 0.28/0.48 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.48 => Xphi )
% 0.28/0.48 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionCommutes_type,type,
% 0.28/0.48 breln1unionCommutes: $o ).
% 0.28/0.48
% 0.28/0.48 thf(breln1unionCommutes,definition,
% 0.28/0.48 ( breln1unionCommutes
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ( ( binunion @ R @ S )
% 0.28/0.48 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz2Ex_type,type,
% 0.28/0.48 woz2Ex: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz2Ex,definition,
% 0.28/0.48 ( woz2Ex
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ( R
% 0.28/0.48 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz2W_type,type,
% 0.28/0.48 woz2W: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz2W,definition,
% 0.28/0.48 ( woz2W
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.28/0.48 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz2A_type,type,
% 0.28/0.48 woz2A: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz2A,definition,
% 0.28/0.48 ( woz2A
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [T: $i] :
% 0.28/0.48 ( ( breln1 @ A @ T )
% 0.28/0.48 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.28/0.48 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz2B_type,type,
% 0.28/0.48 woz2B: $o ).
% 0.28/0.48
% 0.28/0.48 thf(woz2B,definition,
% 0.28/0.48 ( woz2B
% 0.28/0.48 = ( ! [A: $i,R: $i] :
% 0.28/0.48 ( ( breln1 @ A @ R )
% 0.28/0.48 => ! [S: $i] :
% 0.28/0.48 ( ( breln1 @ A @ S )
% 0.28/0.48 => ! [T: $i] :
% 0.28/0.48 ( ( breln1 @ A @ T )
% 0.28/0.48 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.28/0.48 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(image1Ex_type,type,
% 0.28/0.48 image1Ex: $o ).
% 0.28/0.48
% 0.28/0.48 thf(image1Ex,definition,
% 0.28/0.48 ( image1Ex
% 0.28/0.48 = ( ! [A: $i,Xf: $i > $i] :
% 0.28/0.48 ? [B: $i] :
% 0.28/0.48 ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ B )
% 0.28/0.48 <=> ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( Xx
% 0.28/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(image1Ex1_type,type,
% 0.28/0.48 image1Ex1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(image1Ex1,definition,
% 0.28/0.48 ( image1Ex1
% 0.28/0.48 = ( ! [A: $i,Xf: $i > $i] :
% 0.28/0.48 ( exu
% 0.28/0.48 @ ^ [B: $i] :
% 0.28/0.48 ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ B )
% 0.28/0.48 <=> ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( Xx
% 0.28/0.48 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(image1_type,type,
% 0.28/0.48 image1: $i > ( $i > $i ) > $i ).
% 0.28/0.48
% 0.28/0.48 thf(image1Equiv_type,type,
% 0.28/0.48 image1Equiv: $o ).
% 0.28/0.48
% 0.28/0.48 thf(image1Equiv,definition,
% 0.28/0.48 ( image1Equiv
% 0.28/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx
% 0.28/0.48 @ ( image1 @ A
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.28/0.48 <=> ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( Xx
% 0.28/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(image1E_type,type,
% 0.28/0.48 image1E: $o ).
% 0.28/0.48
% 0.28/0.48 thf(image1E,definition,
% 0.28/0.48 ( image1E
% 0.28/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx
% 0.28/0.48 @ ( image1 @ A
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.28/0.48 => ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( Xx
% 0.28/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(image1I_type,type,
% 0.28/0.48 image1I: $o ).
% 0.28/0.48
% 0.28/0.48 thf(image1I,definition,
% 0.28/0.48 ( image1I
% 0.28/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.28/0.48 ( ? [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 & ( Xx
% 0.28/0.48 = ( Xf @ Xy ) ) )
% 0.28/0.48 => ( in @ Xx
% 0.28/0.48 @ ( image1 @ A
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(injective_type,type,
% 0.28/0.48 injective: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSet_type,type,
% 0.28/0.48 injFuncSet: $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncInInjFuncSet_type,type,
% 0.28/0.48 injFuncInInjFuncSet: $o ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncInInjFuncSet,definition,
% 0.28/0.48 ( injFuncInInjFuncSet
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ( ( injective @ A @ B @ Xf )
% 0.28/0.48 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSetFuncIn_type,type,
% 0.28/0.48 injFuncSetFuncIn: $o ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSetFuncIn,definition,
% 0.28/0.48 ( injFuncSetFuncIn
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.28/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSetFuncInj_type,type,
% 0.28/0.48 injFuncSetFuncInj: $o ).
% 0.28/0.48
% 0.28/0.48 thf(injFuncSetFuncInj,definition,
% 0.28/0.48 ( injFuncSetFuncInj
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.28/0.48 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(surjective_type,type,
% 0.28/0.48 surjective: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(surjFuncSet_type,type,
% 0.28/0.48 surjFuncSet: $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(surjFuncSetFuncIn_type,type,
% 0.28/0.48 surjFuncSetFuncIn: $o ).
% 0.28/0.48
% 0.28/0.48 thf(surjFuncSetFuncIn,definition,
% 0.28/0.48 ( surjFuncSetFuncIn
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.28/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(surjFuncSetFuncSurj_type,type,
% 0.28/0.48 surjFuncSetFuncSurj: $o ).
% 0.28/0.48
% 0.28/0.48 thf(surjFuncSetFuncSurj,definition,
% 0.28/0.48 ( surjFuncSetFuncSurj
% 0.28/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.28/0.48 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(leftInvIsSurj_type,type,
% 0.28/0.48 leftInvIsSurj: $o ).
% 0.28/0.48
% 0.28/0.48 thf(leftInvIsSurj,definition,
% 0.28/0.48 ( leftInvIsSurj
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ! [Xg: $i] :
% 0.28/0.48 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.28/0.48 = Xx ) )
% 0.28/0.48 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(surjCantorThm_type,type,
% 0.28/0.48 surjCantorThm: $o ).
% 0.28/0.48
% 0.28/0.48 thf(surjCantorThm,definition,
% 0.28/0.48 ( surjCantorThm
% 0.28/0.48 = ( ! [A: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.28/0.48 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(foundation2_type,type,
% 0.28/0.48 foundation2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(foundation2,definition,
% 0.28/0.48 ( foundation2
% 0.28/0.48 = ( ! [A: $i] :
% 0.28/0.48 ( ( nonempty @ A )
% 0.28/0.48 => ? [X: $i] :
% 0.28/0.48 ( ( in @ X @ A )
% 0.28/0.48 & ( ( binintersect @ X @ A )
% 0.28/0.48 = emptyset ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(notinself_type,type,
% 0.28/0.48 notinself: $o ).
% 0.28/0.48
% 0.28/0.48 thf(notinself,definition,
% 0.28/0.48 ( notinself
% 0.28/0.48 = ( ! [A: $i] :
% 0.28/0.48 ~ ( in @ A @ A ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(notinself2_type,type,
% 0.28/0.48 notinself2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(notinself2,definition,
% 0.28/0.48 ( notinself2
% 0.28/0.48 = ( ! [A: $i,B: $i] :
% 0.28/0.48 ( ( in @ A @ B )
% 0.28/0.48 => ~ ( in @ B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(omegaS_type,type,
% 0.28/0.48 omegaS: $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(omegaSp_type,type,
% 0.28/0.48 omegaSp: $o ).
% 0.28/0.48
% 0.28/0.48 thf(omegaSp,definition,
% 0.28/0.48 ( omegaSp
% 0.28/0.48 = ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ omega )
% 0.28/0.48 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(omegaSclos_type,type,
% 0.28/0.48 omegaSclos: $o ).
% 0.28/0.48
% 0.28/0.48 thf(omegaSclos,definition,
% 0.28/0.48 ( omegaSclos
% 0.28/0.48 = ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ omega )
% 0.28/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(peano0notS_type,type,
% 0.28/0.48 peano0notS: $o ).
% 0.28/0.48
% 0.28/0.48 thf(peano0notS,definition,
% 0.28/0.48 ( peano0notS
% 0.28/0.48 = ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ omega )
% 0.28/0.48 => ( ( omegaS @ Xx )
% 0.28/0.48 != emptyset ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(peano3_type,type,
% 0.28/0.48 peano3: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(peano4_type,type,
% 0.28/0.48 peano4: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(peano5_type,type,
% 0.28/0.48 peano5: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(peanoSinj_type,type,
% 0.28/0.48 peanoSinj: $o ).
% 0.28/0.48
% 0.28/0.48 thf(peanoSinj,definition,
% 0.28/0.48 ( peanoSinj
% 0.28/0.48 = ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ omega )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ omega )
% 0.28/0.48 => ( ( ( omegaS @ Xx )
% 0.28/0.48 = ( omegaS @ Xy ) )
% 0.28/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(peano_type,type,
% 0.28/0.48 peano: $i > $i > $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(transitiveset_type,type,
% 0.28/0.48 transitiveset: $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(transitivesetOp1_type,type,
% 0.28/0.48 transitivesetOp1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(transitivesetOp1,definition,
% 0.28/0.48 ( transitivesetOp1
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ( transitiveset @ X )
% 0.28/0.48 => ! [A: $i] :
% 0.28/0.48 ( ( in @ A @ X )
% 0.28/0.48 => ( subset @ A @ X ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binintTransitive_type,type,
% 0.28/0.48 binintTransitive: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binintTransitive,definition,
% 0.28/0.48 ( binintTransitive
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ( transitiveset @ X )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( transitiveset @ Y )
% 0.28/0.48 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(transitivesetOp2_type,type,
% 0.28/0.48 transitivesetOp2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(transitivesetOp2,definition,
% 0.28/0.48 ( transitivesetOp2
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ( transitiveset @ X )
% 0.28/0.48 => ! [A: $i,Xx: $i] :
% 0.28/0.48 ( ( in @ A @ X )
% 0.28/0.48 => ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setunionTransitive_type,type,
% 0.28/0.48 setunionTransitive: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setunionTransitive,definition,
% 0.28/0.48 ( setunionTransitive
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ X )
% 0.28/0.48 => ( transitiveset @ Xx ) )
% 0.28/0.48 => ( transitiveset @ ( setunion @ X ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(stricttotalorderedByIn_type,type,
% 0.28/0.48 stricttotalorderedByIn: $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(stricttotalorderedByIn,definition,
% 0.28/0.48 ( stricttotalorderedByIn
% 0.28/0.48 = ( ^ [A: $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [X: $i] :
% 0.28/0.48 ( ( in @ X @ A )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ A )
% 0.28/0.48 => ( ( ( in @ Xx @ X )
% 0.28/0.48 & ( in @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ Y ) ) ) ) )
% 0.28/0.48 & ! [X: $i] :
% 0.28/0.48 ( ( in @ X @ A )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ A )
% 0.28/0.48 => ( ( X = Y )
% 0.28/0.48 | ( in @ X @ Y )
% 0.28/0.48 | ( in @ Y @ X ) ) ) )
% 0.28/0.48 & ! [X: $i] :
% 0.28/0.48 ( ( in @ X @ A )
% 0.28/0.48 => ~ ( in @ X @ X ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(wellorderedByIn_type,type,
% 0.28/0.48 wellorderedByIn: $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(wellorderedByIn,definition,
% 0.28/0.48 ( wellorderedByIn
% 0.28/0.48 = ( ^ [A: $i] :
% 0.28/0.48 ( ( stricttotalorderedByIn @ A )
% 0.28/0.48 & ! [X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( ( nonempty @ X )
% 0.28/0.48 => ? [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ X )
% 0.28/0.48 & ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ X )
% 0.28/0.48 => ( ( Xx = Y )
% 0.28/0.48 | ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ordinal_type,type,
% 0.28/0.48 ordinal: $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(ordinal,definition,
% 0.28/0.48 ( ordinal
% 0.28/0.48 = ( ^ [Xx: $i] :
% 0.28/0.48 ( ( transitiveset @ Xx )
% 0.28/0.48 & ( wellorderedByIn @ Xx ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(limitOrdinal_type,type,
% 0.28/0.48 limitOrdinal: $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(ordinalMinLem1_type,type,
% 0.28/0.48 ordinalMinLem1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ordinalMinLem1,definition,
% 0.28/0.48 ( ordinalMinLem1
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ( ordinal @ X )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( ordinal @ Y )
% 0.28/0.48 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ordinalTransSet_type,type,
% 0.28/0.48 ordinalTransSet: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ordinalTransSet,definition,
% 0.28/0.48 ( ordinalTransSet
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ( ordinal @ X )
% 0.28/0.48 => ! [Xx: $i,A: $i] :
% 0.28/0.48 ( ( in @ A @ X )
% 0.28/0.48 => ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ordinalTransSet1_type,type,
% 0.28/0.48 ordinalTransSet1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ordinalTransSet1,definition,
% 0.28/0.48 ( ordinalTransSet1
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ( ordinal @ X )
% 0.28/0.48 => ! [A: $i] :
% 0.28/0.48 ( ( in @ A @ X )
% 0.28/0.48 => ( subset @ A @ X ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setunionOrdinalLem1_type,type,
% 0.28/0.48 setunionOrdinalLem1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setunionOrdinalLem1,definition,
% 0.28/0.48 ( setunionOrdinalLem1
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ X )
% 0.28/0.48 => ( ordinal @ Xx ) )
% 0.28/0.48 => ( transitiveset @ ( setunion @ X ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(succOrdinal_type,type,
% 0.28/0.48 succOrdinal: $i > $o ).
% 0.28/0.48
% 0.28/0.48 thf(emptysetOrdinal_type,type,
% 0.28/0.48 emptysetOrdinal: $o ).
% 0.28/0.48
% 0.28/0.48 thf(emptysetOrdinal,definition,
% 0.28/0.48 ( emptysetOrdinal
% 0.28/0.48 = ( ordinal @ emptyset ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ordinalIrrefl,conjecture,
% 0.28/0.48 ( setextAx
% 0.28/0.48 => ( emptysetAx
% 0.28/0.48 => ( setadjoinAx
% 0.28/0.48 => ( powersetAx
% 0.28/0.48 => ( setunionAx
% 0.28/0.48 => ( omega0Ax
% 0.28/0.48 => ( omegaSAx
% 0.28/0.48 => ( omegaIndAx
% 0.28/0.48 => ( replAx
% 0.28/0.48 => ( foundationAx
% 0.28/0.48 => ( wellorderingAx
% 0.28/0.48 => ( descrp
% 0.28/0.48 => ( dsetconstrI
% 0.28/0.48 => ( dsetconstrEL
% 0.28/0.48 => ( dsetconstrER
% 0.28/0.48 => ( exuE1
% 0.28/0.48 => ( prop2setE
% 0.28/0.48 => ( emptysetE
% 0.28/0.48 => ( emptysetimpfalse
% 0.28/0.48 => ( notinemptyset
% 0.28/0.48 => ( exuE3e
% 0.28/0.48 => ( setext
% 0.28/0.48 => ( emptyI
% 0.28/0.48 => ( noeltsimpempty
% 0.28/0.48 => ( setbeta
% 0.28/0.48 => ( nonemptyE1
% 0.28/0.48 => ( nonemptyI
% 0.28/0.48 => ( nonemptyI1
% 0.28/0.48 => ( setadjoinIL
% 0.28/0.48 => ( emptyinunitempty
% 0.28/0.48 => ( setadjoinIR
% 0.28/0.48 => ( setadjoinE
% 0.28/0.48 => ( setadjoinOr
% 0.28/0.48 => ( setoftrueEq
% 0.28/0.48 => ( powersetI
% 0.28/0.48 => ( emptyinPowerset
% 0.28/0.48 => ( emptyInPowerset
% 0.28/0.48 => ( powersetE
% 0.28/0.48 => ( setunionI
% 0.28/0.48 => ( setunionE
% 0.28/0.48 => ( subPowSU
% 0.28/0.48 => ( exuE2
% 0.28/0.48 => ( nonemptyImpWitness
% 0.28/0.48 => ( uniqinunit
% 0.28/0.48 => ( notinsingleton
% 0.28/0.48 => ( eqinunit
% 0.28/0.48 => ( singletonsswitch
% 0.28/0.48 => ( upairsetE
% 0.28/0.48 => ( upairsetIL
% 0.28/0.48 => ( upairsetIR
% 0.28/0.48 => ( emptyE1
% 0.28/0.48 => ( vacuousDall
% 0.28/0.48 => ( quantDeMorgan1
% 0.28/0.48 => ( quantDeMorgan2
% 0.28/0.48 => ( quantDeMorgan3
% 0.28/0.48 => ( quantDeMorgan4
% 0.28/0.48 => ( prop2setI
% 0.28/0.48 => ( prop2set2propI
% 0.28/0.48 => ( notdexE
% 0.28/0.48 => ( notdallE
% 0.28/0.48 => ( exuI1
% 0.28/0.48 => ( exuI3
% 0.28/0.48 => ( exuI2
% 0.28/0.48 => ( inCongP
% 0.28/0.48 => ( in__Cong
% 0.28/0.48 => ( exuE3u
% 0.28/0.48 => ( exu__Cong
% 0.28/0.48 => ( emptyset__Cong
% 0.28/0.48 => ( setadjoin__Cong
% 0.28/0.48 => ( powerset__Cong
% 0.28/0.48 => ( setunion__Cong
% 0.28/0.48 => ( omega__Cong
% 0.28/0.48 => ( exuEu
% 0.28/0.48 => ( descr__Cong
% 0.28/0.48 => ( dsetconstr__Cong
% 0.28/0.48 => ( subsetI1
% 0.28/0.48 => ( eqimpsubset2
% 0.28/0.48 => ( eqimpsubset1
% 0.28/0.48 => ( subsetI2
% 0.28/0.48 => ( emptysetsubset
% 0.28/0.48 => ( subsetE
% 0.28/0.48 => ( subsetE2
% 0.28/0.48 => ( notsubsetI
% 0.28/0.48 => ( notequalI1
% 0.28/0.48 => ( notequalI2
% 0.28/0.48 => ( subsetRefl
% 0.28/0.48 => ( subsetTrans
% 0.28/0.48 => ( setadjoinSub
% 0.28/0.48 => ( setadjoinSub2
% 0.28/0.48 => ( subset2powerset
% 0.28/0.48 => ( setextsub
% 0.28/0.48 => ( subsetemptysetimpeq
% 0.28/0.48 => ( powersetI1
% 0.28/0.48 => ( powersetE1
% 0.28/0.48 => ( inPowerset
% 0.28/0.48 => ( powersetsubset
% 0.28/0.48 => ( sepInPowerset
% 0.28/0.48 => ( sepSubset
% 0.28/0.48 => ( binunionIL
% 0.28/0.48 => ( upairset2IR
% 0.28/0.48 => ( binunionIR
% 0.28/0.48 => ( binunionEcases
% 0.28/0.48 => ( binunionE
% 0.28/0.48 => ( binunionLsub
% 0.28/0.48 => ( binunionRsub
% 0.28/0.48 => ( binintersectI
% 0.28/0.48 => ( binintersectSubset5
% 0.28/0.48 => ( binintersectEL
% 0.28/0.48 => ( binintersectLsub
% 0.28/0.48 => ( binintersectSubset2
% 0.28/0.48 => ( binintersectSubset3
% 0.28/0.48 => ( binintersectER
% 0.28/0.48 => ( disjointsetsI1
% 0.28/0.48 => ( binintersectRsub
% 0.28/0.48 => ( binintersectSubset4
% 0.28/0.48 => ( binintersectSubset1
% 0.28/0.48 => ( bs114d
% 0.28/0.48 => ( setminusI
% 0.28/0.48 => ( setminusEL
% 0.28/0.48 => ( setminusER
% 0.28/0.48 => ( setminusSubset2
% 0.28/0.48 => ( setminusERneg
% 0.28/0.48 => ( setminusELneg
% 0.28/0.48 => ( setminusILneg
% 0.28/0.48 => ( setminusIRneg
% 0.28/0.48 => ( setminusLsub
% 0.28/0.48 => ( setminusSubset1
% 0.28/0.48 => ( symdiffE
% 0.28/0.48 => ( symdiffI1
% 0.28/0.48 => ( symdiffI2
% 0.28/0.48 => ( symdiffIneg1
% 0.28/0.48 => ( symdiffIneg2
% 0.28/0.48 => ( secondinupair
% 0.28/0.48 => ( setukpairIL
% 0.28/0.48 => ( setukpairIR
% 0.28/0.48 => ( kpairiskpair
% 0.28/0.48 => ( kpairp
% 0.28/0.48 => ( singletonsubset
% 0.28/0.48 => ( singletoninpowerset
% 0.28/0.48 => ( singletoninpowunion
% 0.28/0.48 => ( upairset2E
% 0.28/0.48 => ( upairsubunion
% 0.28/0.48 => ( upairinpowunion
% 0.28/0.48 => ( ubforcartprodlem1
% 0.28/0.48 => ( ubforcartprodlem2
% 0.28/0.48 => ( ubforcartprodlem3
% 0.28/0.48 => ( cartprodpairin
% 0.28/0.48 => ( cartprodmempair1
% 0.28/0.48 => ( cartprodmempair
% 0.28/0.48 => ( setunionE2
% 0.28/0.48 => ( setunionsingleton1
% 0.28/0.48 => ( setunionsingleton2
% 0.28/0.48 => ( setunionsingleton
% 0.28/0.48 => ( singletonprop
% 0.28/0.48 => ( ex1E1
% 0.28/0.48 => ( ex1I
% 0.28/0.48 => ( ex1I2
% 0.28/0.48 => ( singletonsuniq
% 0.28/0.48 => ( setukpairinjL1
% 0.28/0.48 => ( kfstsingleton
% 0.28/0.48 => ( theprop
% 0.28/0.48 => ( kfstpairEq
% 0.28/0.48 => ( cartprodfstin
% 0.28/0.48 => ( setukpairinjL2
% 0.28/0.48 => ( setukpairinjL
% 0.28/0.48 => ( setukpairinjR11
% 0.28/0.48 => ( setukpairinjR12
% 0.28/0.48 => ( setukpairinjR1
% 0.28/0.48 => ( upairequniteq
% 0.28/0.48 => ( setukpairinjR2
% 0.28/0.48 => ( setukpairinjR
% 0.28/0.48 => ( ksndsingleton
% 0.28/0.48 => ( ksndpairEq
% 0.28/0.48 => ( kpairsurjEq
% 0.28/0.48 => ( cartprodsndin
% 0.28/0.48 => ( cartprodpairmemEL
% 0.28/0.48 => ( cartprodpairmemER
% 0.28/0.48 => ( cartprodmempaircEq
% 0.28/0.48 => ( cartprodfstpairEq
% 0.28/0.48 => ( cartprodsndpairEq
% 0.28/0.48 => ( cartprodpairsurjEq
% 0.28/0.48 => ( dpsetconstrI
% 0.28/0.48 => ( dpsetconstrSub
% 0.28/0.48 => ( setOfPairsIsBReln
% 0.28/0.48 => ( dpsetconstrERa
% 0.28/0.48 => ( dpsetconstrEL1
% 0.28/0.48 => ( dpsetconstrEL2
% 0.28/0.48 => ( dpsetconstrER
% 0.28/0.48 => ( funcImageSingleton
% 0.28/0.48 => ( apProp
% 0.28/0.48 => ( app
% 0.28/0.48 => ( infuncsetfunc
% 0.28/0.48 => ( ap2p
% 0.28/0.48 => ( funcinfuncset
% 0.28/0.48 => ( lamProp
% 0.28/0.48 => ( lamp
% 0.28/0.48 => ( lam2p
% 0.28/0.48 => ( brelnall1
% 0.28/0.48 => ( brelnall2
% 0.28/0.48 => ( ex1E2
% 0.28/0.48 => ( funcGraphProp1
% 0.28/0.48 => ( funcGraphProp3
% 0.28/0.48 => ( funcGraphProp2
% 0.28/0.48 => ( funcextLem
% 0.28/0.48 => ( funcGraphProp4
% 0.28/0.48 => ( subbreln
% 0.28/0.48 => ( eqbreln
% 0.28/0.48 => ( funcext
% 0.28/0.48 => ( funcext2
% 0.28/0.48 => ( ap2apEq1
% 0.28/0.48 => ( ap2apEq2
% 0.28/0.48 => ( beta1
% 0.28/0.48 => ( eta1
% 0.28/0.48 => ( lam2lamEq
% 0.28/0.48 => ( beta2
% 0.28/0.48 => ( eta2
% 0.28/0.48 => ( iffalseProp1
% 0.28/0.48 => ( iffalseProp2
% 0.28/0.48 => ( iftrueProp1
% 0.28/0.48 => ( iftrueProp2
% 0.28/0.48 => ( ifSingleton
% 0.28/0.48 => ( ifp
% 0.28/0.48 => ( theeq
% 0.28/0.48 => ( iftrue
% 0.28/0.48 => ( iffalse
% 0.28/0.48 => ( iftrueorfalse
% 0.28/0.48 => ( binintersectT_lem
% 0.28/0.48 => ( binunionT_lem
% 0.28/0.48 => ( powersetT_lem
% 0.28/0.48 => ( setminusT_lem
% 0.28/0.48 => ( complementT_lem
% 0.28/0.48 => ( setextT
% 0.28/0.48 => ( subsetTI
% 0.28/0.48 => ( powersetTI1
% 0.28/0.48 => ( powersetTE1
% 0.28/0.48 => ( complementTI1
% 0.28/0.48 => ( complementTE1
% 0.28/0.48 => ( binintersectTELcontra
% 0.28/0.48 => ( binintersectTERcontra
% 0.28/0.48 => ( contrasubsetT
% 0.28/0.48 => ( contrasubsetT1
% 0.28/0.48 => ( contrasubsetT2
% 0.28/0.48 => ( contrasubsetT3
% 0.28/0.48 => ( doubleComplementI1
% 0.28/0.48 => ( doubleComplementE1
% 0.28/0.48 => ( doubleComplementSub1
% 0.28/0.48 => ( doubleComplementSub2
% 0.28/0.48 => ( doubleComplementEq
% 0.28/0.48 => ( complementTnotintersectT
% 0.28/0.48 => ( complementImpComplementIntersect
% 0.28/0.48 => ( complementSubsetComplementIntersect
% 0.28/0.48 => ( complementInPowersetComplementIntersect
% 0.28/0.48 => ( contraSubsetComplement
% 0.28/0.48 => ( complementTcontraSubset
% 0.28/0.48 => ( binunionTILcontra
% 0.28/0.48 => ( binunionTIRcontra
% 0.28/0.48 => ( inIntersectImpInUnion
% 0.28/0.48 => ( inIntersectImpInUnion2
% 0.28/0.48 => ( inIntersectImpInIntersectUnions
% 0.28/0.48 => ( intersectInPowersetIntersectUnions
% 0.28/0.48 => ( inComplementUnionImpNotIn1
% 0.28/0.48 => ( inComplementUnionImpInComplement1
% 0.28/0.48 => ( binunionTE
% 0.28/0.48 => ( binunionTEcontra
% 0.28/0.48 => ( demorgan2a1
% 0.28/0.48 => ( complementUnionInPowersetComplement
% 0.28/0.48 => ( demorgan2a2
% 0.28/0.48 => ( demorgan1a
% 0.28/0.48 => ( demorgan1b
% 0.28/0.48 => ( demorgan1
% 0.28/0.48 => ( demorgan2a
% 0.28/0.48 => ( demorgan2b2
% 0.28/0.48 => ( demorgan2b
% 0.28/0.48 => ( demorgan2
% 0.28/0.48 => ( woz13rule0
% 0.28/0.48 => ( woz13rule1
% 0.28/0.48 => ( woz13rule2
% 0.28/0.48 => ( woz13rule3
% 0.28/0.48 => ( woz13rule4
% 0.28/0.48 => ( woz1_1
% 0.28/0.48 => ( woz1_2
% 0.28/0.48 => ( woz1_3
% 0.28/0.48 => ( woz1_4
% 0.28/0.48 => ( woz1_5
% 0.28/0.48 => ( breln1all2
% 0.28/0.48 => ( breln1SetBreln1
% 0.28/0.48 => ( choice2fnsingleton
% 0.28/0.48 => ( setOfPairsIsBReln1
% 0.28/0.48 => ( breln1all1
% 0.28/0.48 => ( subbreln1
% 0.28/0.48 => ( eqbreln1
% 0.28/0.48 => ( breln1invprop
% 0.28/0.48 => ( breln1invI
% 0.28/0.48 => ( breln1invE
% 0.28/0.48 => ( breln1compprop
% 0.28/0.48 => ( breln1compI
% 0.28/0.48 => ( breln1compE
% 0.28/0.48 => ( breln1compEex
% 0.28/0.48 => ( breln1unionprop
% 0.28/0.48 => ( breln1unionIL
% 0.28/0.48 => ( breln1unionIR
% 0.28/0.48 => ( breln1unionI
% 0.28/0.48 => ( breln1unionE
% 0.28/0.48 => ( breln1unionEcases
% 0.28/0.48 => ( breln1unionCommutes
% 0.28/0.48 => ( woz2Ex
% 0.28/0.48 => ( woz2W
% 0.28/0.48 => ( woz2A
% 0.28/0.48 => ( woz2B
% 0.28/0.48 => ( image1Ex
% 0.28/0.48 => ( image1Ex1
% 0.28/0.48 => ( image1Equiv
% 0.28/0.48 => ( image1E
% 0.28/0.48 => ( image1I
% 0.28/0.48 => ( injFuncInInjFuncSet
% 0.28/0.48 => ( injFuncSetFuncIn
% 0.28/0.48 => ( injFuncSetFuncInj
% 0.28/0.48 => ( surjFuncSetFuncIn
% 0.28/0.48 => ( surjFuncSetFuncSurj
% 0.28/0.48 => ( leftInvIsSurj
% 0.28/0.48 => ( surjCantorThm
% 0.28/0.48 => ( foundation2
% 0.28/0.48 => ( notinself
% 0.28/0.48 => ( notinself2
% 0.28/0.48 => ( omegaSp
% 0.28/0.48 => ( omegaSclos
% 0.28/0.48 => ( peano0notS
% 0.28/0.48 => ( peanoSinj
% 0.28/0.48 => ( transitivesetOp1
% 0.28/0.48 => ( binintTransitive
% 0.28/0.48 => ( transitivesetOp2
% 0.28/0.48 => ( setunionTransitive
% 0.28/0.48 => ( ordinalMinLem1
% 0.28/0.48 => ( ordinalTransSet
% 0.28/0.48 => ( ordinalTransSet1
% 0.28/0.48 => ( setunionOrdinalLem1
% 0.28/0.48 => ( emptysetOrdinal
% 0.28/0.48 => ! [X: $i] :
% 0.28/0.48 ( ( ordinal @ X )
% 0.28/0.48 => ! [A: $i] :
% 0.28/0.48 ( ( in @ A @ X )
% 0.28/0.48 => ~ ( in @ A @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 %------------------------------------------------------------------------------
% 0.28/0.48 ------------------------
% 3.02/1.51 ---- Embedded in THF ---
% 3.02/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.02/1.51 %%% Generated on Tue Feb 24 12:34:48 EST 2026
% 3.02/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 3.02/1.51 %%% Logic specification used:
% 3.02/1.51 %%% thf(spec, logic, $$dhol).
% 3.02/1.51
% 3.02/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.zpLud4Dut7/DTF2DTF5045.p
% See solution above
% 3.02/1.51 ------------------------
% 3.02/1.56 ---- Cleaned THF ---
% 3.02/1.56 thf(in_type,type,
% 3.02/1.56 in: $i > $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(exu_type,type,
% 3.02/1.56 exu: ( $i > $o ) > $o ).
% 3.02/1.56
% 3.02/1.56 thf(exu,definition,
% 3.02/1.56 ( exu
% 3.02/1.56 = ( ^ [Xphi: $i > $o] :
% 3.02/1.56 ? [Xx: $i] :
% 3.02/1.56 ( ( Xphi @ Xx )
% 3.02/1.56 & ! [Xy: $i] :
% 3.02/1.56 ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setextAx_type,type,
% 3.02/1.56 setextAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setextAx,definition,
% 3.02/1.56 ( setextAx
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 <=> ( in @ Xx @ B ) )
% 3.02/1.56 => ( A = B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptyset_type,type,
% 3.02/1.56 emptyset: $i ).
% 3.02/1.56
% 3.02/1.56 thf(emptysetAx_type,type,
% 3.02/1.56 emptysetAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptysetAx,definition,
% 3.02/1.56 ( emptysetAx
% 3.02/1.56 = ( ! [Xx: $i] :
% 3.02/1.56 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoin_type,type,
% 3.02/1.56 setadjoin: $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinAx_type,type,
% 3.02/1.56 setadjoinAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinAx,definition,
% 3.02/1.56 ( setadjoinAx
% 3.02/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.02/1.56 <=> ( ( Xy = Xx )
% 3.02/1.56 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(powerset_type,type,
% 3.02/1.56 powerset: $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(powersetAx_type,type,
% 3.02/1.56 powersetAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(powersetAx,definition,
% 3.02/1.56 ( powersetAx
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( in @ B @ ( powerset @ A ) )
% 3.02/1.56 <=> ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setunion_type,type,
% 3.02/1.56 setunion: $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(setunionAx_type,type,
% 3.02/1.56 setunionAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setunionAx,definition,
% 3.02/1.56 ( setunionAx
% 3.02/1.56 = ( ! [A: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 3.02/1.56 <=> ? [B: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 & ( in @ B @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(omega_type,type,
% 3.02/1.56 omega: $i ).
% 3.02/1.56
% 3.02/1.56 thf(omega0Ax_type,type,
% 3.02/1.56 omega0Ax: $o ).
% 3.02/1.56
% 3.02/1.56 thf(omega0Ax,definition,
% 3.02/1.56 ( omega0Ax
% 3.02/1.56 = ( in @ emptyset @ omega ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(omegaSAx_type,type,
% 3.02/1.56 omegaSAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(omegaSAx,definition,
% 3.02/1.56 ( omegaSAx
% 3.02/1.56 = ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ omega )
% 3.02/1.56 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(omegaIndAx_type,type,
% 3.02/1.56 omegaIndAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(omegaIndAx,definition,
% 3.02/1.56 ( omegaIndAx
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ( ( in @ emptyset @ A )
% 3.02/1.56 & ! [Xx: $i] :
% 3.02/1.56 ( ( ( in @ Xx @ omega )
% 3.02/1.56 & ( in @ Xx @ A ) )
% 3.02/1.56 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ omega )
% 3.02/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(replAx_type,type,
% 3.02/1.56 replAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(replAx,definition,
% 3.02/1.56 ( replAx
% 3.02/1.56 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( exu
% 3.02/1.56 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.02/1.56 => ? [B: $i] :
% 3.02/1.56 ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 <=> ? [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ A )
% 3.02/1.56 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(foundationAx_type,type,
% 3.02/1.56 foundationAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(foundationAx,definition,
% 3.02/1.56 ( foundationAx
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.02/1.56 => ? [B: $i] :
% 3.02/1.56 ( ( in @ B @ A )
% 3.02/1.56 & ~ ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(wellorderingAx_type,type,
% 3.02/1.56 wellorderingAx: $o ).
% 3.02/1.56
% 3.02/1.56 thf(wellorderingAx,definition,
% 3.02/1.56 ( wellorderingAx
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ? [B: $i] :
% 3.02/1.56 ( ! [C: $i] :
% 3.02/1.56 ( ( in @ C @ B )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ C )
% 3.02/1.56 => ( in @ Xx @ A ) ) )
% 3.02/1.56 & ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( ( in @ Xx @ A )
% 3.02/1.56 & ( in @ Xy @ A ) )
% 3.02/1.56 => ( ! [C: $i] :
% 3.02/1.56 ( ( in @ C @ B )
% 3.02/1.56 => ( ( in @ Xx @ C )
% 3.02/1.56 <=> ( in @ Xy @ C ) ) )
% 3.02/1.56 => ( Xx = Xy ) ) )
% 3.02/1.56 & ! [C: $i,D: $i] :
% 3.02/1.56 ( ( ( in @ C @ B )
% 3.02/1.56 & ( in @ D @ B ) )
% 3.02/1.56 => ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ C )
% 3.02/1.56 => ( in @ Xx @ D ) )
% 3.02/1.56 | ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ D )
% 3.02/1.56 => ( in @ Xx @ C ) ) ) )
% 3.02/1.56 & ! [C: $i] :
% 3.02/1.56 ( ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ C )
% 3.02/1.56 => ( in @ Xx @ A ) )
% 3.02/1.56 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.02/1.56 => ? [D: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ D @ B )
% 3.02/1.56 & ( in @ Xx @ C )
% 3.02/1.56 & ~ ? [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ D )
% 3.02/1.56 & ( in @ Xy @ C ) )
% 3.02/1.56 & ! [E: $i] :
% 3.02/1.56 ( ( in @ E @ B )
% 3.02/1.56 => ( ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ E )
% 3.02/1.56 => ( in @ Xy @ D ) )
% 3.02/1.56 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(descr_type,type,
% 3.02/1.56 descr: ( $i > $o ) > $i ).
% 3.02/1.56
% 3.02/1.56 thf(descrp_type,type,
% 3.02/1.56 descrp: $o ).
% 3.02/1.56
% 3.02/1.56 thf(descrp,definition,
% 3.02/1.56 ( descrp
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ( Xphi
% 3.02/1.56 @ ( descr
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstr_type,type,
% 3.02/1.56 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstrI_type,type,
% 3.02/1.56 dsetconstrI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstrI,definition,
% 3.02/1.56 ( dsetconstrI
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 => ( in @ Xx
% 3.02/1.56 @ ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstrEL_type,type,
% 3.02/1.56 dsetconstrEL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstrEL,definition,
% 3.02/1.56 ( dsetconstrEL
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx
% 3.02/1.56 @ ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.02/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstrER_type,type,
% 3.02/1.56 dsetconstrER: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstrER,definition,
% 3.02/1.56 ( dsetconstrER
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx
% 3.02/1.56 @ ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.02/1.56 => ( Xphi @ Xx ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exuE1_type,type,
% 3.02/1.56 exuE1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exuE1,definition,
% 3.02/1.56 ( exuE1
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ? [Xx: $i] :
% 3.02/1.56 ( ( Xphi @ Xx )
% 3.02/1.56 & ! [Xy: $i] :
% 3.02/1.56 ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(prop2set_type,type,
% 3.02/1.56 prop2set: $o > $i ).
% 3.02/1.56
% 3.02/1.56 thf(prop2setE_type,type,
% 3.02/1.56 prop2setE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(prop2setE,definition,
% 3.02/1.56 ( prop2setE
% 3.02/1.56 = ( ! [Xphi: $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.02/1.56 => Xphi ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptysetE_type,type,
% 3.02/1.56 emptysetE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptysetE,definition,
% 3.02/1.56 ( emptysetE
% 3.02/1.56 = ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ emptyset )
% 3.02/1.56 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptysetimpfalse_type,type,
% 3.02/1.56 emptysetimpfalse: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptysetimpfalse,definition,
% 3.02/1.56 ( emptysetimpfalse
% 3.02/1.56 = ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ emptyset )
% 3.02/1.56 => $false ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(notinemptyset_type,type,
% 3.02/1.56 notinemptyset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(notinemptyset,definition,
% 3.02/1.56 ( notinemptyset
% 3.02/1.56 = ( ! [Xx: $i] :
% 3.02/1.56 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exuE3e_type,type,
% 3.02/1.56 exuE3e: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exuE3e,definition,
% 3.02/1.56 ( exuE3e
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setext_type,type,
% 3.02/1.56 setext: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setext,definition,
% 3.02/1.56 ( setext
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ Xx @ B ) )
% 3.02/1.56 => ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ A ) )
% 3.02/1.56 => ( A = B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptyI_type,type,
% 3.02/1.56 emptyI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptyI,definition,
% 3.02/1.56 ( emptyI
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ~ ( in @ Xx @ A )
% 3.02/1.56 => ( A = emptyset ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(noeltsimpempty_type,type,
% 3.02/1.56 noeltsimpempty: $o ).
% 3.02/1.56
% 3.02/1.56 thf(noeltsimpempty,definition,
% 3.02/1.56 ( noeltsimpempty
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ~ ( in @ Xx @ A )
% 3.02/1.56 => ( A = emptyset ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setbeta_type,type,
% 3.02/1.56 setbeta: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setbeta,definition,
% 3.02/1.56 ( setbeta
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( in @ Xx
% 3.02/1.56 @ ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.02/1.56 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(nonempty_type,type,
% 3.02/1.56 nonempty: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(nonemptyE1_type,type,
% 3.02/1.56 nonemptyE1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(nonemptyE1,definition,
% 3.02/1.56 ( nonemptyE1
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ( nonempty @ A )
% 3.02/1.56 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(nonemptyI_type,type,
% 3.02/1.56 nonemptyI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(nonemptyI,definition,
% 3.02/1.56 ( nonemptyI
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 => ( nonempty
% 3.02/1.56 @ ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(nonemptyI1_type,type,
% 3.02/1.56 nonemptyI1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(nonemptyI1,definition,
% 3.02/1.56 ( nonemptyI1
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.02/1.56 => ( nonempty @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinIL_type,type,
% 3.02/1.56 setadjoinIL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinIL,definition,
% 3.02/1.56 ( setadjoinIL
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptyinunitempty_type,type,
% 3.02/1.56 emptyinunitempty: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptyinunitempty,definition,
% 3.02/1.56 ( emptyinunitempty
% 3.02/1.56 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinIR_type,type,
% 3.02/1.56 setadjoinIR: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinIR,definition,
% 3.02/1.56 ( setadjoinIR
% 3.02/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ A )
% 3.02/1.56 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinE_type,type,
% 3.02/1.56 setadjoinE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinE,definition,
% 3.02/1.56 ( setadjoinE
% 3.02/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.02/1.56 => ! [Xphi: $o] :
% 3.02/1.56 ( ( ( Xy = Xx )
% 3.02/1.56 => Xphi )
% 3.02/1.56 => ( ( ( in @ Xy @ A )
% 3.02/1.56 => Xphi )
% 3.02/1.56 => Xphi ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinOr_type,type,
% 3.02/1.56 setadjoinOr: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinOr,definition,
% 3.02/1.56 ( setadjoinOr
% 3.02/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.02/1.56 => ( ( Xy = Xx )
% 3.02/1.56 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setoftrueEq_type,type,
% 3.02/1.56 setoftrueEq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setoftrueEq,definition,
% 3.02/1.56 ( setoftrueEq
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xx: $i] : $true )
% 3.02/1.56 = A ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(powersetI_type,type,
% 3.02/1.56 powersetI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(powersetI,definition,
% 3.02/1.56 ( powersetI
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ A ) )
% 3.02/1.56 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptyinPowerset_type,type,
% 3.02/1.56 emptyinPowerset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptyinPowerset,definition,
% 3.02/1.56 ( emptyinPowerset
% 3.02/1.56 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptyInPowerset_type,type,
% 3.02/1.56 emptyInPowerset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptyInPowerset,definition,
% 3.02/1.56 ( emptyInPowerset
% 3.02/1.56 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(powersetE_type,type,
% 3.02/1.56 powersetE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(powersetE,definition,
% 3.02/1.56 ( powersetE
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ B @ ( powerset @ A ) )
% 3.02/1.56 => ( ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setunionI_type,type,
% 3.02/1.56 setunionI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setunionI,definition,
% 3.02/1.56 ( setunionI
% 3.02/1.56 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 => ( ( in @ B @ A )
% 3.02/1.56 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setunionE_type,type,
% 3.02/1.56 setunionE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setunionE,definition,
% 3.02/1.56 ( setunionE
% 3.02/1.56 = ( ! [A: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 3.02/1.56 => ! [Xphi: $o] :
% 3.02/1.56 ( ! [B: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 => ( ( in @ B @ A )
% 3.02/1.56 => Xphi ) )
% 3.02/1.56 => Xphi ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subPowSU_type,type,
% 3.02/1.56 subPowSU: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subPowSU,definition,
% 3.02/1.56 ( subPowSU
% 3.02/1.56 = ( ! [A: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exuE2_type,type,
% 3.02/1.56 exuE2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exuE2,definition,
% 3.02/1.56 ( exuE2
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ? [Xx: $i] :
% 3.02/1.56 ! [Xy: $i] :
% 3.02/1.56 ( ( Xphi @ Xy )
% 3.02/1.56 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(nonemptyImpWitness_type,type,
% 3.02/1.56 nonemptyImpWitness: $o ).
% 3.02/1.56
% 3.02/1.56 thf(nonemptyImpWitness,definition,
% 3.02/1.56 ( nonemptyImpWitness
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ( nonempty @ A )
% 3.02/1.56 => ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & $true ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(uniqinunit_type,type,
% 3.02/1.56 uniqinunit: $o ).
% 3.02/1.56
% 3.02/1.56 thf(uniqinunit,definition,
% 3.02/1.56 ( uniqinunit
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(notinsingleton_type,type,
% 3.02/1.56 notinsingleton: $o ).
% 3.02/1.56
% 3.02/1.56 thf(notinsingleton,definition,
% 3.02/1.56 ( notinsingleton
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx != Xy )
% 3.02/1.56 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(eqinunit_type,type,
% 3.02/1.56 eqinunit: $o ).
% 3.02/1.56
% 3.02/1.56 thf(eqinunit,definition,
% 3.02/1.56 ( eqinunit
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx = Xy )
% 3.02/1.56 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(singletonsswitch_type,type,
% 3.02/1.56 singletonsswitch: $o ).
% 3.02/1.56
% 3.02/1.56 thf(singletonsswitch,definition,
% 3.02/1.56 ( singletonsswitch
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.56 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(upairsetE_type,type,
% 3.02/1.56 upairsetE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairsetE,definition,
% 3.02/1.56 ( upairsetE
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.02/1.56 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.02/1.56 => ( ( Xz = Xx )
% 3.02/1.56 | ( Xz = Xy ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(upairsetIL_type,type,
% 3.02/1.56 upairsetIL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairsetIL,definition,
% 3.02/1.56 ( upairsetIL
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(upairsetIR_type,type,
% 3.02/1.56 upairsetIR: $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairsetIR,definition,
% 3.02/1.56 ( upairsetIR
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptyE1_type,type,
% 3.02/1.56 emptyE1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptyE1,definition,
% 3.02/1.56 ( emptyE1
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ( Xphi @ Xx ) )
% 3.02/1.56 => ( ( ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 = emptyset )
% 3.02/1.56 => $false ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(vacuousDall_type,type,
% 3.02/1.56 vacuousDall: $o ).
% 3.02/1.56
% 3.02/1.56 thf(vacuousDall,definition,
% 3.02/1.56 ( vacuousDall
% 3.02/1.56 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ emptyset )
% 3.02/1.56 => ( Xphi @ Xx ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(quantDeMorgan1_type,type,
% 3.02/1.56 quantDeMorgan1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(quantDeMorgan1,definition,
% 3.02/1.56 ( quantDeMorgan1
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ~ ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( Xphi @ Xx ) )
% 3.02/1.56 => ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(quantDeMorgan2_type,type,
% 3.02/1.56 quantDeMorgan2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(quantDeMorgan2,definition,
% 3.02/1.56 ( quantDeMorgan2
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ~ ( Xphi @ Xx ) )
% 3.02/1.56 => ~ ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(quantDeMorgan3_type,type,
% 3.02/1.56 quantDeMorgan3: $o ).
% 3.02/1.56
% 3.02/1.56 thf(quantDeMorgan3,definition,
% 3.02/1.56 ( quantDeMorgan3
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ~ ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ( Xphi @ Xx ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(quantDeMorgan4_type,type,
% 3.02/1.56 quantDeMorgan4: $o ).
% 3.02/1.56
% 3.02/1.56 thf(quantDeMorgan4,definition,
% 3.02/1.56 ( quantDeMorgan4
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ~ ( Xphi @ Xx ) )
% 3.02/1.56 => ~ ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(prop2setI_type,type,
% 3.02/1.56 prop2setI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(prop2setI,definition,
% 3.02/1.56 ( prop2setI
% 3.02/1.56 = ( ! [Xphi: $o] :
% 3.02/1.56 ( Xphi
% 3.02/1.56 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(set2prop_type,type,
% 3.02/1.56 set2prop: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(prop2set2propI_type,type,
% 3.02/1.56 prop2set2propI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(prop2set2propI,definition,
% 3.02/1.56 ( prop2set2propI
% 3.02/1.56 = ( ! [Xphi: $o] :
% 3.02/1.56 ( Xphi
% 3.02/1.56 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(notdexE_type,type,
% 3.02/1.56 notdexE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(notdexE,definition,
% 3.02/1.56 ( notdexE
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ~ ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ( Xphi @ Xx ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(notdallE_type,type,
% 3.02/1.56 notdallE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(notdallE,definition,
% 3.02/1.56 ( notdallE
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ~ ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( Xphi @ Xx ) )
% 3.02/1.56 => ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exuI1_type,type,
% 3.02/1.56 exuI1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exuI1,definition,
% 3.02/1.56 ( exuI1
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ? [Xx: $i] :
% 3.02/1.56 ( ( Xphi @ Xx )
% 3.02/1.56 & ! [Xy: $i] :
% 3.02/1.56 ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) )
% 3.02/1.56 => ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exuI3_type,type,
% 3.02/1.56 exuI3: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exuI3,definition,
% 3.02/1.56 ( exuI3
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.02/1.56 => ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xphi @ Xx )
% 3.02/1.56 => ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) )
% 3.02/1.56 => ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exuI2_type,type,
% 3.02/1.56 exuI2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exuI2,definition,
% 3.02/1.56 ( exuI2
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ? [Xx: $i] :
% 3.02/1.56 ! [Xy: $i] :
% 3.02/1.56 ( ( Xphi @ Xy )
% 3.02/1.56 <=> ( Xy = Xx ) )
% 3.02/1.56 => ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(inCongP_type,type,
% 3.02/1.56 inCongP: $o ).
% 3.02/1.56
% 3.02/1.56 thf(inCongP,definition,
% 3.02/1.56 ( inCongP
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( A = B )
% 3.02/1.56 => ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx = Xy )
% 3.02/1.56 => ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(in__Cong_type,type,
% 3.02/1.56 in__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(in__Cong,definition,
% 3.02/1.56 ( in__Cong
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( A = B )
% 3.02/1.56 => ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx = Xy )
% 3.02/1.56 => ( ( in @ Xx @ A )
% 3.02/1.56 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exuE3u_type,type,
% 3.02/1.56 exuE3u: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exuE3u,definition,
% 3.02/1.56 ( exuE3u
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xphi @ Xx )
% 3.02/1.56 => ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exu__Cong_type,type,
% 3.02/1.56 exu__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exu__Cong,definition,
% 3.02/1.56 ( exu__Cong
% 3.02/1.56 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.02/1.56 ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx = Xy )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 <=> ( Xpsi @ Xy ) ) )
% 3.02/1.56 => ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 <=> ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptyset__Cong_type,type,
% 3.02/1.56 emptyset__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptyset__Cong,definition,
% 3.02/1.56 ( emptyset__Cong
% 3.02/1.56 = ( emptyset = emptyset ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoin__Cong_type,type,
% 3.02/1.56 setadjoin__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoin__Cong,definition,
% 3.02/1.56 ( setadjoin__Cong
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx = Xy )
% 3.02/1.56 => ! [Xz: $i,Xu: $i] :
% 3.02/1.56 ( ( Xz = Xu )
% 3.02/1.56 => ( ( setadjoin @ Xx @ Xz )
% 3.02/1.56 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(powerset__Cong_type,type,
% 3.02/1.56 powerset__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(powerset__Cong,definition,
% 3.02/1.56 ( powerset__Cong
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( A = B )
% 3.02/1.56 => ( ( powerset @ A )
% 3.02/1.56 = ( powerset @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setunion__Cong_type,type,
% 3.02/1.56 setunion__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setunion__Cong,definition,
% 3.02/1.56 ( setunion__Cong
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( A = B )
% 3.02/1.56 => ( ( setunion @ A )
% 3.02/1.56 = ( setunion @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(omega__Cong_type,type,
% 3.02/1.56 omega__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(omega__Cong,definition,
% 3.02/1.56 ( omega__Cong
% 3.02/1.56 = ( omega = omega ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(exuEu_type,type,
% 3.02/1.56 exuEu: $o ).
% 3.02/1.56
% 3.02/1.56 thf(exuEu,definition,
% 3.02/1.56 ( exuEu
% 3.02/1.56 = ( ! [Xphi: $i > $o] :
% 3.02/1.56 ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xphi @ Xx )
% 3.02/1.56 => ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(descr__Cong_type,type,
% 3.02/1.56 descr__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(descr__Cong,definition,
% 3.02/1.56 ( descr__Cong
% 3.02/1.56 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.02/1.56 ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx = Xy )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 <=> ( Xpsi @ Xy ) ) )
% 3.02/1.56 => ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ( ( exu
% 3.02/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.02/1.56 => ( ( descr
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 = ( descr
% 3.02/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstr__Cong_type,type,
% 3.02/1.56 dsetconstr__Cong: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dsetconstr__Cong,definition,
% 3.02/1.56 ( dsetconstr__Cong
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( A = B )
% 3.02/1.56 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( Xx = Xy )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.02/1.56 => ( ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 = ( dsetconstr @ B
% 3.02/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subset_type,type,
% 3.02/1.56 subset: $i > $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(disjoint_type,type,
% 3.02/1.56 disjoint: $i > $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(setsmeet_type,type,
% 3.02/1.56 setsmeet: $i > $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(subsetI1_type,type,
% 3.02/1.56 subsetI1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subsetI1,definition,
% 3.02/1.56 ( subsetI1
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ Xx @ B ) )
% 3.02/1.56 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(eqimpsubset2_type,type,
% 3.02/1.56 eqimpsubset2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(eqimpsubset2,definition,
% 3.02/1.56 ( eqimpsubset2
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( A = B )
% 3.02/1.56 => ( subset @ B @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(eqimpsubset1_type,type,
% 3.02/1.56 eqimpsubset1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(eqimpsubset1,definition,
% 3.02/1.56 ( eqimpsubset1
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( A = B )
% 3.02/1.56 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subsetI2_type,type,
% 3.02/1.56 subsetI2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subsetI2,definition,
% 3.02/1.56 ( subsetI2
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ Xx @ B ) )
% 3.02/1.56 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(emptysetsubset_type,type,
% 3.02/1.56 emptysetsubset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(emptysetsubset,definition,
% 3.02/1.56 ( emptysetsubset
% 3.02/1.56 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subsetE_type,type,
% 3.02/1.56 subsetE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subsetE,definition,
% 3.02/1.56 ( subsetE
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subsetE2_type,type,
% 3.02/1.56 subsetE2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subsetE2,definition,
% 3.02/1.56 ( subsetE2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( ~ ( in @ Xx @ B )
% 3.02/1.56 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(notsubsetI_type,type,
% 3.02/1.56 notsubsetI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(notsubsetI,definition,
% 3.02/1.56 ( notsubsetI
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ~ ( in @ Xx @ B )
% 3.02/1.56 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(notequalI1_type,type,
% 3.02/1.56 notequalI1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(notequalI1,definition,
% 3.02/1.56 ( notequalI1
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ~ ( subset @ A @ B )
% 3.02/1.56 => ( A != B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(notequalI2_type,type,
% 3.02/1.56 notequalI2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(notequalI2,definition,
% 3.02/1.56 ( notequalI2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ~ ( in @ Xx @ B )
% 3.02/1.56 => ( A != B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subsetRefl_type,type,
% 3.02/1.56 subsetRefl: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subsetRefl,definition,
% 3.02/1.56 ( subsetRefl
% 3.02/1.56 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subsetTrans_type,type,
% 3.02/1.56 subsetTrans: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subsetTrans,definition,
% 3.02/1.56 ( subsetTrans
% 3.02/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( ( subset @ B @ C )
% 3.02/1.56 => ( subset @ A @ C ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinSub_type,type,
% 3.02/1.56 setadjoinSub: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinSub,definition,
% 3.02/1.56 ( setadjoinSub
% 3.02/1.56 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinSub2_type,type,
% 3.02/1.56 setadjoinSub2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setadjoinSub2,definition,
% 3.02/1.56 ( setadjoinSub2
% 3.02/1.56 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subset2powerset_type,type,
% 3.02/1.56 subset2powerset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subset2powerset,definition,
% 3.02/1.56 ( subset2powerset
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setextsub_type,type,
% 3.02/1.56 setextsub: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setextsub,definition,
% 3.02/1.56 ( setextsub
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( ( subset @ B @ A )
% 3.02/1.56 => ( A = B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subsetemptysetimpeq_type,type,
% 3.02/1.56 subsetemptysetimpeq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subsetemptysetimpeq,definition,
% 3.02/1.56 ( subsetemptysetimpeq
% 3.02/1.56 = ( ! [A: $i] :
% 3.02/1.56 ( ( subset @ A @ emptyset )
% 3.02/1.56 => ( A = emptyset ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(powersetI1_type,type,
% 3.02/1.56 powersetI1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(powersetI1,definition,
% 3.02/1.56 ( powersetI1
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( subset @ B @ A )
% 3.02/1.56 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(powersetE1_type,type,
% 3.02/1.56 powersetE1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(powersetE1,definition,
% 3.02/1.56 ( powersetE1
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( in @ B @ ( powerset @ A ) )
% 3.02/1.56 => ( subset @ B @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(inPowerset_type,type,
% 3.02/1.56 inPowerset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(inPowerset,definition,
% 3.02/1.56 ( inPowerset
% 3.02/1.56 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(powersetsubset_type,type,
% 3.02/1.56 powersetsubset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(powersetsubset,definition,
% 3.02/1.56 ( powersetsubset
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(sepInPowerset_type,type,
% 3.02/1.56 sepInPowerset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(sepInPowerset,definition,
% 3.02/1.56 ( sepInPowerset
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( in
% 3.02/1.56 @ ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 @ ( powerset @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(sepSubset_type,type,
% 3.02/1.56 sepSubset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(sepSubset,definition,
% 3.02/1.56 ( sepSubset
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( subset
% 3.02/1.56 @ ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 @ A ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binunion_type,type,
% 3.02/1.56 binunion: $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(binunionIL_type,type,
% 3.02/1.56 binunionIL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binunionIL,definition,
% 3.02/1.56 ( binunionIL
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(upairset2IR_type,type,
% 3.02/1.56 upairset2IR: $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairset2IR,definition,
% 3.02/1.56 ( upairset2IR
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binunionIR_type,type,
% 3.02/1.56 binunionIR: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binunionIR,definition,
% 3.02/1.56 ( binunionIR
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binunionEcases_type,type,
% 3.02/1.56 binunionEcases: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binunionEcases,definition,
% 3.02/1.56 ( binunionEcases
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.02/1.56 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.02/1.56 => ( ( ( in @ Xx @ A )
% 3.02/1.56 => Xphi )
% 3.02/1.56 => ( ( ( in @ Xx @ B )
% 3.02/1.56 => Xphi )
% 3.02/1.56 => Xphi ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binunionE_type,type,
% 3.02/1.56 binunionE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binunionE,definition,
% 3.02/1.56 ( binunionE
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.02/1.56 => ( ( in @ Xx @ A )
% 3.02/1.56 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binunionLsub_type,type,
% 3.02/1.56 binunionLsub: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binunionLsub,definition,
% 3.02/1.56 ( binunionLsub
% 3.02/1.56 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binunionRsub_type,type,
% 3.02/1.56 binunionRsub: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binunionRsub,definition,
% 3.02/1.56 ( binunionRsub
% 3.02/1.56 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersect_type,type,
% 3.02/1.56 binintersect: $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectI_type,type,
% 3.02/1.56 binintersectI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectI,definition,
% 3.02/1.56 ( binintersectI
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset5_type,type,
% 3.02/1.56 binintersectSubset5: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset5,definition,
% 3.02/1.56 ( binintersectSubset5
% 3.02/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 3.02/1.56 ( ( subset @ C @ A )
% 3.02/1.56 => ( ( subset @ C @ B )
% 3.02/1.56 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectEL_type,type,
% 3.02/1.56 binintersectEL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectEL,definition,
% 3.02/1.56 ( binintersectEL
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.02/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectLsub_type,type,
% 3.02/1.56 binintersectLsub: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectLsub,definition,
% 3.02/1.56 ( binintersectLsub
% 3.02/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset2_type,type,
% 3.02/1.56 binintersectSubset2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset2,definition,
% 3.02/1.56 ( binintersectSubset2
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( ( binintersect @ A @ B )
% 3.02/1.56 = A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset3_type,type,
% 3.02/1.56 binintersectSubset3: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset3,definition,
% 3.02/1.56 ( binintersectSubset3
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( ( binintersect @ A @ B )
% 3.02/1.56 = B )
% 3.02/1.56 => ( subset @ B @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectER_type,type,
% 3.02/1.56 binintersectER: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectER,definition,
% 3.02/1.56 ( binintersectER
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.02/1.56 => ( in @ Xx @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(disjointsetsI1_type,type,
% 3.02/1.56 disjointsetsI1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(disjointsetsI1,definition,
% 3.02/1.56 ( disjointsetsI1
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ~ ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ( in @ Xx @ B ) )
% 3.02/1.56 => ( ( binintersect @ A @ B )
% 3.02/1.56 = emptyset ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectRsub_type,type,
% 3.02/1.56 binintersectRsub: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectRsub,definition,
% 3.02/1.56 ( binintersectRsub
% 3.02/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset4_type,type,
% 3.02/1.56 binintersectSubset4: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset4,definition,
% 3.02/1.56 ( binintersectSubset4
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( subset @ B @ A )
% 3.02/1.56 => ( ( binintersect @ A @ B )
% 3.02/1.56 = B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset1_type,type,
% 3.02/1.56 binintersectSubset1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(binintersectSubset1,definition,
% 3.02/1.56 ( binintersectSubset1
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( ( binintersect @ A @ B )
% 3.02/1.56 = A )
% 3.02/1.56 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(bs114d_type,type,
% 3.02/1.56 bs114d: $o ).
% 3.02/1.56
% 3.02/1.56 thf(bs114d,definition,
% 3.02/1.56 ( bs114d
% 3.02/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 3.02/1.56 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.02/1.56 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(regular_type,type,
% 3.02/1.56 regular: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminus_type,type,
% 3.02/1.56 setminus: $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(setminusI_type,type,
% 3.02/1.56 setminusI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusI,definition,
% 3.02/1.56 ( setminusI
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ~ ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusEL_type,type,
% 3.02/1.56 setminusEL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusEL,definition,
% 3.02/1.56 ( setminusEL
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.02/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusER_type,type,
% 3.02/1.56 setminusER: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusER,definition,
% 3.02/1.56 ( setminusER
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.02/1.56 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusSubset2_type,type,
% 3.02/1.56 setminusSubset2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusSubset2,definition,
% 3.02/1.56 ( setminusSubset2
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( subset @ A @ B )
% 3.02/1.56 => ( ( setminus @ A @ B )
% 3.02/1.56 = emptyset ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusERneg_type,type,
% 3.02/1.56 setminusERneg: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusERneg,definition,
% 3.02/1.56 ( setminusERneg
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.02/1.56 => ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusELneg_type,type,
% 3.02/1.56 setminusELneg: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusELneg,definition,
% 3.02/1.56 ( setminusELneg
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.02/1.56 => ( ~ ( in @ Xx @ B )
% 3.02/1.56 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusILneg_type,type,
% 3.02/1.56 setminusILneg: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusILneg,definition,
% 3.02/1.56 ( setminusILneg
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ~ ( in @ Xx @ A )
% 3.02/1.56 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusIRneg_type,type,
% 3.02/1.56 setminusIRneg: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusIRneg,definition,
% 3.02/1.56 ( setminusIRneg
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ B )
% 3.02/1.56 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusLsub_type,type,
% 3.02/1.56 setminusLsub: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusLsub,definition,
% 3.02/1.56 ( setminusLsub
% 3.02/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setminusSubset1_type,type,
% 3.02/1.56 setminusSubset1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setminusSubset1,definition,
% 3.02/1.56 ( setminusSubset1
% 3.02/1.56 = ( ! [A: $i,B: $i] :
% 3.02/1.56 ( ( ( setminus @ A @ B )
% 3.02/1.56 = emptyset )
% 3.02/1.56 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(symdiff_type,type,
% 3.02/1.56 symdiff: $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffE_type,type,
% 3.02/1.56 symdiffE: $o ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffE,definition,
% 3.02/1.56 ( symdiffE
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.02/1.56 => ! [Xphi: $o] :
% 3.02/1.56 ( ( ( in @ Xx @ A )
% 3.02/1.56 => ( ~ ( in @ Xx @ B )
% 3.02/1.56 => Xphi ) )
% 3.02/1.56 => ( ( ~ ( in @ Xx @ A )
% 3.02/1.56 => ( ( in @ Xx @ B )
% 3.02/1.56 => Xphi ) )
% 3.02/1.56 => Xphi ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffI1_type,type,
% 3.02/1.56 symdiffI1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffI1,definition,
% 3.02/1.56 ( symdiffI1
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ~ ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffI2_type,type,
% 3.02/1.56 symdiffI2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffI2,definition,
% 3.02/1.56 ( symdiffI2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ~ ( in @ Xx @ A )
% 3.02/1.56 => ( ( in @ Xx @ B )
% 3.02/1.56 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffIneg1_type,type,
% 3.02/1.56 symdiffIneg1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffIneg1,definition,
% 3.02/1.56 ( symdiffIneg1
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( in @ Xx @ B )
% 3.02/1.56 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffIneg2_type,type,
% 3.02/1.56 symdiffIneg2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(symdiffIneg2,definition,
% 3.02/1.56 ( symdiffIneg2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ~ ( in @ Xx @ A )
% 3.02/1.56 => ( ~ ( in @ Xx @ B )
% 3.02/1.56 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(iskpair_type,type,
% 3.02/1.56 iskpair: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(secondinupair_type,type,
% 3.02/1.56 secondinupair: $o ).
% 3.02/1.56
% 3.02/1.56 thf(secondinupair,definition,
% 3.02/1.56 ( secondinupair
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairIL_type,type,
% 3.02/1.56 setukpairIL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairIL,definition,
% 3.02/1.56 ( setukpairIL
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairIR_type,type,
% 3.02/1.56 setukpairIR: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairIR,definition,
% 3.02/1.56 ( setukpairIR
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(kpairiskpair_type,type,
% 3.02/1.56 kpairiskpair: $o ).
% 3.02/1.56
% 3.02/1.56 thf(kpairiskpair,definition,
% 3.02/1.56 ( kpairiskpair
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(kpair_type,type,
% 3.02/1.56 kpair: $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(kpairp_type,type,
% 3.02/1.56 kpairp: $o ).
% 3.02/1.56
% 3.02/1.56 thf(kpairp,definition,
% 3.02/1.56 ( kpairp
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprod_type,type,
% 3.02/1.56 cartprod: $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(singletonsubset_type,type,
% 3.02/1.56 singletonsubset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(singletonsubset,definition,
% 3.02/1.56 ( singletonsubset
% 3.02/1.56 = ( ! [A: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(singletoninpowerset_type,type,
% 3.02/1.56 singletoninpowerset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(singletoninpowerset,definition,
% 3.02/1.56 ( singletoninpowerset
% 3.02/1.56 = ( ! [A: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(singletoninpowunion_type,type,
% 3.02/1.56 singletoninpowunion: $o ).
% 3.02/1.56
% 3.02/1.56 thf(singletoninpowunion,definition,
% 3.02/1.56 ( singletoninpowunion
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(upairset2E_type,type,
% 3.02/1.56 upairset2E: $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairset2E,definition,
% 3.02/1.56 ( upairset2E
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.02/1.56 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.02/1.56 => ( ( Xz = Xx )
% 3.02/1.56 | ( Xz = Xy ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(upairsubunion_type,type,
% 3.02/1.56 upairsubunion: $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairsubunion,definition,
% 3.02/1.56 ( upairsubunion
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(upairinpowunion_type,type,
% 3.02/1.56 upairinpowunion: $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairinpowunion,definition,
% 3.02/1.56 ( upairinpowunion
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ubforcartprodlem1_type,type,
% 3.02/1.56 ubforcartprodlem1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ubforcartprodlem1,definition,
% 3.02/1.56 ( ubforcartprodlem1
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ubforcartprodlem2_type,type,
% 3.02/1.56 ubforcartprodlem2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ubforcartprodlem2,definition,
% 3.02/1.56 ( ubforcartprodlem2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ubforcartprodlem3_type,type,
% 3.02/1.56 ubforcartprodlem3: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ubforcartprodlem3,definition,
% 3.02/1.56 ( ubforcartprodlem3
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodpairin_type,type,
% 3.02/1.56 cartprodpairin: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodpairin,definition,
% 3.02/1.56 ( cartprodpairin
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodmempair1_type,type,
% 3.02/1.56 cartprodmempair1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodmempair1,definition,
% 3.02/1.56 ( cartprodmempair1
% 3.02/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.56 => ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ? [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 & ( Xu
% 3.02/1.56 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodmempair_type,type,
% 3.02/1.56 cartprodmempair: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodmempair,definition,
% 3.02/1.56 ( cartprodmempair
% 3.02/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.56 => ( iskpair @ Xu ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setunionE2_type,type,
% 3.02/1.56 setunionE2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setunionE2,definition,
% 3.02/1.56 ( setunionE2
% 3.02/1.56 = ( ! [A: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 3.02/1.56 => ? [X: $i] :
% 3.02/1.56 ( ( in @ X @ A )
% 3.02/1.56 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setunionsingleton1_type,type,
% 3.02/1.56 setunionsingleton1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setunionsingleton1,definition,
% 3.02/1.56 ( setunionsingleton1
% 3.02/1.56 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setunionsingleton2_type,type,
% 3.02/1.56 setunionsingleton2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setunionsingleton2,definition,
% 3.02/1.56 ( setunionsingleton2
% 3.02/1.56 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setunionsingleton_type,type,
% 3.02/1.56 setunionsingleton: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setunionsingleton,definition,
% 3.02/1.56 ( setunionsingleton
% 3.02/1.56 = ( ! [Xx: $i] :
% 3.02/1.56 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.02/1.56 = Xx ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(singleton_type,type,
% 3.02/1.56 singleton: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(singletonprop_type,type,
% 3.02/1.56 singletonprop: $o ).
% 3.02/1.56
% 3.02/1.56 thf(singletonprop,definition,
% 3.02/1.56 ( singletonprop
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ A )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 => ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) )
% 3.02/1.56 => ( ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ( Xphi @ Xx ) )
% 3.02/1.56 => ( singleton
% 3.02/1.56 @ ( dsetconstr @ A
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ex1_type,type,
% 3.02/1.56 ex1: $i > ( $i > $o ) > $o ).
% 3.02/1.56
% 3.02/1.56 thf(ex1E1_type,type,
% 3.02/1.56 ex1E1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ex1E1,definition,
% 3.02/1.56 ( ex1E1
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ( ex1 @ A
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ex1I_type,type,
% 3.02/1.56 ex1I: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ex1I,definition,
% 3.02/1.56 ( ex1I
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 => ( ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ A )
% 3.02/1.56 => ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xy = Xx ) ) )
% 3.02/1.56 => ( ex1 @ A
% 3.02/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ex1I2_type,type,
% 3.02/1.56 ex1I2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ex1I2,definition,
% 3.02/1.56 ( ex1I2
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ A )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 => ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) )
% 3.02/1.56 => ( ? [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 & ( Xphi @ Xx ) )
% 3.02/1.56 => ( ex1 @ A
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(singletonsuniq_type,type,
% 3.02/1.56 singletonsuniq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(singletonsuniq,definition,
% 3.02/1.56 ( singletonsuniq
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( ( setadjoin @ Xx @ emptyset )
% 3.02/1.56 = ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(atmost1p_type,type,
% 3.02/1.56 atmost1p: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(atleast2p_type,type,
% 3.02/1.56 atleast2p: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(atmost2p_type,type,
% 3.02/1.56 atmost2p: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairsetp_type,type,
% 3.02/1.56 upairsetp: $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjL1_type,type,
% 3.02/1.56 setukpairinjL1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjL1,definition,
% 3.02/1.56 ( setukpairinjL1
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.02/1.56 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.02/1.56 => ( Xx = Xz ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(kfstsingleton_type,type,
% 3.02/1.56 kfstsingleton: $o ).
% 3.02/1.56
% 3.02/1.56 thf(kfstsingleton,definition,
% 3.02/1.56 ( kfstsingleton
% 3.02/1.56 = ( ! [Xu: $i] :
% 3.02/1.56 ( ( iskpair @ Xu )
% 3.02/1.56 => ( singleton
% 3.02/1.56 @ ( dsetconstr @ ( setunion @ Xu )
% 3.02/1.56 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(theprop_type,type,
% 3.02/1.56 theprop: $o ).
% 3.02/1.56
% 3.02/1.56 thf(theprop,definition,
% 3.02/1.56 ( theprop
% 3.02/1.56 = ( ! [X: $i] :
% 3.02/1.56 ( ( singleton @ X )
% 3.02/1.56 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(kfst_type,type,
% 3.02/1.56 kfst: $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(kfstpairEq_type,type,
% 3.02/1.56 kfstpairEq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(kfstpairEq,definition,
% 3.02/1.56 ( kfstpairEq
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.02/1.56 = Xx ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodfstin_type,type,
% 3.02/1.56 cartprodfstin: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodfstin,definition,
% 3.02/1.56 ( cartprodfstin
% 3.02/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.56 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjL2_type,type,
% 3.02/1.56 setukpairinjL2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjL2,definition,
% 3.02/1.56 ( setukpairinjL2
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.02/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.02/1.56 => ( Xx = Xz ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjL_type,type,
% 3.02/1.56 setukpairinjL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjL,definition,
% 3.02/1.56 ( setukpairinjL
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.56 ( ( ( kpair @ Xx @ Xy )
% 3.02/1.56 = ( kpair @ Xz @ Xu ) )
% 3.02/1.56 => ( Xx = Xz ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR11_type,type,
% 3.02/1.56 setukpairinjR11: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR11,definition,
% 3.02/1.56 ( setukpairinjR11
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx = Xy )
% 3.02/1.56 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.56 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR12_type,type,
% 3.02/1.56 setukpairinjR12: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR12,definition,
% 3.02/1.56 ( setukpairinjR12
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xx = Xy )
% 3.02/1.56 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.02/1.56 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR1_type,type,
% 3.02/1.56 setukpairinjR1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR1,definition,
% 3.02/1.56 ( setukpairinjR1
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.02/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.02/1.56 => ( ( Xz = Xu )
% 3.02/1.56 => ( Xy = Xu ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(upairequniteq_type,type,
% 3.02/1.56 upairequniteq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(upairequniteq,definition,
% 3.02/1.56 ( upairequniteq
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.02/1.56 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.56 = ( setadjoin @ Xz @ emptyset ) )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR2_type,type,
% 3.02/1.56 setukpairinjR2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR2,definition,
% 3.02/1.56 ( setukpairinjR2
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.02/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.02/1.56 => ( Xy = Xu ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR_type,type,
% 3.02/1.56 setukpairinjR: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setukpairinjR,definition,
% 3.02/1.56 ( setukpairinjR
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.56 ( ( ( kpair @ Xx @ Xy )
% 3.02/1.56 = ( kpair @ Xz @ Xu ) )
% 3.02/1.56 => ( Xy = Xu ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ksndsingleton_type,type,
% 3.02/1.56 ksndsingleton: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ksndsingleton,definition,
% 3.02/1.56 ( ksndsingleton
% 3.02/1.56 = ( ! [Xu: $i] :
% 3.02/1.56 ( ( iskpair @ Xu )
% 3.02/1.56 => ( singleton
% 3.02/1.56 @ ( dsetconstr @ ( setunion @ Xu )
% 3.02/1.56 @ ^ [Xx: $i] :
% 3.02/1.56 ( Xu
% 3.02/1.56 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ksnd_type,type,
% 3.02/1.56 ksnd: $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(ksndpairEq_type,type,
% 3.02/1.56 ksndpairEq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ksndpairEq,definition,
% 3.02/1.56 ( ksndpairEq
% 3.02/1.56 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.02/1.56 = Xy ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(kpairsurjEq_type,type,
% 3.02/1.56 kpairsurjEq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(kpairsurjEq,definition,
% 3.02/1.56 ( kpairsurjEq
% 3.02/1.56 = ( ! [Xu: $i] :
% 3.02/1.56 ( ( iskpair @ Xu )
% 3.02/1.56 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.02/1.56 = Xu ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodsndin_type,type,
% 3.02/1.56 cartprodsndin: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodsndin,definition,
% 3.02/1.56 ( cartprodsndin
% 3.02/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.56 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodpairmemEL_type,type,
% 3.02/1.56 cartprodpairmemEL: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodpairmemEL,definition,
% 3.02/1.56 ( cartprodpairmemEL
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.02/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodpairmemER_type,type,
% 3.02/1.56 cartprodpairmemER: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodpairmemER,definition,
% 3.02/1.56 ( cartprodpairmemER
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.02/1.56 => ( in @ Xy @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodmempaircEq_type,type,
% 3.02/1.56 cartprodmempaircEq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodmempaircEq,definition,
% 3.02/1.56 ( cartprodmempaircEq
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( kpair @ Xx @ Xy )
% 3.02/1.56 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodfstpairEq_type,type,
% 3.02/1.56 cartprodfstpairEq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodfstpairEq,definition,
% 3.02/1.56 ( cartprodfstpairEq
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.02/1.56 = Xx ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodsndpairEq_type,type,
% 3.02/1.56 cartprodsndpairEq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodsndpairEq,definition,
% 3.02/1.56 ( cartprodsndpairEq
% 3.02/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.02/1.56 = Xy ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodpairsurjEq_type,type,
% 3.02/1.56 cartprodpairsurjEq: $o ).
% 3.02/1.56
% 3.02/1.56 thf(cartprodpairsurjEq,definition,
% 3.02/1.56 ( cartprodpairsurjEq
% 3.02/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.56 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.02/1.56 = Xu ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(breln_type,type,
% 3.02/1.56 breln: $i > $i > $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstr_type,type,
% 3.02/1.56 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrI_type,type,
% 3.02/1.56 dpsetconstrI: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrI,definition,
% 3.02/1.56 ( dpsetconstrI
% 3.02/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( Xphi @ Xx @ Xy )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ Xy )
% 3.02/1.56 @ ( dpsetconstr @ A @ B
% 3.02/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrSub_type,type,
% 3.02/1.56 dpsetconstrSub: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrSub,definition,
% 3.02/1.56 ( dpsetconstrSub
% 3.02/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.02/1.56 ( subset
% 3.02/1.56 @ ( dpsetconstr @ A @ B
% 3.02/1.56 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.02/1.56 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(setOfPairsIsBReln_type,type,
% 3.02/1.56 setOfPairsIsBReln: $o ).
% 3.02/1.56
% 3.02/1.56 thf(setOfPairsIsBReln,definition,
% 3.02/1.56 ( setOfPairsIsBReln
% 3.02/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.02/1.56 ( breln @ A @ B
% 3.02/1.56 @ ( dpsetconstr @ A @ B
% 3.02/1.56 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrERa_type,type,
% 3.02/1.56 dpsetconstrERa: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrERa,definition,
% 3.02/1.56 ( dpsetconstrERa
% 3.02/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.02/1.56 @ ( dpsetconstr @ A @ B
% 3.02/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.02/1.56 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrEL1_type,type,
% 3.02/1.56 dpsetconstrEL1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrEL1,definition,
% 3.02/1.56 ( dpsetconstrEL1
% 3.02/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 3.02/1.56 @ ( dpsetconstr @ A @ B
% 3.02/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.02/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrEL2_type,type,
% 3.02/1.56 dpsetconstrEL2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrEL2,definition,
% 3.02/1.56 ( dpsetconstrEL2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 3.02/1.56 @ ( dpsetconstr @ A @ B
% 3.02/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.02/1.56 => ( in @ Xy @ B ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrER_type,type,
% 3.02/1.56 dpsetconstrER: $o ).
% 3.02/1.56
% 3.02/1.56 thf(dpsetconstrER,definition,
% 3.02/1.56 ( dpsetconstrER
% 3.02/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 3.02/1.56 @ ( dpsetconstr @ A @ B
% 3.02/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.02/1.56 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(func_type,type,
% 3.02/1.56 func: $i > $i > $i > $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcSet_type,type,
% 3.02/1.56 funcSet: $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(funcImageSingleton_type,type,
% 3.02/1.56 funcImageSingleton: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcImageSingleton,definition,
% 3.02/1.56 ( funcImageSingleton
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( singleton
% 3.02/1.56 @ ( dsetconstr @ B
% 3.02/1.56 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(apProp_type,type,
% 3.02/1.56 apProp: $o ).
% 3.02/1.56
% 3.02/1.56 thf(apProp,definition,
% 3.02/1.56 ( apProp
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in
% 3.02/1.56 @ ( setunion
% 3.02/1.56 @ ( dsetconstr @ B
% 3.02/1.56 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.02/1.56 @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ap_type,type,
% 3.02/1.56 ap: $i > $i > $i > $i > $i ).
% 3.02/1.56
% 3.02/1.56 thf(app_type,type,
% 3.02/1.56 app: $o ).
% 3.02/1.56
% 3.02/1.56 thf(app,definition,
% 3.02/1.56 ( app
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(infuncsetfunc_type,type,
% 3.02/1.56 infuncsetfunc: $o ).
% 3.02/1.56
% 3.02/1.56 thf(infuncsetfunc,definition,
% 3.02/1.56 ( infuncsetfunc
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.02/1.56 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ap2p_type,type,
% 3.02/1.56 ap2p: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ap2p,definition,
% 3.02/1.56 ( ap2p
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(funcinfuncset_type,type,
% 3.02/1.56 funcinfuncset: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcinfuncset,definition,
% 3.02/1.56 ( funcinfuncset
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(lamProp_type,type,
% 3.02/1.56 lamProp: $o ).
% 3.02/1.56
% 3.02/1.56 thf(lamProp,definition,
% 3.02/1.56 ( lamProp
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.02/1.56 => ( func @ A @ B
% 3.02/1.56 @ ( dpsetconstr @ A @ B
% 3.02/1.56 @ ^ [Xx: $i,Xy: $i] :
% 3.02/1.56 ( ( Xf @ Xx )
% 3.02/1.56 = Xy ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(lam_type,type,
% 3.02/1.56 lam: $i > $i > ( $i > $i ) > $i ).
% 3.02/1.56
% 3.02/1.56 thf(lamp_type,type,
% 3.02/1.56 lamp: $o ).
% 3.02/1.56
% 3.02/1.56 thf(lamp,definition,
% 3.02/1.56 ( lamp
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.02/1.56 => ( func @ A @ B
% 3.02/1.56 @ ( lam @ A @ B
% 3.02/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(lam2p_type,type,
% 3.02/1.56 lam2p: $o ).
% 3.02/1.56
% 3.02/1.56 thf(lam2p,definition,
% 3.02/1.56 ( lam2p
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.02/1.56 => ( in
% 3.02/1.56 @ ( lam @ A @ B
% 3.02/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.02/1.56 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(brelnall1_type,type,
% 3.02/1.56 brelnall1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(brelnall1,definition,
% 3.02/1.56 ( brelnall1
% 3.02/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 3.02/1.56 ( ( breln @ A @ B @ R )
% 3.02/1.56 => ! [Xphi: $i > $o] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.56 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ R )
% 3.02/1.56 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(brelnall2_type,type,
% 3.02/1.56 brelnall2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(brelnall2,definition,
% 3.02/1.56 ( brelnall2
% 3.02/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 3.02/1.56 ( ( breln @ A @ B @ R )
% 3.02/1.56 => ! [Xphi: $i > $o] :
% 3.02/1.56 ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.56 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ R )
% 3.02/1.56 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ex1E2_type,type,
% 3.02/1.56 ex1E2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ex1E2,definition,
% 3.02/1.56 ( ex1E2
% 3.02/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.56 ( ( ex1 @ A
% 3.02/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ A )
% 3.02/1.56 => ( ( Xphi @ Xx )
% 3.02/1.56 => ( ( Xphi @ Xy )
% 3.02/1.56 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(funcGraphProp1_type,type,
% 3.02/1.56 funcGraphProp1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcGraphProp1,definition,
% 3.02/1.56 ( funcGraphProp1
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(funcGraphProp3_type,type,
% 3.02/1.56 funcGraphProp3: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcGraphProp3,definition,
% 3.02/1.56 ( funcGraphProp3
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(funcGraphProp2_type,type,
% 3.02/1.56 funcGraphProp2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcGraphProp2,definition,
% 3.02/1.56 ( funcGraphProp2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.02/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.02/1.56 = Xy ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(funcextLem_type,type,
% 3.02/1.56 funcextLem: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcextLem,definition,
% 3.02/1.56 ( funcextLem
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ! [Xg: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xg )
% 3.02/1.56 => ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.02/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(funcGraphProp4_type,type,
% 3.02/1.56 funcGraphProp4: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcGraphProp4,definition,
% 3.02/1.56 ( funcGraphProp4
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.02/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.02/1.56 = Xy ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(subbreln_type,type,
% 3.02/1.56 subbreln: $o ).
% 3.02/1.56
% 3.02/1.56 thf(subbreln,definition,
% 3.02/1.56 ( subbreln
% 3.02/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 3.02/1.56 ( ( breln @ A @ B @ R )
% 3.02/1.56 => ! [S: $i] :
% 3.02/1.56 ( ( breln @ A @ B @ S )
% 3.02/1.56 => ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.02/1.56 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(eqbreln_type,type,
% 3.02/1.56 eqbreln: $o ).
% 3.02/1.56
% 3.02/1.56 thf(eqbreln,definition,
% 3.02/1.56 ( eqbreln
% 3.02/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 3.02/1.56 ( ( breln @ A @ B @ R )
% 3.02/1.56 => ! [S: $i] :
% 3.02/1.56 ( ( breln @ A @ B @ S )
% 3.02/1.56 => ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.02/1.56 => ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ! [Xy: $i] :
% 3.02/1.56 ( ( in @ Xy @ B )
% 3.02/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.02/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.02/1.56 => ( R = S ) ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(funcext_type,type,
% 3.02/1.56 funcext: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcext,definition,
% 3.02/1.56 ( funcext
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ! [Xg: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xg )
% 3.02/1.56 => ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.02/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.02/1.56 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(funcext2_type,type,
% 3.02/1.56 funcext2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(funcext2,definition,
% 3.02/1.56 ( funcext2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.02/1.56 => ! [Xg: $i] :
% 3.02/1.56 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.02/1.56 => ( ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.02/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.02/1.56 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ap2apEq1_type,type,
% 3.02/1.56 ap2apEq1: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ap2apEq1,definition,
% 3.02/1.56 ( ap2apEq1
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.02/1.56 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.02/1.56
% 3.02/1.56 thf(ap2apEq2_type,type,
% 3.02/1.56 ap2apEq2: $o ).
% 3.02/1.56
% 3.02/1.56 thf(ap2apEq2,definition,
% 3.02/1.56 ( ap2apEq2
% 3.02/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.56 ( ( func @ A @ B @ Xf )
% 3.02/1.56 => ! [Xx: $i] :
% 3.02/1.56 ( ( in @ Xx @ A )
% 3.02/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.02/1.57 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(beta1_type,type,
% 3.02/1.57 beta1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(beta1,definition,
% 3.02/1.57 ( beta1
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( ap @ A @ B
% 3.02/1.57 @ ( lam @ A @ B
% 3.02/1.57 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.02/1.57 @ Xx )
% 3.02/1.57 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(eta1_type,type,
% 3.02/1.57 eta1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(eta1,definition,
% 3.02/1.57 ( eta1
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.57 ( ( func @ A @ B @ Xf )
% 3.02/1.57 => ( ( lam @ A @ B
% 3.02/1.57 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.02/1.57 = Xf ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(lam2lamEq_type,type,
% 3.02/1.57 lam2lamEq: $o ).
% 3.02/1.57
% 3.02/1.57 thf(lam2lamEq,definition,
% 3.02/1.57 ( lam2lamEq
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.02/1.57 => ( ( lam @ A @ B
% 3.02/1.57 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.02/1.57 = ( lam @ A @ B
% 3.02/1.57 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(beta2_type,type,
% 3.02/1.57 beta2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(beta2,definition,
% 3.02/1.57 ( beta2
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( ap @ A @ B
% 3.02/1.57 @ ( lam @ A @ B
% 3.02/1.57 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.02/1.57 @ Xx )
% 3.02/1.57 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(eta2_type,type,
% 3.02/1.57 eta2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(eta2,definition,
% 3.02/1.57 ( eta2
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.02/1.57 => ( ( lam @ A @ B
% 3.02/1.57 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.02/1.57 = Xf ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(iffalseProp1_type,type,
% 3.02/1.57 iffalseProp1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(iffalseProp1,definition,
% 3.02/1.57 ( iffalseProp1
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ~ Xphi
% 3.02/1.57 => ( in @ Xy
% 3.02/1.57 @ ( dsetconstr @ A
% 3.02/1.57 @ ^ [Xz: $i] :
% 3.02/1.57 ( ( Xphi
% 3.02/1.57 & ( Xz = Xx ) )
% 3.02/1.57 | ( ~ Xphi
% 3.02/1.57 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(iffalseProp2_type,type,
% 3.02/1.57 iffalseProp2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(iffalseProp2,definition,
% 3.02/1.57 ( iffalseProp2
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ~ Xphi
% 3.02/1.57 => ( ( dsetconstr @ A
% 3.02/1.57 @ ^ [Xz: $i] :
% 3.02/1.57 ( ( Xphi
% 3.02/1.57 & ( Xz = Xx ) )
% 3.02/1.57 | ( ~ Xphi
% 3.02/1.57 & ( Xz = Xy ) ) ) )
% 3.02/1.57 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(iftrueProp1_type,type,
% 3.02/1.57 iftrueProp1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(iftrueProp1,definition,
% 3.02/1.57 ( iftrueProp1
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( Xphi
% 3.02/1.57 => ( in @ Xx
% 3.02/1.57 @ ( dsetconstr @ A
% 3.02/1.57 @ ^ [Xz: $i] :
% 3.02/1.57 ( ( Xphi
% 3.02/1.57 & ( Xz = Xx ) )
% 3.02/1.57 | ( ~ Xphi
% 3.02/1.57 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(iftrueProp2_type,type,
% 3.02/1.57 iftrueProp2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(iftrueProp2,definition,
% 3.02/1.57 ( iftrueProp2
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( Xphi
% 3.02/1.57 => ( ( dsetconstr @ A
% 3.02/1.57 @ ^ [Xz: $i] :
% 3.02/1.57 ( ( Xphi
% 3.02/1.57 & ( Xz = Xx ) )
% 3.02/1.57 | ( ~ Xphi
% 3.02/1.57 & ( Xz = Xy ) ) ) )
% 3.02/1.57 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(ifSingleton_type,type,
% 3.02/1.57 ifSingleton: $o ).
% 3.02/1.57
% 3.02/1.57 thf(ifSingleton,definition,
% 3.02/1.57 ( ifSingleton
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( singleton
% 3.02/1.57 @ ( dsetconstr @ A
% 3.02/1.57 @ ^ [Xz: $i] :
% 3.02/1.57 ( ( Xphi
% 3.02/1.57 & ( Xz = Xx ) )
% 3.02/1.57 | ( ~ Xphi
% 3.02/1.57 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(if_type,type,
% 3.02/1.57 if: $i > $o > $i > $i > $i ).
% 3.02/1.57
% 3.02/1.57 thf(ifp_type,type,
% 3.02/1.57 ifp: $o ).
% 3.02/1.57
% 3.02/1.57 thf(ifp,definition,
% 3.02/1.57 ( ifp
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(theeq_type,type,
% 3.02/1.57 theeq: $o ).
% 3.02/1.57
% 3.02/1.57 thf(theeq,definition,
% 3.02/1.57 ( theeq
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ( singleton @ X )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ X )
% 3.02/1.57 => ( ( setunion @ X )
% 3.02/1.57 = Xx ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(iftrue_type,type,
% 3.02/1.57 iftrue: $o ).
% 3.02/1.57
% 3.02/1.57 thf(iftrue,definition,
% 3.02/1.57 ( iftrue
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( Xphi
% 3.02/1.57 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.02/1.57 = Xx ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(iffalse_type,type,
% 3.02/1.57 iffalse: $o ).
% 3.02/1.57
% 3.02/1.57 thf(iffalse,definition,
% 3.02/1.57 ( iffalse
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ~ Xphi
% 3.02/1.57 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.02/1.57 = Xy ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(iftrueorfalse_type,type,
% 3.02/1.57 iftrueorfalse: $o ).
% 3.02/1.57
% 3.02/1.57 thf(iftrueorfalse,definition,
% 3.02/1.57 ( iftrueorfalse
% 3.02/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binintersectT_lem_type,type,
% 3.02/1.57 binintersectT_lem: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binintersectT_lem,definition,
% 3.02/1.57 ( binintersectT_lem
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binunionT_lem_type,type,
% 3.02/1.57 binunionT_lem: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binunionT_lem,definition,
% 3.02/1.57 ( binunionT_lem
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(powersetT_lem_type,type,
% 3.02/1.57 powersetT_lem: $o ).
% 3.02/1.57
% 3.02/1.57 thf(powersetT_lem,definition,
% 3.02/1.57 ( powersetT_lem
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(setminusT_lem_type,type,
% 3.02/1.57 setminusT_lem: $o ).
% 3.02/1.57
% 3.02/1.57 thf(setminusT_lem,definition,
% 3.02/1.57 ( setminusT_lem
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementT_lem_type,type,
% 3.02/1.57 complementT_lem: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementT_lem,definition,
% 3.02/1.57 ( complementT_lem
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(setextT_type,type,
% 3.02/1.57 setextT: $o ).
% 3.02/1.57
% 3.02/1.57 thf(setextT,definition,
% 3.02/1.57 ( setextT
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ X )
% 3.02/1.57 => ( in @ Xx @ Y ) ) )
% 3.02/1.57 => ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ Y )
% 3.02/1.57 => ( in @ Xx @ X ) ) )
% 3.02/1.57 => ( X = Y ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(subsetTI_type,type,
% 3.02/1.57 subsetTI: $o ).
% 3.02/1.57
% 3.02/1.57 thf(subsetTI,definition,
% 3.02/1.57 ( subsetTI
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ X )
% 3.02/1.57 => ( in @ Xx @ Y ) ) )
% 3.02/1.57 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(powersetTI1_type,type,
% 3.02/1.57 powersetTI1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(powersetTI1,definition,
% 3.02/1.57 ( powersetTI1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ X )
% 3.02/1.57 => ( in @ Xx @ Y ) ) )
% 3.02/1.57 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(powersetTE1_type,type,
% 3.02/1.57 powersetTE1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(powersetTE1,definition,
% 3.02/1.57 ( powersetTE1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ X @ ( powerset @ Y ) )
% 3.02/1.57 => ( ( in @ Xx @ X )
% 3.02/1.57 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementTI1_type,type,
% 3.02/1.57 complementTI1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementTI1,definition,
% 3.02/1.57 ( complementTI1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ X )
% 3.02/1.57 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementTE1_type,type,
% 3.02/1.57 complementTE1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementTE1,definition,
% 3.02/1.57 ( complementTE1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.02/1.57 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binintersectTELcontra_type,type,
% 3.02/1.57 binintersectTELcontra: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binintersectTELcontra,definition,
% 3.02/1.57 ( binintersectTELcontra
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ~ ( in @ Xx @ X )
% 3.02/1.57 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binintersectTERcontra_type,type,
% 3.02/1.57 binintersectTERcontra: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binintersectTERcontra,definition,
% 3.02/1.57 ( binintersectTERcontra
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ~ ( in @ Xx @ Y )
% 3.02/1.57 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(contrasubsetT_type,type,
% 3.02/1.57 contrasubsetT: $o ).
% 3.02/1.57
% 3.02/1.57 thf(contrasubsetT,definition,
% 3.02/1.57 ( contrasubsetT
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.02/1.57 => ( ( in @ Xx @ Y )
% 3.02/1.57 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(contrasubsetT1_type,type,
% 3.02/1.57 contrasubsetT1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(contrasubsetT1,definition,
% 3.02/1.57 ( contrasubsetT1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( subset @ X @ Y )
% 3.02/1.57 => ( ~ ( in @ Xx @ Y )
% 3.02/1.57 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(contrasubsetT2_type,type,
% 3.02/1.57 contrasubsetT2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(contrasubsetT2,definition,
% 3.02/1.57 ( contrasubsetT2
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ X @ Y )
% 3.02/1.57 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(contrasubsetT3_type,type,
% 3.02/1.57 contrasubsetT3: $o ).
% 3.02/1.57
% 3.02/1.57 thf(contrasubsetT3,definition,
% 3.02/1.57 ( contrasubsetT3
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.02/1.57 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementI1_type,type,
% 3.02/1.57 doubleComplementI1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementI1,definition,
% 3.02/1.57 ( doubleComplementI1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ X )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementE1_type,type,
% 3.02/1.57 doubleComplementE1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementE1,definition,
% 3.02/1.57 ( doubleComplementE1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.02/1.57 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementSub1_type,type,
% 3.02/1.57 doubleComplementSub1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementSub1,definition,
% 3.02/1.57 ( doubleComplementSub1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementSub2_type,type,
% 3.02/1.57 doubleComplementSub2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementSub2,definition,
% 3.02/1.57 ( doubleComplementSub2
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementEq_type,type,
% 3.02/1.57 doubleComplementEq: $o ).
% 3.02/1.57
% 3.02/1.57 thf(doubleComplementEq,definition,
% 3.02/1.57 ( doubleComplementEq
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ( X
% 3.02/1.57 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementTnotintersectT_type,type,
% 3.02/1.57 complementTnotintersectT: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementTnotintersectT,definition,
% 3.02/1.57 ( complementTnotintersectT
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.02/1.57 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementImpComplementIntersect_type,type,
% 3.02/1.57 complementImpComplementIntersect: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementImpComplementIntersect,definition,
% 3.02/1.57 ( complementImpComplementIntersect
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementSubsetComplementIntersect_type,type,
% 3.02/1.57 complementSubsetComplementIntersect: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementSubsetComplementIntersect,definition,
% 3.02/1.57 ( complementSubsetComplementIntersect
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementInPowersetComplementIntersect_type,type,
% 3.02/1.57 complementInPowersetComplementIntersect: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementInPowersetComplementIntersect,definition,
% 3.02/1.57 ( complementInPowersetComplementIntersect
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(contraSubsetComplement_type,type,
% 3.02/1.57 contraSubsetComplement: $o ).
% 3.02/1.57
% 3.02/1.57 thf(contraSubsetComplement,definition,
% 3.02/1.57 ( contraSubsetComplement
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ Y )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementTcontraSubset_type,type,
% 3.02/1.57 complementTcontraSubset: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementTcontraSubset,definition,
% 3.02/1.57 ( complementTcontraSubset
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.02/1.57 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binunionTILcontra_type,type,
% 3.02/1.57 binunionTILcontra: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binunionTILcontra,definition,
% 3.02/1.57 ( binunionTILcontra
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.02/1.57 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binunionTIRcontra_type,type,
% 3.02/1.57 binunionTIRcontra: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binunionTIRcontra,definition,
% 3.02/1.57 ( binunionTIRcontra
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.02/1.57 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(inIntersectImpInUnion_type,type,
% 3.02/1.57 inIntersectImpInUnion: $o ).
% 3.02/1.57
% 3.02/1.57 thf(inIntersectImpInUnion,definition,
% 3.02/1.57 ( inIntersectImpInUnion
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.02/1.57 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(inIntersectImpInUnion2_type,type,
% 3.02/1.57 inIntersectImpInUnion2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(inIntersectImpInUnion2,definition,
% 3.02/1.57 ( inIntersectImpInUnion2
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.02/1.57 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(inIntersectImpInIntersectUnions_type,type,
% 3.02/1.57 inIntersectImpInIntersectUnions: $o ).
% 3.02/1.57
% 3.02/1.57 thf(inIntersectImpInIntersectUnions,definition,
% 3.02/1.57 ( inIntersectImpInIntersectUnions
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.02/1.57 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(intersectInPowersetIntersectUnions_type,type,
% 3.02/1.57 intersectInPowersetIntersectUnions: $o ).
% 3.02/1.57
% 3.02/1.57 thf(intersectInPowersetIntersectUnions,definition,
% 3.02/1.57 ( intersectInPowersetIntersectUnions
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(inComplementUnionImpNotIn1_type,type,
% 3.02/1.57 inComplementUnionImpNotIn1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(inComplementUnionImpNotIn1,definition,
% 3.02/1.57 ( inComplementUnionImpNotIn1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.02/1.57 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(inComplementUnionImpInComplement1_type,type,
% 3.02/1.57 inComplementUnionImpInComplement1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(inComplementUnionImpInComplement1,definition,
% 3.02/1.57 ( inComplementUnionImpInComplement1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binunionTE_type,type,
% 3.02/1.57 binunionTE: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binunionTE,definition,
% 3.02/1.57 ( binunionTE
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xphi: $o,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.02/1.57 => ( ( ( in @ Xx @ X )
% 3.02/1.57 => Xphi )
% 3.02/1.57 => ( ( ( in @ Xx @ Y )
% 3.02/1.57 => Xphi )
% 3.02/1.57 => Xphi ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binunionTEcontra_type,type,
% 3.02/1.57 binunionTEcontra: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binunionTEcontra,definition,
% 3.02/1.57 ( binunionTEcontra
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ~ ( in @ Xx @ X )
% 3.02/1.57 => ( ~ ( in @ Xx @ Y )
% 3.02/1.57 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2a1_type,type,
% 3.02/1.57 demorgan2a1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2a1,definition,
% 3.02/1.57 ( demorgan2a1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(complementUnionInPowersetComplement_type,type,
% 3.02/1.57 complementUnionInPowersetComplement: $o ).
% 3.02/1.57
% 3.02/1.57 thf(complementUnionInPowersetComplement,definition,
% 3.02/1.57 ( complementUnionInPowersetComplement
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2a2_type,type,
% 3.02/1.57 demorgan2a2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2a2,definition,
% 3.02/1.57 ( demorgan2a2
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan1a_type,type,
% 3.02/1.57 demorgan1a: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan1a,definition,
% 3.02/1.57 ( demorgan1a
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.02/1.57 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan1b_type,type,
% 3.02/1.57 demorgan1b: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan1b,definition,
% 3.02/1.57 ( demorgan1b
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan1_type,type,
% 3.02/1.57 demorgan1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan1,definition,
% 3.02/1.57 ( demorgan1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.02/1.57 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2a_type,type,
% 3.02/1.57 demorgan2a: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2a,definition,
% 3.02/1.57 ( demorgan2a
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.02/1.57 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2b2_type,type,
% 3.02/1.57 demorgan2b2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2b2,definition,
% 3.02/1.57 ( demorgan2b2
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.02/1.57 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2b_type,type,
% 3.02/1.57 demorgan2b: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2b,definition,
% 3.02/1.57 ( demorgan2b
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.02/1.57 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2_type,type,
% 3.02/1.57 demorgan2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(demorgan2,definition,
% 3.02/1.57 ( demorgan2
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.02/1.57 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule0_type,type,
% 3.02/1.57 woz13rule0: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule0,definition,
% 3.02/1.57 ( woz13rule0
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.02/1.57 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule1_type,type,
% 3.02/1.57 woz13rule1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule1,definition,
% 3.02/1.57 ( woz13rule1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ X @ Z )
% 3.02/1.57 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule2_type,type,
% 3.02/1.57 woz13rule2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule2,definition,
% 3.02/1.57 ( woz13rule2
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ Y @ Z )
% 3.02/1.57 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule3_type,type,
% 3.02/1.57 woz13rule3: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule3,definition,
% 3.02/1.57 ( woz13rule3
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ X @ Y )
% 3.02/1.57 => ( ( subset @ X @ Z )
% 3.02/1.57 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule4_type,type,
% 3.02/1.57 woz13rule4: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz13rule4,definition,
% 3.02/1.57 ( woz13rule4
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ! [W: $i] :
% 3.02/1.57 ( ( in @ W @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ X @ Z )
% 3.02/1.57 => ( ( subset @ Y @ W )
% 3.02/1.57 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_1_type,type,
% 3.02/1.57 woz1_1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_1,definition,
% 3.02/1.57 ( woz1_1
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_2_type,type,
% 3.02/1.57 woz1_2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_2,definition,
% 3.02/1.57 ( woz1_2
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ! [W: $i] :
% 3.02/1.57 ( ( in @ W @ ( powerset @ A ) )
% 3.02/1.57 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.02/1.57 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_3_type,type,
% 3.02/1.57 woz1_3: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_3,definition,
% 3.02/1.57 ( woz1_3
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ! [Z: $i] :
% 3.02/1.57 ( ( in @ Z @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_4_type,type,
% 3.02/1.57 woz1_4: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_4,definition,
% 3.02/1.57 ( woz1_4
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.02/1.57 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_5_type,type,
% 3.02/1.57 woz1_5: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz1_5,definition,
% 3.02/1.57 ( woz1_5
% 3.02/1.57 = ( ! [A: $i,X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 3.02/1.57 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1_type,type,
% 3.02/1.57 breln1: $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1all2_type,type,
% 3.02/1.57 breln1all2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1all2,definition,
% 3.02/1.57 ( breln1all2
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [Xphi: $i > $o] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ R )
% 3.02/1.57 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1Set_type,type,
% 3.02/1.57 breln1Set: $i > $i ).
% 3.02/1.57
% 3.02/1.57 thf(breln1SetBreln1_type,type,
% 3.02/1.57 breln1SetBreln1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1SetBreln1,definition,
% 3.02/1.57 ( breln1SetBreln1
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( in @ R @ ( breln1Set @ A ) )
% 3.02/1.57 => ( breln1 @ A @ R ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(transitive_type,type,
% 3.02/1.57 transitive: $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(antisymmetric_type,type,
% 3.02/1.57 antisymmetric: $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(reflexive_type,type,
% 3.02/1.57 reflexive: $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(refltransitive_type,type,
% 3.02/1.57 refltransitive: $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(refllinearorder_type,type,
% 3.02/1.57 refllinearorder: $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(reflwellordering_type,type,
% 3.02/1.57 reflwellordering: $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(choice2fnsingleton_type,type,
% 3.02/1.57 choice2fnsingleton: $o ).
% 3.02/1.57
% 3.02/1.57 thf(choice2fnsingleton,definition,
% 3.02/1.57 ( choice2fnsingleton
% 3.02/1.57 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ? [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ B )
% 3.02/1.57 & ( Xphi @ Xx @ Xy ) ) )
% 3.02/1.57 => ! [R: $i] :
% 3.02/1.57 ( ( in @ R @ ( breln1Set @ B ) )
% 3.02/1.57 => ( ( reflwellordering @ B @ R )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( singleton
% 3.02/1.57 @ ( dsetconstr @ B
% 3.02/1.57 @ ^ [Xy: $i] :
% 3.02/1.57 ( ( Xphi @ Xx @ Xy )
% 3.02/1.57 & ! [Xz: $i] :
% 3.02/1.57 ( ( in @ Xz @ B )
% 3.02/1.57 => ( ( Xphi @ Xx @ Xz )
% 3.02/1.57 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(setOfPairsIsBReln1_type,type,
% 3.02/1.57 setOfPairsIsBReln1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(setOfPairsIsBReln1,definition,
% 3.02/1.57 ( setOfPairsIsBReln1
% 3.02/1.57 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.02/1.57 ( breln1 @ A
% 3.02/1.57 @ ( dpsetconstr @ A @ A
% 3.02/1.57 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1all1_type,type,
% 3.02/1.57 breln1all1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1all1,definition,
% 3.02/1.57 ( breln1all1
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [Xphi: $i > $o] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ R )
% 3.02/1.57 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(subbreln1_type,type,
% 3.02/1.57 subbreln1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(subbreln1,definition,
% 3.02/1.57 ( subbreln1
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.02/1.57 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(eqbreln1_type,type,
% 3.02/1.57 eqbreln1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(eqbreln1,definition,
% 3.02/1.57 ( eqbreln1
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.02/1.57 => ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.02/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.02/1.57 => ( R = S ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1invset_type,type,
% 3.02/1.57 breln1invset: $i > $i > $i ).
% 3.02/1.57
% 3.02/1.57 thf(breln1invprop_type,type,
% 3.02/1.57 breln1invprop: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1invprop,definition,
% 3.02/1.57 ( breln1invprop
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1invI_type,type,
% 3.02/1.57 breln1invI: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1invI,definition,
% 3.02/1.57 ( breln1invI
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1invE_type,type,
% 3.02/1.57 breln1invE: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1invE,definition,
% 3.02/1.57 ( breln1invE
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.02/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compset_type,type,
% 3.02/1.57 breln1compset: $i > $i > $i > $i ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compprop_type,type,
% 3.02/1.57 breln1compprop: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compprop,definition,
% 3.02/1.57 ( breln1compprop
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compI_type,type,
% 3.02/1.57 breln1compI: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compI,definition,
% 3.02/1.57 ( breln1compI
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ! [Xz: $i] :
% 3.02/1.57 ( ( in @ Xz @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.02/1.57 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.02/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compE_type,type,
% 3.02/1.57 breln1compE: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compE,definition,
% 3.02/1.57 ( breln1compE
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.02/1.57 => ? [Xz: $i] :
% 3.02/1.57 ( ( in @ Xz @ A )
% 3.02/1.57 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.02/1.57 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compEex_type,type,
% 3.02/1.57 breln1compEex: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1compEex,definition,
% 3.02/1.57 ( breln1compEex
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.02/1.57 => ! [Xphi: $o] :
% 3.02/1.57 ( ! [Xz: $i] :
% 3.02/1.57 ( ( in @ Xz @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.02/1.57 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.02/1.57 => Xphi ) ) )
% 3.02/1.57 => Xphi ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionprop_type,type,
% 3.02/1.57 breln1unionprop: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionprop,definition,
% 3.02/1.57 ( breln1unionprop
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionIL_type,type,
% 3.02/1.57 breln1unionIL: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionIL,definition,
% 3.02/1.57 ( breln1unionIL
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionIR_type,type,
% 3.02/1.57 breln1unionIR: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionIR,definition,
% 3.02/1.57 ( breln1unionIR
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.02/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionI_type,type,
% 3.02/1.57 breln1unionI: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionI,definition,
% 3.02/1.57 ( breln1unionI
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.02/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionE_type,type,
% 3.02/1.57 breln1unionE: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionE,definition,
% 3.02/1.57 ( breln1unionE
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionEcases_type,type,
% 3.02/1.57 breln1unionEcases: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionEcases,definition,
% 3.02/1.57 ( breln1unionEcases
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.02/1.57 => ! [Xphi: $o] :
% 3.02/1.57 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.02/1.57 => Xphi )
% 3.02/1.57 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.02/1.57 => Xphi )
% 3.02/1.57 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionCommutes_type,type,
% 3.02/1.57 breln1unionCommutes: $o ).
% 3.02/1.57
% 3.02/1.57 thf(breln1unionCommutes,definition,
% 3.02/1.57 ( breln1unionCommutes
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ( ( binunion @ R @ S )
% 3.02/1.57 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz2Ex_type,type,
% 3.02/1.57 woz2Ex: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz2Ex,definition,
% 3.02/1.57 ( woz2Ex
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ( R
% 3.02/1.57 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz2W_type,type,
% 3.02/1.57 woz2W: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz2W,definition,
% 3.02/1.57 ( woz2W
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.02/1.57 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz2A_type,type,
% 3.02/1.57 woz2A: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz2A,definition,
% 3.02/1.57 ( woz2A
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [T: $i] :
% 3.02/1.57 ( ( breln1 @ A @ T )
% 3.02/1.57 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.02/1.57 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(woz2B_type,type,
% 3.02/1.57 woz2B: $o ).
% 3.02/1.57
% 3.02/1.57 thf(woz2B,definition,
% 3.02/1.57 ( woz2B
% 3.02/1.57 = ( ! [A: $i,R: $i] :
% 3.02/1.57 ( ( breln1 @ A @ R )
% 3.02/1.57 => ! [S: $i] :
% 3.02/1.57 ( ( breln1 @ A @ S )
% 3.02/1.57 => ! [T: $i] :
% 3.02/1.57 ( ( breln1 @ A @ T )
% 3.02/1.57 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.02/1.57 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(image1Ex_type,type,
% 3.02/1.57 image1Ex: $o ).
% 3.02/1.57
% 3.02/1.57 thf(image1Ex,definition,
% 3.02/1.57 ( image1Ex
% 3.02/1.57 = ( ! [A: $i,Xf: $i > $i] :
% 3.02/1.57 ? [B: $i] :
% 3.02/1.57 ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ B )
% 3.02/1.57 <=> ? [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 & ( Xx
% 3.02/1.57 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(image1Ex1_type,type,
% 3.02/1.57 image1Ex1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(image1Ex1,definition,
% 3.02/1.57 ( image1Ex1
% 3.02/1.57 = ( ! [A: $i,Xf: $i > $i] :
% 3.02/1.57 ( exu
% 3.02/1.57 @ ^ [B: $i] :
% 3.02/1.57 ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ B )
% 3.02/1.57 <=> ? [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 & ( Xx
% 3.02/1.57 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(image1_type,type,
% 3.02/1.57 image1: $i > ( $i > $i ) > $i ).
% 3.02/1.57
% 3.02/1.57 thf(image1Equiv_type,type,
% 3.02/1.57 image1Equiv: $o ).
% 3.02/1.57
% 3.02/1.57 thf(image1Equiv,definition,
% 3.02/1.57 ( image1Equiv
% 3.02/1.57 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx
% 3.02/1.57 @ ( image1 @ A
% 3.02/1.57 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.02/1.57 <=> ? [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 & ( Xx
% 3.02/1.57 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(image1E_type,type,
% 3.02/1.57 image1E: $o ).
% 3.02/1.57
% 3.02/1.57 thf(image1E,definition,
% 3.02/1.57 ( image1E
% 3.02/1.57 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.02/1.57 ( ( in @ Xx
% 3.02/1.57 @ ( image1 @ A
% 3.02/1.57 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.02/1.57 => ? [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 & ( Xx
% 3.02/1.57 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(image1I_type,type,
% 3.02/1.57 image1I: $o ).
% 3.02/1.57
% 3.02/1.57 thf(image1I,definition,
% 3.02/1.57 ( image1I
% 3.02/1.57 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.02/1.57 ( ? [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ A )
% 3.02/1.57 & ( Xx
% 3.02/1.57 = ( Xf @ Xy ) ) )
% 3.02/1.57 => ( in @ Xx
% 3.02/1.57 @ ( image1 @ A
% 3.02/1.57 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(injective_type,type,
% 3.02/1.57 injective: $i > $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(injFuncSet_type,type,
% 3.02/1.57 injFuncSet: $i > $i > $i ).
% 3.02/1.57
% 3.02/1.57 thf(injFuncInInjFuncSet_type,type,
% 3.02/1.57 injFuncInInjFuncSet: $o ).
% 3.02/1.57
% 3.02/1.57 thf(injFuncInInjFuncSet,definition,
% 3.02/1.57 ( injFuncInInjFuncSet
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.02/1.57 => ( ( injective @ A @ B @ Xf )
% 3.02/1.57 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(injFuncSetFuncIn_type,type,
% 3.02/1.57 injFuncSetFuncIn: $o ).
% 3.02/1.57
% 3.02/1.57 thf(injFuncSetFuncIn,definition,
% 3.02/1.57 ( injFuncSetFuncIn
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.57 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.02/1.57 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(injFuncSetFuncInj_type,type,
% 3.02/1.57 injFuncSetFuncInj: $o ).
% 3.02/1.57
% 3.02/1.57 thf(injFuncSetFuncInj,definition,
% 3.02/1.57 ( injFuncSetFuncInj
% 3.02/1.57 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.02/1.57 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.02/1.57 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(surjective_type,type,
% 3.02/1.57 surjective: $i > $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(surjFuncSet_type,type,
% 3.02/1.57 surjFuncSet: $i > $i > $i ).
% 3.02/1.57
% 3.02/1.57 thf(surjFuncSetFuncIn_type,type,
% 3.02/1.57 surjFuncSetFuncIn: $o ).
% 3.02/1.57
% 3.02/1.57 thf(surjFuncSetFuncIn,definition,
% 3.02/1.57 ( surjFuncSetFuncIn
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.02/1.57 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 3.02/1.57 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(surjFuncSetFuncSurj_type,type,
% 3.02/1.57 surjFuncSetFuncSurj: $o ).
% 3.02/1.57
% 3.02/1.57 thf(surjFuncSetFuncSurj,definition,
% 3.02/1.57 ( surjFuncSetFuncSurj
% 3.02/1.57 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.02/1.57 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 3.02/1.57 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(leftInvIsSurj_type,type,
% 3.02/1.57 leftInvIsSurj: $o ).
% 3.02/1.57
% 3.02/1.57 thf(leftInvIsSurj,definition,
% 3.02/1.57 ( leftInvIsSurj
% 3.02/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.02/1.57 => ! [Xg: $i] :
% 3.02/1.57 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 3.02/1.57 => ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 3.02/1.57 = Xx ) )
% 3.02/1.57 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(surjCantorThm_type,type,
% 3.02/1.57 surjCantorThm: $o ).
% 3.02/1.57
% 3.02/1.57 thf(surjCantorThm,definition,
% 3.02/1.57 ( surjCantorThm
% 3.02/1.57 = ( ! [A: $i,Xf: $i] :
% 3.02/1.57 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 3.02/1.57 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(foundation2_type,type,
% 3.02/1.57 foundation2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(foundation2,definition,
% 3.02/1.57 ( foundation2
% 3.02/1.57 = ( ! [A: $i] :
% 3.02/1.57 ( ( nonempty @ A )
% 3.02/1.57 => ? [X: $i] :
% 3.02/1.57 ( ( in @ X @ A )
% 3.02/1.57 & ( ( binintersect @ X @ A )
% 3.02/1.57 = emptyset ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(notinself_type,type,
% 3.02/1.57 notinself: $o ).
% 3.02/1.57
% 3.02/1.57 thf(notinself,definition,
% 3.02/1.57 ( notinself
% 3.02/1.57 = ( ! [A: $i] :
% 3.02/1.57 ~ ( in @ A @ A ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(notinself2_type,type,
% 3.02/1.57 notinself2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(notinself2,definition,
% 3.02/1.57 ( notinself2
% 3.02/1.57 = ( ! [A: $i,B: $i] :
% 3.02/1.57 ( ( in @ A @ B )
% 3.02/1.57 => ~ ( in @ B @ A ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(omegaS_type,type,
% 3.02/1.57 omegaS: $i > $i ).
% 3.02/1.57
% 3.02/1.57 thf(omegaSp_type,type,
% 3.02/1.57 omegaSp: $o ).
% 3.02/1.57
% 3.02/1.57 thf(omegaSp,definition,
% 3.02/1.57 ( omegaSp
% 3.02/1.57 = ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ omega )
% 3.02/1.57 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(omegaSclos_type,type,
% 3.02/1.57 omegaSclos: $o ).
% 3.02/1.57
% 3.02/1.57 thf(omegaSclos,definition,
% 3.02/1.57 ( omegaSclos
% 3.02/1.57 = ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ omega )
% 3.02/1.57 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(peano0notS_type,type,
% 3.02/1.57 peano0notS: $o ).
% 3.02/1.57
% 3.02/1.57 thf(peano0notS,definition,
% 3.02/1.57 ( peano0notS
% 3.02/1.57 = ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ omega )
% 3.02/1.57 => ( ( omegaS @ Xx )
% 3.02/1.57 != emptyset ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(peano3_type,type,
% 3.02/1.57 peano3: $i > $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(peano4_type,type,
% 3.02/1.57 peano4: $i > $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(peano5_type,type,
% 3.02/1.57 peano5: $i > $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(peanoSinj_type,type,
% 3.02/1.57 peanoSinj: $o ).
% 3.02/1.57
% 3.02/1.57 thf(peanoSinj,definition,
% 3.02/1.57 ( peanoSinj
% 3.02/1.57 = ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ omega )
% 3.02/1.57 => ! [Xy: $i] :
% 3.02/1.57 ( ( in @ Xy @ omega )
% 3.02/1.57 => ( ( ( omegaS @ Xx )
% 3.02/1.57 = ( omegaS @ Xy ) )
% 3.02/1.57 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(peano_type,type,
% 3.02/1.57 peano: $i > $i > $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(transitiveset_type,type,
% 3.02/1.57 transitiveset: $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(transitivesetOp1_type,type,
% 3.02/1.57 transitivesetOp1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(transitivesetOp1,definition,
% 3.02/1.57 ( transitivesetOp1
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ( transitiveset @ X )
% 3.02/1.57 => ! [A: $i] :
% 3.02/1.57 ( ( in @ A @ X )
% 3.02/1.57 => ( subset @ A @ X ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(binintTransitive_type,type,
% 3.02/1.57 binintTransitive: $o ).
% 3.02/1.57
% 3.02/1.57 thf(binintTransitive,definition,
% 3.02/1.57 ( binintTransitive
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ( transitiveset @ X )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( transitiveset @ Y )
% 3.02/1.57 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(transitivesetOp2_type,type,
% 3.02/1.57 transitivesetOp2: $o ).
% 3.02/1.57
% 3.02/1.57 thf(transitivesetOp2,definition,
% 3.02/1.57 ( transitivesetOp2
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ( transitiveset @ X )
% 3.02/1.57 => ! [A: $i,Xx: $i] :
% 3.02/1.57 ( ( in @ A @ X )
% 3.02/1.57 => ( ( in @ Xx @ A )
% 3.02/1.57 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(setunionTransitive_type,type,
% 3.02/1.57 setunionTransitive: $o ).
% 3.02/1.57
% 3.02/1.57 thf(setunionTransitive,definition,
% 3.02/1.57 ( setunionTransitive
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ X )
% 3.02/1.57 => ( transitiveset @ Xx ) )
% 3.02/1.57 => ( transitiveset @ ( setunion @ X ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(stricttotalorderedByIn_type,type,
% 3.02/1.57 stricttotalorderedByIn: $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(stricttotalorderedByIn,definition,
% 3.02/1.57 ( stricttotalorderedByIn
% 3.02/1.57 = ( ^ [A: $i] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ A )
% 3.02/1.57 => ! [X: $i] :
% 3.02/1.57 ( ( in @ X @ A )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ A )
% 3.02/1.57 => ( ( ( in @ Xx @ X )
% 3.02/1.57 & ( in @ X @ Y ) )
% 3.02/1.57 => ( in @ Xx @ Y ) ) ) ) )
% 3.02/1.57 & ! [X: $i] :
% 3.02/1.57 ( ( in @ X @ A )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ A )
% 3.02/1.57 => ( ( X = Y )
% 3.02/1.57 | ( in @ X @ Y )
% 3.02/1.57 | ( in @ Y @ X ) ) ) )
% 3.02/1.57 & ! [X: $i] :
% 3.02/1.57 ( ( in @ X @ A )
% 3.02/1.57 => ~ ( in @ X @ X ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(wellorderedByIn_type,type,
% 3.02/1.57 wellorderedByIn: $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(wellorderedByIn,definition,
% 3.02/1.57 ( wellorderedByIn
% 3.02/1.57 = ( ^ [A: $i] :
% 3.02/1.57 ( ( stricttotalorderedByIn @ A )
% 3.02/1.57 & ! [X: $i] :
% 3.02/1.57 ( ( in @ X @ ( powerset @ A ) )
% 3.02/1.57 => ( ( nonempty @ X )
% 3.02/1.57 => ? [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ X )
% 3.02/1.57 & ! [Y: $i] :
% 3.02/1.57 ( ( in @ Y @ X )
% 3.02/1.57 => ( ( Xx = Y )
% 3.02/1.57 | ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(ordinal_type,type,
% 3.02/1.57 ordinal: $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(ordinal,definition,
% 3.02/1.57 ( ordinal
% 3.02/1.57 = ( ^ [Xx: $i] :
% 3.02/1.57 ( ( transitiveset @ Xx )
% 3.02/1.57 & ( wellorderedByIn @ Xx ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(limitOrdinal_type,type,
% 3.02/1.57 limitOrdinal: $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(ordinalMinLem1_type,type,
% 3.02/1.57 ordinalMinLem1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(ordinalMinLem1,definition,
% 3.02/1.57 ( ordinalMinLem1
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ( ordinal @ X )
% 3.02/1.57 => ! [Y: $i] :
% 3.02/1.57 ( ( ordinal @ Y )
% 3.02/1.57 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(ordinalTransSet_type,type,
% 3.02/1.57 ordinalTransSet: $o ).
% 3.02/1.57
% 3.02/1.57 thf(ordinalTransSet,definition,
% 3.02/1.57 ( ordinalTransSet
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ( ordinal @ X )
% 3.02/1.57 => ! [Xx: $i,A: $i] :
% 3.02/1.57 ( ( in @ A @ X )
% 3.02/1.57 => ( ( in @ Xx @ A )
% 3.02/1.57 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(ordinalTransSet1_type,type,
% 3.02/1.57 ordinalTransSet1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(ordinalTransSet1,definition,
% 3.02/1.57 ( ordinalTransSet1
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ( ordinal @ X )
% 3.02/1.57 => ! [A: $i] :
% 3.02/1.57 ( ( in @ A @ X )
% 3.02/1.57 => ( subset @ A @ X ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(setunionOrdinalLem1_type,type,
% 3.02/1.57 setunionOrdinalLem1: $o ).
% 3.02/1.57
% 3.02/1.57 thf(setunionOrdinalLem1,definition,
% 3.02/1.57 ( setunionOrdinalLem1
% 3.02/1.57 = ( ! [X: $i] :
% 3.02/1.57 ( ! [Xx: $i] :
% 3.02/1.57 ( ( in @ Xx @ X )
% 3.02/1.57 => ( ordinal @ Xx ) )
% 3.02/1.57 => ( transitiveset @ ( setunion @ X ) ) ) ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(succOrdinal_type,type,
% 3.02/1.57 succOrdinal: $i > $o ).
% 3.02/1.57
% 3.02/1.57 thf(emptysetOrdinal_type,type,
% 3.02/1.57 emptysetOrdinal: $o ).
% 3.02/1.57
% 3.02/1.57 thf(emptysetOrdinal,definition,
% 3.02/1.57 ( emptysetOrdinal
% 3.02/1.57 = ( ordinal @ emptyset ) ) ).
% 3.02/1.57
% 3.02/1.57 thf(ordinalIrrefl,conjecture,
% 3.02/1.57 ( setextAx
% 3.02/1.57 => ( emptysetAx
% 3.02/1.57 => ( setadjoinAx
% 3.02/1.57 => ( powersetAx
% 3.02/1.57 => ( setunionAx
% 3.02/1.57 => ( omega0Ax
% 3.02/1.57 => ( omegaSAx
% 3.02/1.57 => ( omegaIndAx
% 3.02/1.57 => ( replAx
% 3.02/1.57 => ( foundationAx
% 3.02/1.57 => ( wellorderingAx
% 3.02/1.57 => ( descrp
% 3.02/1.57 => ( dsetconstrI
% 3.02/1.57 => ( dsetconstrEL
% 3.02/1.57 => ( dsetconstrER
% 3.02/1.57 => ( exuE1
% 3.02/1.57 => ( prop2setE
% 3.02/1.57 => ( emptysetE
% 3.02/1.57 => ( emptysetimpfalse
% 3.02/1.57 => ( notinemptyset
% 3.02/1.57 => ( exuE3e
% 3.02/1.57 => ( setext
% 3.02/1.57 => ( emptyI
% 3.02/1.57 => ( noeltsimpempty
% 3.02/1.57 => ( setbeta
% 3.02/1.57 => ( nonemptyE1
% 3.02/1.57 => ( nonemptyI
% 3.02/1.57 => ( nonemptyI1
% 3.02/1.57 => ( setadjoinIL
% 3.02/1.57 => ( emptyinunitempty
% 3.02/1.57 => ( setadjoinIR
% 3.02/1.57 => ( setadjoinE
% 3.02/1.57 => ( setadjoinOr
% 3.02/1.57 => ( setoftrueEq
% 3.02/1.57 => ( powersetI
% 3.02/1.57 => ( emptyinPowerset
% 3.02/1.57 => ( emptyInPowerset
% 3.02/1.57 => ( powersetE
% 3.02/1.57 => ( setunionI
% 3.02/1.57 => ( setunionE
% 3.02/1.57 => ( subPowSU
% 3.02/1.57 => ( exuE2
% 3.02/1.57 => ( nonemptyImpWitness
% 3.02/1.57 => ( uniqinunit
% 3.02/1.57 => ( notinsingleton
% 3.02/1.57 => ( eqinunit
% 3.02/1.57 => ( singletonsswitch
% 3.02/1.57 => ( upairsetE
% 3.02/1.57 => ( upairsetIL
% 3.02/1.57 => ( upairsetIR
% 3.02/1.57 => ( emptyE1
% 3.02/1.57 => ( vacuousDall
% 3.02/1.57 => ( quantDeMorgan1
% 3.02/1.57 => ( quantDeMorgan2
% 3.02/1.57 => ( quantDeMorgan3
% 3.02/1.57 => ( quantDeMorgan4
% 3.02/1.57 => ( prop2setI
% 3.02/1.57 => ( prop2set2propI
% 3.02/1.57 => ( notdexE
% 3.02/1.57 => ( notdallE
% 3.02/1.57 => ( exuI1
% 3.02/1.57 => ( exuI3
% 3.02/1.57 => ( exuI2
% 3.02/1.57 => ( inCongP
% 3.02/1.57 => ( in__Cong
% 3.02/1.57 => ( exuE3u
% 3.02/1.57 => ( exu__Cong
% 3.02/1.57 => ( emptyset__Cong
% 3.02/1.57 => ( setadjoin__Cong
% 3.02/1.57 => ( powerset__Cong
% 3.02/1.57 => ( setunion__Cong
% 3.02/1.57 => ( omega__Cong
% 3.02/1.57 => ( exuEu
% 3.02/1.57 => ( descr__Cong
% 3.02/1.57 => ( dsetconstr__Cong
% 3.02/1.57 => ( subsetI1
% 3.02/1.57 => ( eqimpsubset2
% 3.02/1.57 => ( eqimpsubset1
% 3.02/1.57 => ( subsetI2
% 3.02/1.57 => ( emptysetsubset
% 3.02/1.57 => ( subsetE
% 3.02/1.57 => ( subsetE2
% 3.02/1.57 => ( notsubsetI
% 3.02/1.57 => ( notequalI1
% 3.02/1.57 => ( notequalI2
% 3.02/1.57 => ( subsetRefl
% 3.02/1.57 => ( subsetTrans
% 3.02/1.57 => ( setadjoinSub
% 3.02/1.57 => ( setadjoinSub2
% 3.02/1.57 => ( subset2powerset
% 3.02/1.57 => ( setextsub
% 3.02/1.57 => ( subsetemptysetimpeq
% 3.02/1.57 => ( powersetI1
% 3.02/1.57 => ( powersetE1
% 3.02/1.57 => ( inPowerset
% 3.02/1.57 => ( powersetsubset
% 3.02/1.57 => ( sepInPowerset
% 3.02/1.57 => ( sepSubset
% 3.02/1.57 => ( binunionIL
% 3.02/1.57 => ( upairset2IR
% 3.02/1.57 => ( binunionIR
% 3.02/1.57 => ( binunionEcases
% 3.02/1.57 => ( binunionE
% 3.02/1.57 => ( binunionLsub
% 3.02/1.57 => ( binunionRsub
% 3.02/1.57 => ( binintersectI
% 3.02/1.57 => ( binintersectSubset5
% 3.02/1.57 => ( binintersectEL
% 3.02/1.57 => ( binintersectLsub
% 3.02/1.57 => ( binintersectSubset2
% 3.02/1.57 => ( binintersectSubset3
% 3.02/1.57 => ( binintersectER
% 3.02/1.57 => ( disjointsetsI1
% 3.02/1.57 => ( binintersectRsub
% 3.02/1.57 => ( binintersectSubset4
% 3.02/1.57 => ( binintersectSubset1
% 3.02/1.57 => ( bs114d
% 3.02/1.57 => ( setminusI
% 3.02/1.57 => ( setminusEL
% 3.02/1.57 => ( setminusER
% 3.02/1.57 => ( setminusSubset2
% 3.02/1.57 => ( setminusERneg
% 3.02/1.57 => ( setminusELneg
% 3.02/1.57 => ( setminusILneg
% 3.02/1.57 => ( setminusIRneg
% 3.02/1.57 => ( setminusLsub
% 3.02/1.57 => ( setminusSubset1
% 3.02/1.57 => ( symdiffE
% 3.02/1.57 => ( symdiffI1
% 3.02/1.57 => ( symdiffI2
% 3.02/1.57 => ( symdiffIneg1
% 3.02/1.57 => ( symdiffIneg2
% 3.02/1.57 => ( secondinupair
% 3.02/1.57 => ( setukpairIL
% 3.02/1.57 => ( setukpairIR
% 3.02/1.57 => ( kpairiskpair
% 3.02/1.57 => ( kpairp
% 3.02/1.57 => ( singletonsubset
% 3.02/1.57 => ( singletoninpowerset
% 3.02/1.57 => ( singletoninpowunion
% 3.02/1.57 => ( upairset2E
% 3.02/1.57 => ( upairsubunion
% 3.02/1.57 => ( upairinpowunion
% 3.02/1.57 => ( ubforcartprodlem1
% 3.02/1.57 => ( ubforcartprodlem2
% 3.02/1.57 => ( ubforcartprodlem3
% 3.02/1.57 => ( cartprodpairin
% 3.02/1.57 => ( cartprodmempair1
% 3.02/1.57 => ( cartprodmempair
% 3.02/1.57 => ( setunionE2
% 3.02/1.57 => ( setunionsingleton1
% 3.02/1.57 => ( setunionsingleton2
% 3.02/1.57 => ( setunionsingleton
% 3.02/1.57 => ( singletonprop
% 3.02/1.57 => ( ex1E1
% 3.02/1.57 => ( ex1I
% 3.02/1.57 => ( ex1I2
% 3.02/1.57 => ( singletonsuniq
% 3.02/1.57 => ( setukpairinjL1
% 3.02/1.57 => ( kfstsingleton
% 3.02/1.57 => ( theprop
% 3.02/1.57 => ( kfstpairEq
% 3.02/1.57 => ( cartprodfstin
% 3.02/1.57 => ( setukpairinjL2
% 3.02/1.57 => ( setukpairinjL
% 3.02/1.57 => ( setukpairinjR11
% 3.02/1.57 => ( setukpairinjR12
% 3.02/1.57 => ( setukpairinjR1
% 3.02/1.57 => ( upairequniteq
% 3.02/1.57 => ( setukpairinjR2
% 3.02/1.57 => ( setukpairinjR
% 3.02/1.57 => ( ksndsingleton
% 3.02/1.57 => ( ksndpairEq
% 3.02/1.57 => ( kpairsurjEq
% 3.02/1.57 => ( cartprodsndin
% 3.02/1.57 => ( cartprodpairmemEL
% 3.02/1.57 => ( cartprodpairmemER
% 3.02/1.57 => ( cartprodmempaircEq
% 3.02/1.57 => ( cartprodfstpairEq
% 3.02/1.57 => ( cartprodsndpairEq
% 3.02/1.57 => ( cartprodpairsurjEq
% 3.02/1.57 => ( dpsetconstrI
% 3.02/1.57 => ( dpsetconstrSub
% 3.02/1.57 => ( setOfPairsIsBReln
% 3.02/1.57 => ( dpsetconstrERa
% 3.02/1.57 => ( dpsetconstrEL1
% 3.02/1.57 => ( dpsetconstrEL2
% 3.02/1.57 => ( dpsetconstrER
% 3.02/1.57 => ( funcImageSingleton
% 3.02/1.57 => ( apProp
% 3.02/1.57 => ( app
% 3.02/1.57 => ( infuncsetfunc
% 3.02/1.57 => ( ap2p
% 3.02/1.57 => ( funcinfuncset
% 3.02/1.57 => ( lamProp
% 3.02/1.57 => ( lamp
% 3.02/1.57 => ( lam2p
% 3.02/1.57 => ( brelnall1
% 3.02/1.57 => ( brelnall2
% 3.02/1.57 => ( ex1E2
% 3.02/1.57 => ( funcGraphProp1
% 3.02/1.57 => ( funcGraphProp3
% 3.02/1.57 => ( funcGraphProp2
% 3.02/1.57 => ( funcextLem
% 3.02/1.57 => ( funcGraphProp4
% 3.02/1.57 => ( subbreln
% 3.02/1.57 => ( eqbreln
% 3.02/1.57 => ( funcext
% 3.02/1.57 => ( funcext2
% 3.02/1.57 => ( ap2apEq1
% 3.02/1.57 => ( ap2apEq2
% 3.02/1.57 => ( beta1
% 3.02/1.57 => ( eta1
% 3.02/1.57 => ( lam2lamEq
% 3.02/1.57 => ( beta2
% 3.02/1.57 => ( eta2
% 3.02/1.57 => ( iffalseProp1
% 3.02/1.57 => ( iffalseProp2
% 3.02/1.57 => ( iftrueProp1
% 3.02/1.57 => ( iftrueProp2
% 3.02/1.57 => ( ifSingleton
% 3.02/1.57 => ( ifp
% 3.02/1.57 => ( theeq
% 3.02/1.57 => ( iftrue
% 3.02/1.57 => ( iffalse
% 3.02/1.57 => ( iftrueorfalse
% 3.02/1.57 => ( binintersectT_lem
% 3.02/1.57 => ( binunionT_lem
% 3.02/1.57 => ( powersetT_lem
% 3.02/1.57 => ( setminusT_lem
% 3.02/1.57 => ( complementT_lem
% 3.02/1.57 => ( setextT
% 3.02/1.57 => ( subsetTI
% 3.02/1.57 => ( powersetTI1
% 3.02/1.57 => ( powersetTE1
% 3.02/1.57 => ( complementTI1
% 3.02/1.57 => ( complementTE1
% 3.02/1.57 => ( binintersectTELcontra
% 3.02/1.57 => ( binintersectTERcontra
% 3.02/1.57 => ( contrasubsetT
% 3.02/1.57 => ( contrasubsetT1
% 3.02/1.57 => ( contrasubsetT2
% 3.02/1.57 => ( contrasubsetT3
% 3.02/1.57 => ( doubleComplementI1
% 3.02/1.57 => ( doubleComplementE1
% 3.02/1.57 => ( doubleComplementSub1
% 3.02/1.57 => ( doubleComplementSub2
% 3.02/1.57 => ( doubleComplementEq
% 3.02/1.57 => ( complementTnotintersectT
% 3.02/1.57 => ( complementImpComplementIntersect
% 3.02/1.57 => ( complementSubsetComplementIntersect
% 3.02/1.57 => ( complementInPowersetComplementIntersect
% 3.02/1.57 => ( contraSubsetComplement
% 3.02/1.57 => ( complementTcontraSubset
% 3.02/1.57 => ( binunionTILcontra
% 3.02/1.57 => ( binunionTIRcontra
% 3.02/1.57 => ( inIntersectImpInUnion
% 3.02/1.57 => ( inIntersectImpInUnion2
% 3.02/1.57 => ( inIntersectImpInIntersectUnions
% 3.02/1.57 => ( intersectInPowersetIntersectUnions
% 3.02/1.57 => ( inComplementUnionImpNotIn1
% 3.02/1.57 => ( inComplementUnionImpInComplement1
% 3.02/1.57 => ( binunionTE
% 3.02/1.57 => ( binunionTEcontra
% 3.02/1.57 => ( demorgan2a1
% 3.02/1.57 => ( complementUnionInPowersetComplement
% 3.02/1.57 => ( demorgan2a2
% 3.02/1.57 => ( demorgan1a
% 3.02/1.57 => ( demorgan1b
% 3.02/1.57 => ( demorgan1
% 3.02/1.57 => ( demorgan2a
% 3.02/1.57 => ( demorgan2b2
% 3.02/1.57 => ( demorgan2b
% 3.02/1.57 => ( demorgan2
% 3.02/1.57 => ( woz13rule0
% 3.02/1.57 => ( woz13rule1
% 3.02/1.57 => ( woz13rule2
% 3.02/1.57 => ( woz13rule3
% 3.02/1.57 => ( woz13rule4
% 3.02/1.57 => ( woz1_1
% 3.02/1.57 => ( woz1_2
% 3.02/1.57 => ( woz1_3
% 3.02/1.57 => ( woz1_4
% 3.02/1.57 => ( woz1_5
% 3.02/1.57 => ( breln1all2
% 3.02/1.57 => ( breln1SetBreln1
% 3.02/1.57 => ( choice2fnsingleton
% 3.02/1.57 => ( setOfPairsIsBReln1
% 3.02/1.57 => ( breln1all1
% 3.02/1.57 => ( subbreln1
% 3.02/1.57 => ( eqbreln1
% 3.02/1.57 => ( breln1invprop
% 3.02/1.57 => ( breln1invI
% 3.02/1.57 => ( breln1invE
% 3.02/1.57 => ( breln1compprop
% 3.02/1.57 => ( breln1compI
% 3.02/1.57 => ( breln1compE
% 3.02/1.57 => ( breln1compEex
% 3.02/1.57 => ( breln1unionprop
% 3.02/1.57 => ( breln1unionIL
% 3.02/1.57 => ( breln1unionIR
% 3.02/1.57 => ( breln1unionI
% 3.02/1.57 => ( breln1unionE
% 3.02/1.57 => ( breln1unionEcases
% 3.02/1.57 => ( breln1unionCommutes
% 3.02/1.57 => ( woz2Ex
% 3.02/1.57 => ( woz2W
% 3.02/1.57 => ( woz2A
% 3.02/1.57 => ( woz2B
% 3.02/1.57 => ( image1Ex
% 3.02/1.57 => ( image1Ex1
% 3.02/1.57 => ( image1Equiv
% 3.02/1.57 => ( image1E
% 3.02/1.57 => ( image1I
% 3.02/1.57 => ( injFuncInInjFuncSet
% 3.02/1.57 => ( injFuncSetFuncIn
% 3.02/1.57 => ( injFuncSetFuncInj
% 3.02/1.57 => ( surjFuncSetFuncIn
% 3.02/1.57 => ( surjFuncSetFuncSurj
% 3.02/1.57 => ( leftInvIsSurj
% 3.02/1.57 => ( surjCantorThm
% 3.02/1.57 => ( foundation2
% 3.02/1.57 => ( notinself
% 3.02/1.57 => ( notinself2
% 3.02/1.57 => ( omegaSp
% 3.02/1.57 => ( omegaSclos
% 3.02/1.57 => ( peano0notS
% 3.02/1.57 => ( peanoSinj
% 3.02/1.57 => ( transitivesetOp1
% 3.02/1.57 => ( binintTransitive
% 3.02/1.57 => ( transitivesetOp2
% 3.02/1.57 => ( setunionTransitive
% 3.02/1.57 => ( ordinalMinLem1
% 3.02/1.57 => ( ordinalTransSet
% 3.02/1.57 => ( ordinalTransSet1
% 3.02/1.57 => ( setunionOrdinalLem1
% 3.02/1.57 => ( emptysetOrdinal
% 3.02/1.57 => ! [X: $i] :
% 3.02/1.57 ( ( ordinal @ X )
% 3.02/1.57 => ! [A: $i] :
% 3.02/1.57 ( ( in @ A @ X )
% 3.02/1.57 => ~ ( in @ A @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.57 ------------------------
% 3.09/1.60 % (5150)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_5045 for (2999ds/275Mi)
% 3.09/1.60 % (5151)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/18Mi)
% 3.09/1.60 % (5149)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_5045 for (2999ds/2Mi)
% 3.09/1.60 % (5146)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_5045 for (2999ds/4Mi)
% 3.09/1.60 % (5147)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_5045 for (2999ds/27Mi)
% 3.09/1.60 % (5149)Instruction limit reached!
% 3.09/1.60 % (5149)------------------------------
% 3.09/1.60 % (5149)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.60 % (5149)Termination reason: Unknown
% 3.09/1.60 % (5149)Termination phase: shuffling
% 3.09/1.60
% 3.09/1.60 % (5149)Memory used [KB]: 1663
% 3.09/1.60 % (5149)Time elapsed: 0.003 s
% 3.09/1.60 % (5149)Instructions burned: 3 (million)
% 3.09/1.60 % (5149)------------------------------
% 3.09/1.60 % (5149)------------------------------
% 3.09/1.60 % (5146)Instruction limit reached!
% 3.09/1.60 % (5146)------------------------------
% 3.09/1.60 % (5146)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.60 % (5146)Termination reason: Unknown
% 3.09/1.60 % (5146)Termination phase: shuffling
% 3.09/1.60
% 3.09/1.60 % (5146)Memory used [KB]: 1791
% 3.09/1.60 % (5146)Time elapsed: 0.003 s
% 3.09/1.60 % (5146)Instructions burned: 5 (million)
% 3.09/1.60 % (5146)------------------------------
% 3.09/1.60 % (5146)------------------------------
% 3.09/1.61 % (5148)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/2Mi)
% 3.09/1.61 % (5145)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/183Mi)
% 3.09/1.61 % (5148)Instruction limit reached!
% 3.09/1.61 % (5148)------------------------------
% 3.09/1.61 % (5148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.61 % (5148)Termination reason: Unknown
% 3.09/1.61 % (5148)Termination phase: shuffling
% 3.09/1.61
% 3.09/1.61 % (5148)Memory used [KB]: 1663
% 3.09/1.61 % (5148)Time elapsed: 0.004 s
% 3.09/1.61 % (5148)Instructions burned: 2 (million)
% 3.09/1.61 % (5148)------------------------------
% 3.09/1.61 % (5148)------------------------------
% 3.09/1.61 % (5147)Instruction limit reached!
% 3.09/1.61 % (5147)------------------------------
% 3.09/1.61 % (5147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.61 % (5147)Termination reason: Unknown
% 3.09/1.61 % (5147)Termination phase: shuffling
% 3.09/1.61
% 3.09/1.61 % (5147)Memory used [KB]: 2174
% 3.09/1.61 % (5147)Time elapsed: 0.011 s
% 3.09/1.61 % (5147)Instructions burned: 30 (million)
% 3.09/1.61 % (5147)------------------------------
% 3.09/1.61 % (5147)------------------------------
% 3.09/1.61 % (5152)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/3Mi)
% 3.09/1.61 % (5153)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_5045 for (2999ds/37Mi)
% 3.09/1.61 % (5152)Instruction limit reached!
% 3.09/1.61 % (5152)------------------------------
% 3.09/1.61 % (5152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.61 % (5152)Termination reason: Unknown
% 3.09/1.61 % (5152)Termination phase: shuffling
% 3.09/1.61
% 3.09/1.61 % (5152)Memory used [KB]: 1791
% 3.09/1.61 % (5152)Time elapsed: 0.003 s
% 3.09/1.61 % (5152)Instructions burned: 5 (million)
% 3.09/1.61 % (5152)------------------------------
% 3.09/1.61 % (5152)------------------------------
% 3.09/1.62 % (5151)Instruction limit reached!
% 3.09/1.62 % (5151)------------------------------
% 3.09/1.62 % (5151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.62 % (5151)Termination reason: Unknown
% 3.09/1.62 % (5151)Termination phase: shuffling
% 3.09/1.62
% 3.09/1.62 % (5151)Memory used [KB]: 2046
% 3.09/1.62 % (5151)Time elapsed: 0.008 s
% 3.09/1.62 % (5151)Instructions burned: 18 (million)
% 3.09/1.62 % (5151)------------------------------
% 3.09/1.62 % (5151)------------------------------
% 3.09/1.62 % (5155)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_5045 for (2999ds/3Mi)
% 3.09/1.62 % (5155)Instruction limit reached!
% 3.09/1.62 % (5155)------------------------------
% 3.09/1.62 % (5155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.62 % (5155)Termination reason: Unknown
% 3.09/1.62 % (5155)Termination phase: shuffling
% 3.09/1.62
% 3.09/1.62 % (5155)Memory used [KB]: 1663
% 3.09/1.62 % (5155)Time elapsed: 0.003 s
% 3.09/1.62 % (5155)Instructions burned: 4 (million)
% 3.09/1.62 % (5155)------------------------------
% 3.09/1.62 % (5155)------------------------------
% 3.09/1.62 % (5153)Instruction limit reached!
% 3.09/1.62 % (5153)------------------------------
% 3.09/1.62 % (5153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.62 % (5153)Termination reason: Unknown
% 3.09/1.62 % (5153)Termination phase: shuffling
% 3.09/1.62
% 3.09/1.62 % (5153)Memory used [KB]: 2430
% 3.09/1.62 % (5153)Time elapsed: 0.012 s
% 3.09/1.62 % (5153)Instructions burned: 39 (million)
% 3.09/1.62 % (5153)------------------------------
% 3.09/1.62 % (5153)------------------------------
% 3.09/1.62 % (5156)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_5045 for (2999ds/1041Mi)
% 3.09/1.63 % (5154)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_5045 for (2999ds/15Mi)
% 3.09/1.63 % (5157)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/7Mi)
% 3.09/1.63 % (5158)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/16Mi)
% 3.09/1.63 % (5157)Instruction limit reached!
% 3.09/1.63 % (5157)------------------------------
% 3.09/1.63 % (5157)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.63 % (5157)Termination reason: Unknown
% 3.09/1.63 % (5157)Termination phase: shuffling
% 3.09/1.63
% 3.09/1.63 % (5157)Memory used [KB]: 1791
% 3.09/1.63 % (5157)Time elapsed: 0.004 s
% 3.09/1.63 % (5157)Instructions burned: 9 (million)
% 3.09/1.63 % (5157)------------------------------
% 3.09/1.63 % (5157)------------------------------
% 3.09/1.63 % (5159)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_5045 for (2999ds/3Mi)
% 3.09/1.63 % (5159)Instruction limit reached!
% 3.09/1.63 % (5159)------------------------------
% 3.09/1.63 % (5159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.63 % (5159)Termination reason: Unknown
% 3.09/1.63 % (5159)Termination phase: shuffling
% 3.09/1.63
% 3.09/1.63 % (5159)Memory used [KB]: 1791
% 3.09/1.63 % (5159)Time elapsed: 0.003 s
% 3.09/1.63 % (5159)Instructions burned: 6 (million)
% 3.09/1.63 % (5159)------------------------------
% 3.09/1.63 % (5159)------------------------------
% 3.09/1.64 % (5158)Instruction limit reached!
% 3.09/1.64 % (5158)------------------------------
% 3.09/1.64 % (5158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.64 % (5158)Termination reason: Unknown
% 3.09/1.64 % (5158)Termination phase: shuffling
% 3.09/1.64
% 3.09/1.64 % (5158)Memory used [KB]: 1918
% 3.09/1.64 % (5158)Time elapsed: 0.006 s
% 3.09/1.64 % (5158)Instructions burned: 17 (million)
% 3.09/1.64 % (5158)------------------------------
% 3.09/1.64 % (5158)------------------------------
% 3.09/1.64 % (5160)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_5045 for (2999ds/3Mi)
% 3.09/1.64 % (5160)Instruction limit reached!
% 3.09/1.64 % (5160)------------------------------
% 3.09/1.64 % (5160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.64 % (5160)Termination reason: Unknown
% 3.09/1.64 % (5160)Termination phase: shuffling
% 3.09/1.64
% 3.09/1.64 % (5160)Memory used [KB]: 1663
% 3.09/1.64 % (5160)Time elapsed: 0.003 s
% 3.09/1.64 % (5160)Instructions burned: 5 (million)
% 3.09/1.64 % (5160)------------------------------
% 3.09/1.64 % (5160)------------------------------
% 3.09/1.64 % (5161)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/7Mi)
% 3.09/1.65 % (5162)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/3Mi)
% 3.09/1.65 % (5161)Instruction limit reached!
% 3.09/1.65 % (5161)------------------------------
% 3.09/1.65 % (5161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.65 % (5161)Termination reason: Unknown
% 3.09/1.65 % (5161)Termination phase: shuffling
% 3.09/1.65
% 3.09/1.65 % (5161)Memory used [KB]: 1918
% 3.09/1.65 % (5161)Time elapsed: 0.004 s
% 3.09/1.65 % (5161)Instructions burned: 9 (million)
% 3.09/1.65 % (5161)------------------------------
% 3.09/1.65 % (5161)------------------------------
% 3.09/1.65 % (5162)Instruction limit reached!
% 3.09/1.65 % (5162)------------------------------
% 3.09/1.65 % (5162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.65 % (5162)Termination reason: Unknown
% 3.09/1.65 % (5162)Termination phase: shuffling
% 3.09/1.65
% 3.09/1.65 % (5162)Memory used [KB]: 1791
% 3.09/1.65 % (5162)Time elapsed: 0.003 s
% 3.09/1.65 % (5162)Instructions burned: 6 (million)
% 3.09/1.65 % (5162)------------------------------
% 3.09/1.65 % (5162)------------------------------
% 3.09/1.65 % (5154)Instruction limit reached!
% 3.09/1.65 % (5154)------------------------------
% 3.09/1.65 % (5154)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.65 % (5154)Termination reason: Unknown
% 3.09/1.65 % (5154)Termination phase: shuffling
% 3.09/1.65
% 3.09/1.65 % (5154)Memory used [KB]: 1918
% 3.09/1.65 % (5154)Time elapsed: 0.013 s
% 3.09/1.65 % (5154)Instructions burned: 16 (million)
% 3.09/1.65 % (5154)------------------------------
% 3.09/1.65 % (5154)------------------------------
% 3.09/1.65 % (5163)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/4Mi)
% 3.09/1.66 % (5163)Instruction limit reached!
% 3.09/1.66 % (5163)------------------------------
% 3.09/1.66 % (5163)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.66 % (5163)Termination reason: Unknown
% 3.09/1.66 % (5163)Termination phase: shuffling
% 3.09/1.66
% 3.09/1.66 % (5163)Memory used [KB]: 1791
% 3.09/1.66 % (5163)Time elapsed: 0.003 s
% 3.09/1.66 % (5163)Instructions burned: 5 (million)
% 3.09/1.66 % (5163)------------------------------
% 3.09/1.66 % (5163)------------------------------
% 3.09/1.66 % (5164)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/18Mi)
% 3.09/1.66 % (5165)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/710Mi)
% 3.09/1.66 % (5164)Instruction limit reached!
% 3.09/1.66 % (5164)------------------------------
% 3.09/1.66 % (5164)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.66 % (5164)Termination reason: Unknown
% 3.09/1.66 % (5164)Termination phase: shuffling
% 3.09/1.66
% 3.09/1.66 % (5164)Memory used [KB]: 2046
% 3.09/1.66 % (5164)Time elapsed: 0.007 s
% 3.09/1.66 % (5164)Instructions burned: 20 (million)
% 3.09/1.66 % (5164)------------------------------
% 3.09/1.66 % (5164)------------------------------
% 3.09/1.67 % (5167)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_5045 for (2999ds/902Mi)
% 3.09/1.67 % (5166)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_5045 for (2999ds/6Mi)
% 3.09/1.67 % (5166)Instruction limit reached!
% 3.09/1.67 % (5166)------------------------------
% 3.09/1.67 % (5166)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.67 % (5166)Termination reason: Unknown
% 3.09/1.67 % (5166)Termination phase: shuffling
% 3.09/1.67
% 3.09/1.67 % (5166)Memory used [KB]: 1791
% 3.09/1.67 % (5166)Time elapsed: 0.006 s
% 3.09/1.67 % (5166)Instructions burned: 6 (million)
% 3.09/1.67 % (5166)------------------------------
% 3.09/1.67 % (5166)------------------------------
% 3.09/1.67 % (5168)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_5045 for (2999ds/21Mi)
% 3.09/1.67 % (5145)Instruction limit reached!
% 3.09/1.67 % (5145)------------------------------
% 3.09/1.67 % (5145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.67 % (5145)Termination reason: Unknown
% 3.09/1.67 % (5145)Termination phase: Property scanning
% 3.09/1.67
% 3.09/1.67 % (5145)Memory used [KB]: 3198
% 3.09/1.67 % (5145)Time elapsed: 0.049 s
% 3.09/1.67 % (5145)Instructions burned: 183 (million)
% 3.09/1.67 % (5145)------------------------------
% 3.09/1.67 % (5145)------------------------------
% 3.09/1.68 % (5168)Instruction limit reached!
% 3.09/1.68 % (5168)------------------------------
% 3.09/1.68 % (5168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.68 % (5168)Termination reason: Unknown
% 3.09/1.68 % (5168)Termination phase: shuffling
% 3.09/1.68
% 3.09/1.68 % (5168)Memory used [KB]: 2046
% 3.09/1.68 % (5168)Time elapsed: 0.008 s
% 3.09/1.68 % (5168)Instructions burned: 23 (million)
% 3.09/1.68 % (5168)------------------------------
% 3.09/1.68 % (5168)------------------------------
% 3.09/1.68 % (5170)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_5045 for (2998ds/6Mi)
% 3.09/1.69 % (5170)Instruction limit reached!
% 3.09/1.69 % (5170)------------------------------
% 3.09/1.69 % (5170)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.69 % (5170)Termination reason: Unknown
% 3.09/1.69 % (5170)Termination phase: shuffling
% 3.09/1.69
% 3.09/1.69 % (5170)Memory used [KB]: 1791
% 3.09/1.69 % (5170)Time elapsed: 0.004 s
% 3.09/1.69 % (5170)Instructions burned: 9 (million)
% 3.09/1.69 % (5170)------------------------------
% 3.09/1.69 % (5170)------------------------------
% 3.09/1.69 % (5171)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_5045 for (2998ds/377Mi)
% 3.09/1.69 % (5169)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_5045 for (2998ds/5Mi)
% 3.09/1.69 % (5169)Instruction limit reached!
% 3.09/1.69 % (5169)------------------------------
% 3.09/1.69 % (5169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.69 % (5169)Termination reason: Unknown
% 3.09/1.69 % (5169)Termination phase: shuffling
% 3.09/1.69
% 3.09/1.69 % (5169)Memory used [KB]: 1791
% 3.09/1.69 % (5169)Time elapsed: 0.004 s
% 3.09/1.69 % (5169)Instructions burned: 5 (million)
% 3.09/1.69 % (5169)------------------------------
% 3.09/1.69 % (5169)------------------------------
% 3.09/1.70 % (5172)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_5045 for (2998ds/779Mi)
% 3.09/1.70 % (5173)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_5045 for (2998ds/19Mi)
% 3.09/1.71 % (5173)Instruction limit reached!
% 3.09/1.71 % (5173)------------------------------
% 3.09/1.71 % (5173)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.71 % (5173)Termination reason: Unknown
% 3.09/1.71 % (5173)Termination phase: shuffling
% 3.09/1.71
% 3.09/1.71 % (5173)Memory used [KB]: 2046
% 3.09/1.71 % (5173)Time elapsed: 0.008 s
% 3.09/1.71 % (5173)Instructions burned: 20 (million)
% 3.09/1.71 % (5173)------------------------------
% 3.09/1.71 % (5173)------------------------------
% 3.09/1.72 % (5174)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_5045 for (2998ds/879Mi)
% 3.09/1.73 % (5150)Instruction limit reached!
% 3.09/1.73 % (5150)------------------------------
% 3.09/1.73 % (5150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.73 % (5150)Termination reason: Unknown
% 3.09/1.73 % (5150)Termination phase: Property scanning
% 3.09/1.73
% 3.09/1.73 % (5150)Memory used [KB]: 6012
% 3.09/1.73 % (5150)Time elapsed: 0.124 s
% 3.09/1.73 % (5150)Instructions burned: 275 (million)
% 3.09/1.73 % (5150)------------------------------
% 3.09/1.73 % (5150)------------------------------
% 3.09/1.75 % (5175)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_5045 for (2998ds/17Mi)
% 3.09/1.75 % (5175)Instruction limit reached!
% 3.09/1.75 % (5175)------------------------------
% 3.09/1.75 % (5175)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.75 % (5175)Termination reason: Unknown
% 3.09/1.75 % (5175)Termination phase: shuffling
% 3.09/1.75
% 3.09/1.75 % (5175)Memory used [KB]: 1918
% 3.09/1.75 % (5175)Time elapsed: 0.011 s
% 3.09/1.75 % (5175)Instructions burned: 17 (million)
% 3.09/1.75 % (5175)------------------------------
% 3.09/1.75 % (5175)------------------------------
% 3.09/1.77 % (5176)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_5045 for (2998ds/3Mi)
% 3.09/1.78 % (5176)Instruction limit reached!
% 3.09/1.78 % (5176)------------------------------
% 3.09/1.78 % (5176)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.78 % (5176)Termination reason: Unknown
% 3.09/1.78 % (5176)Termination phase: shuffling
% 3.09/1.78
% 3.09/1.78 % (5176)Memory used [KB]: 1663
% 3.09/1.78 % (5176)Time elapsed: 0.004 s
% 3.09/1.78 % (5176)Instructions burned: 4 (million)
% 3.09/1.78 % (5176)------------------------------
% 3.09/1.78 % (5176)------------------------------
% 3.09/1.79 % (5177)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_5045 for (2997ds/30Mi)
% 3.09/1.80 % (5177)Instruction limit reached!
% 3.09/1.80 % (5177)------------------------------
% 3.09/1.80 % (5177)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.80 % (5177)Termination reason: Unknown
% 3.09/1.80 % (5177)Termination phase: shuffling
% 3.09/1.80
% 3.09/1.80 % (5177)Memory used [KB]: 2430
% 3.09/1.80 % (5177)Time elapsed: 0.011 s
% 3.09/1.80 % (5177)Instructions burned: 32 (million)
% 3.09/1.80 % (5177)------------------------------
% 3.09/1.80 % (5177)------------------------------
% 3.09/1.81 % (5165)First to succeed.
% 3.09/1.81 % (5178)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_5045 for (2997ds/127Mi)
% 3.09/1.81 % (5165)Refutation found. Thanks to Tanya!
% 3.09/1.81 % SZS status Theorem for DTF2THF_5045
% 3.09/1.81 % SZS output start Proof for DTF2THF_5045
% 3.09/1.81 thf(func_def_0, type, in: $i > $i > $o).
% 3.09/1.81 thf(func_def_1, type, exu: ($i > $o) > $o).
% 3.09/1.81 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 3.09/1.81 thf(func_def_8, type, powerset: $i > $i).
% 3.09/1.81 thf(func_def_10, type, setunion: $i > $i).
% 3.09/1.81 thf(func_def_19, type, descr: ($i > $o) > $i).
% 3.09/1.81 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_26, type, prop2set: $o > $i).
% 3.09/1.81 thf(func_def_36, type, nonempty: $i > $o).
% 3.09/1.81 thf(func_def_69, type, set2prop: $i > $o).
% 3.09/1.81 thf(func_def_88, type, subset: $i > $i > $o).
% 3.09/1.81 thf(func_def_89, type, disjoint: $i > $i > $o).
% 3.09/1.81 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 3.09/1.81 thf(func_def_114, type, binunion: $i > $i > $i).
% 3.09/1.81 thf(func_def_122, type, binintersect: $i > $i > $i).
% 3.09/1.81 thf(func_def_135, type, regular: $i > $o).
% 3.09/1.81 thf(func_def_136, type, setminus: $i > $i > $i).
% 3.09/1.81 thf(func_def_147, type, symdiff: $i > $i > $i).
% 3.09/1.81 thf(func_def_153, type, iskpair: $i > $o).
% 3.09/1.81 thf(func_def_158, type, kpair: $i > $i > $i).
% 3.09/1.81 thf(func_def_160, type, cartprod: $i > $i > $i).
% 3.09/1.81 thf(func_def_177, type, singleton: $i > $o).
% 3.09/1.81 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 3.09/1.81 thf(func_def_184, type, atmost1p: $i > $o).
% 3.09/1.81 thf(func_def_185, type, atleast2p: $i > $o).
% 3.09/1.81 thf(func_def_186, type, atmost2p: $i > $o).
% 3.09/1.81 thf(func_def_187, type, upairsetp: $i > $o).
% 3.09/1.81 thf(func_def_191, type, kfst: $i > $i).
% 3.09/1.81 thf(func_def_203, type, ksnd: $i > $i).
% 3.09/1.81 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 3.09/1.81 thf(func_def_222, type, func: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_223, type, funcSet: $i > $i > $i).
% 3.09/1.81 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 3.09/1.81 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 3.09/1.81 thf(func_def_323, type, breln1: $i > $i > $o).
% 3.09/1.81 thf(func_def_325, type, breln1Set: $i > $i).
% 3.09/1.81 thf(func_def_327, type, transitive: $i > $i > $o).
% 3.09/1.81 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 3.09/1.81 thf(func_def_329, type, reflexive: $i > $i > $o).
% 3.09/1.81 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 3.09/1.81 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 3.09/1.81 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 3.09/1.81 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 3.09/1.81 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 3.09/1.81 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 3.09/1.81 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 3.09/1.81 thf(func_def_378, type, omegaS: $i > $i).
% 3.09/1.81 thf(func_def_382, type, peano3: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_383, type, peano4: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_384, type, peano5: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_386, type, peano: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_387, type, transitiveset: $i > $o).
% 3.09/1.81 thf(func_def_392, type, stricttotalorderedByIn: $i > $o).
% 3.09/1.81 thf(func_def_393, type, wellorderedByIn: $i > $o).
% 3.09/1.81 thf(func_def_394, type, ordinal: $i > $o).
% 3.09/1.81 thf(func_def_395, type, limitOrdinal: $i > $o).
% 3.09/1.81 thf(func_def_400, type, succOrdinal: $i > $o).
% 3.09/1.81 thf(func_def_421, type, sP4: $i > $i > $o).
% 3.09/1.81 thf(func_def_422, type, sP5: $i > $o).
% 3.09/1.81 thf(func_def_423, type, sP6: $i > $i > $o).
% 3.09/1.81 thf(func_def_424, type, sP7: $i > $i > $o).
% 3.09/1.81 thf(func_def_425, type, sP8: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_428, type, sP11: $i > $i > $i > $o > $o).
% 3.09/1.81 thf(func_def_430, type, sP13: $i > $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_434, type, sP17: $i > $i > $i > $o).
% 3.09/1.81 thf(func_def_439, type, sK22: $i > $i > $i).
% 3.09/1.81 thf(func_def_456, type, sK39: $i > $o).
% 3.09/1.81 thf(func_def_471, type, sK54: $i > $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_473, type, sK56: $i > $i).
% 3.09/1.81 thf(func_def_474, type, sK57: $i > $i).
% 3.09/1.81 thf(func_def_475, type, sK58: $i > $i).
% 3.09/1.81 thf(func_def_476, type, sK59: ($i > $i) > $i > $i).
% 3.09/1.81 thf(func_def_477, type, sK60: $i > ($i > $i) > $i > $i).
% 3.09/1.81 thf(func_def_478, type, sK61: ($i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_480, type, sK63: $i > $o).
% 3.09/1.81 thf(func_def_499, type, sK82: $i > $i).
% 3.09/1.81 thf(func_def_501, type, sK84: ($i > $o) > $i).
% 3.09/1.81 thf(func_def_502, type, sK85: $i > $o).
% 3.09/1.81 thf(func_def_508, type, sK91: $i > $i > $o).
% 3.09/1.81 thf(func_def_512, type, sK95: $i > $i > $o).
% 3.09/1.81 thf(func_def_515, type, sK98: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_538, type, sK121: $i > $o).
% 3.09/1.81 thf(func_def_539, type, sK122: ($i > $o) > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_540, type, sK123: ($i > $o) > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_544, type, sK127: $i > $o).
% 3.09/1.81 thf(func_def_569, type, sK152: $i > $o).
% 3.09/1.81 thf(func_def_570, type, sK153: ($i > $i) > $i > $i > $i).
% 3.09/1.81 thf(func_def_573, type, sK156: $i > $i).
% 3.09/1.81 thf(func_def_581, type, sK164: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_600, type, sK183: $i > $o).
% 3.09/1.81 thf(func_def_614, type, sK197: $i > $i).
% 3.09/1.81 thf(func_def_617, type, sK200: $i > $o).
% 3.09/1.81 thf(func_def_620, type, sK203: ($i > $o) > $i).
% 3.09/1.81 thf(func_def_621, type, sK204: $i > $o).
% 3.09/1.81 thf(func_def_622, type, sK205: $i > $i).
% 3.09/1.81 thf(func_def_646, type, sK229: $i > $o).
% 3.09/1.81 thf(func_def_652, type, sK235: $i > $o).
% 3.09/1.81 thf(func_def_671, type, sK254: $i > $o).
% 3.09/1.81 thf(func_def_672, type, sK255: $i > $o).
% 3.09/1.81 thf(func_def_673, type, sK256: ($i > $o) > ($i > $o) > $i > $i > $i).
% 3.09/1.81 thf(func_def_674, type, sK257: ($i > $o) > ($i > $o) > $i > $i > $i).
% 3.09/1.81 thf(func_def_693, type, sK276: $o > $i > $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_697, type, sK280: $i > $o).
% 3.09/1.81 thf(func_def_700, type, sK283: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_701, type, sK284: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_702, type, sK285: $i > $i).
% 3.09/1.81 thf(func_def_708, type, sK291: $i > $o).
% 3.09/1.81 thf(func_def_714, type, sK297: $i > $i > $i).
% 3.09/1.81 thf(func_def_715, type, sK298: $i > $i).
% 3.09/1.81 thf(func_def_718, type, sK301: $i > $i > ($i > $i) > $i).
% 3.09/1.81 thf(func_def_730, type, sK313: $i > $i > $o).
% 3.09/1.81 thf(func_def_749, type, sK332: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_750, type, sK333: $i > $o).
% 3.09/1.81 thf(func_def_752, type, sK335: $i > $i > $i).
% 3.09/1.81 thf(func_def_775, type, sK358: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_776, type, sK359: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_781, type, sK364: $i > $o).
% 3.09/1.81 thf(func_def_784, type, sK367: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_814, type, sK397: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_818, type, sK401: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_819, type, sK402: $i > $o).
% 3.09/1.81 thf(func_def_826, type, sK409: $i > $i > $i).
% 3.09/1.81 thf(func_def_827, type, sK410: $i > $i > $i).
% 3.09/1.81 thf(func_def_828, type, sK411: $i > $i > $i).
% 3.09/1.81 thf(func_def_829, type, sK412: $i > $i > $i).
% 3.09/1.81 thf(func_def_830, type, sK413: $i > $i > $i).
% 3.09/1.81 thf(func_def_831, type, sK414: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_832, type, sK415: $i > $i).
% 3.09/1.81 thf(func_def_833, type, sK416: $i > $i).
% 3.09/1.81 thf(func_def_834, type, sK417: $i > $i).
% 3.09/1.81 thf(func_def_835, type, sK418: $i > $i).
% 3.09/1.81 thf(func_def_836, type, sK419: $i > $i > $i).
% 3.09/1.81 thf(func_def_837, type, sK420: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_838, type, sK421: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_839, type, sK422: $i > $i > $i).
% 3.09/1.81 thf(func_def_840, type, sK423: $i > $i > $i).
% 3.09/1.81 thf(func_def_841, type, sK424: $i > $i).
% 3.09/1.81 thf(func_def_843, type, sK426: $i > $i).
% 3.09/1.81 thf(func_def_844, type, sK427: $i > $i).
% 3.09/1.81 thf(func_def_852, type, sK435: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_853, type, sK436: $i > $o).
% 3.09/1.81 thf(func_def_860, type, sK443: $i > $i > $o).
% 3.09/1.81 thf(func_def_867, type, sK450: $i > ($i > $i) > $i > $i).
% 3.09/1.81 thf(func_def_869, type, sK452: $i > $i).
% 3.09/1.81 thf(func_def_873, type, sK456: $i > $i > $i).
% 3.09/1.81 thf(func_def_888, type, sK471: $i > $i > $i).
% 3.09/1.81 thf(func_def_893, type, sK476: $i > $i > ($i > $i) > $i).
% 3.09/1.81 thf(func_def_894, type, sK477: $i > $i).
% 3.09/1.81 thf(func_def_898, type, sK481: $i > $i > ($i > $i) > $i).
% 3.09/1.81 thf(func_def_899, type, sK482: $i > $i).
% 3.09/1.81 thf(func_def_910, type, sK493: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_911, type, sK494: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_912, type, sK495: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_913, type, sK496: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_947, type, sK530: $i > $i).
% 3.09/1.81 thf(func_def_951, type, sK534: ($i > $o) > $i > $i > $i).
% 3.09/1.81 thf(func_def_952, type, sK535: ($i > $o) > $i > $i > $i).
% 3.09/1.81 thf(func_def_955, type, sK538: $i > $o).
% 3.09/1.81 thf(func_def_958, type, sK541: $i > $i > $o).
% 3.09/1.81 thf(func_def_959, type, sK542: $i > $i).
% 3.09/1.81 thf(func_def_960, type, sK543: $i > $i).
% 3.09/1.81 thf(func_def_961, type, sK544: ($i > $i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_962, type, sK545: ($i > $i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_963, type, sK546: $i > ($i > $i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_965, type, sK548: ($i > $o) > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_966, type, sK549: ($i > $o) > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_970, type, sK553: $i > $o).
% 3.09/1.81 thf(func_def_974, type, sK557: $i > $i).
% 3.09/1.81 thf(func_def_990, type, sK573: ($i > $o) > $i > $i > $i).
% 3.09/1.81 thf(func_def_991, type, sK574: ($i > $o) > $i > $i > $i).
% 3.09/1.81 thf(func_def_994, type, sK577: $i > $o).
% 3.09/1.81 thf(func_def_997, type, sK580: $i > $i).
% 3.09/1.81 thf(func_def_1015, type, sK598: $i > $i > $o).
% 3.09/1.81 thf(func_def_1021, type, sK604: $i > $i > $o).
% 3.09/1.81 thf(func_def_1024, type, sK607: $i > $i).
% 3.09/1.81 thf(func_def_1026, type, sK609: $i > $i).
% 3.09/1.81 thf(func_def_1037, type, sK620: $o > $i > $i > $i).
% 3.09/1.81 thf(func_def_1048, type, sK631: $i > $i).
% 3.09/1.81 thf(func_def_1051, type, sK634: $i > ($i > $i) > $i > $i).
% 3.09/1.81 thf(func_def_1052, type, sK635: $i > $o).
% 3.09/1.81 thf(func_def_1053, type, sK636: $i > $i).
% 3.09/1.81 thf(func_def_1054, type, sK637: ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1068, type, sK651: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1069, type, sK652: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1080, type, sK663: $i > $o).
% 3.09/1.81 thf(func_def_1088, type, sK671: $i > $i > $o).
% 3.09/1.81 thf(func_def_1093, type, sK676: $i > $o).
% 3.09/1.81 thf(func_def_1094, type, sK677: ($i > $o) > $i > $i > $i).
% 3.09/1.81 thf(func_def_1118, type, sK701: $i > $i).
% 3.09/1.81 thf(func_def_1125, type, sK708: $i > $i > $i).
% 3.09/1.81 thf(func_def_1141, type, sK724: $i > $i > $o).
% 3.09/1.81 thf(func_def_1164, type, sK747: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1165, type, sK748: $i > $o).
% 3.09/1.81 thf(func_def_1177, type, sK760: $i > $i).
% 3.09/1.81 thf(func_def_1183, type, sK766: $i > $i > $i).
% 3.09/1.81 thf(func_def_1184, type, sK767: $i > $i > $i).
% 3.09/1.81 thf(func_def_1187, type, sK770: $i > ($i > $i) > $i > $i).
% 3.09/1.81 thf(func_def_1189, type, sK772: $i > $i).
% 3.09/1.81 thf(func_def_1203, type, sK786: ($i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_1205, type, sK788: $i > $o).
% 3.09/1.81 thf(func_def_1216, type, sK799: $i > ($i > $i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_1218, type, sK801: $i > $i > $o).
% 3.09/1.81 thf(func_def_1222, type, sK805: $i > $i).
% 3.09/1.81 thf(func_def_1228, type, sK811: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1242, type, sK825: $i > $o).
% 3.09/1.81 thf(func_def_1264, type, sK847: $i > $o).
% 3.09/1.81 thf(func_def_1319, type, sK902: $i > $o).
% 3.09/1.81 thf(func_def_1322, type, sK905: $i > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1329, type, sK912: $i > $i).
% 3.09/1.81 thf(func_def_1333, type, sK916: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1334, type, sK917: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1342, type, sK925: $i > $o).
% 3.09/1.81 thf(func_def_1349, type, sK932: $i > $i).
% 3.09/1.81 thf(func_def_1352, type, sK935: $i > ($i > $i) > $i > $i).
% 3.09/1.81 thf(func_def_1359, type, sK942: $i > $i).
% 3.09/1.81 thf(func_def_1378, type, sK961: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1379, type, sK962: $i > $o).
% 3.09/1.81 thf(func_def_1380, type, sK963: $i > $o).
% 3.09/1.81 thf(func_def_1381, type, sK964: ($i > $o) > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1382, type, sK965: ($i > $o) > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1396, type, sK979: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1397, type, sK980: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1398, type, sK981: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1399, type, sK982: $i > $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1414, type, sK997: $i > $o).
% 3.09/1.81 thf(func_def_1416, type, sK999: ($i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_1417, type, sK1000: ($i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_1451, type, sK1034: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1452, type, sK1035: $i > $i > $i > $i).
% 3.09/1.81 thf(func_def_1471, type, sK1054: $i > $i > $i).
% 3.09/1.81 thf(func_def_1474, type, sK1057: $i > $o).
% 3.09/1.81 thf(func_def_1475, type, sK1058: $i > $o).
% 3.09/1.81 thf(func_def_1476, type, sK1059: ($i > $o) > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1477, type, sK1060: ($i > $o) > ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1490, type, sK1073: $i > $o).
% 3.09/1.81 thf(func_def_1493, type, sK1076: ($i > $o) > $i > $i).
% 3.09/1.81 thf(func_def_1495, type, sK1078: $i > $o).
% 3.09/1.81 thf(func_def_1500, type, sK1083: $i > $i).
% 3.09/1.81 thf(func_def_1502, type, sK1085: $i > ($i > $i) > $i > $i > $i).
% 3.09/1.81 thf(func_def_1503, type, sK1086: ($i > $i) > $i > $i > $i).
% 3.09/1.81 thf(func_def_1515, type, sK1098: $i > $i > $i).
% 3.09/1.81 thf(func_def_1525, type, sK1108: ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1526, type, sK1109: ($i > $o) > $i).
% 3.09/1.81 thf(func_def_1527, type, sK1110: $i > $o).
% 3.09/1.81 thf(f7295,plain,(
% 3.09/1.81 $false),
% 3.09/1.81 inference(subsumption_resolution,[status(thm)],[f4827,f6763])).
% 3.09/1.81 thf(f6763,plain,(
% 3.09/1.81 ( ! [X1 : $i] : (($true != (in @ X1 @ X1))) )),
% 3.09/1.81 inference(trivial_inequality_removal,[status(thm)],[f5100])).
% 3.09/1.81 thf(f5100,plain,(
% 3.09/1.81 ( ! [X1 : $i] : (($true != $true) | ($true != (in @ X1 @ X1))) )),
% 3.09/1.81 inference(definition_unfolding,[status(thm)],[f3237,f5021])).
% 3.09/1.81 thf(f5021,plain,(
% 3.09/1.81 (notinself = $true)),
% 3.09/1.81 inference(cnf_transformation,[status(thm)],[f3222])).
% 3.09/1.81 thf(f3222,plain,(
% 3.09/1.81 (breln1all1 = $true) & (quantDeMorgan2 = $true) & (subsetI2 = $true) & (lam2p = $true) & (woz13rule2 = $true) & (eqinunit = $true) & (setminusILneg = $true) & (setadjoinIR = $true) & (complementTI1 = $true) & (binunionTEcontra = $true) & (inIntersectImpInUnion = $true) & (inIntersectImpInUnion2 = $true) & (complementUnionInPowersetComplement = $true) & (setOfPairsIsBReln = $true) & (setminusI = $true) & (dpsetconstrEL2 = $true) & (funcGraphProp1 = $true) & (app = $true) & (breln1unionCommutes = $true) & (inPowerset = $true) & (nonemptyImpWitness = $true) & (setukpairinjR12 = $true) & (breln1all2 = $true) & (setbeta = $true) & (demorgan2a = $true) & (complementInPowersetComplementIntersect = $true) & (powersetAx = $true) & (sepSubset = $true) & (inComplementUnionImpInComplement1 = $true) & (inComplementUnionImpNotIn1 = $true) & (exuE3u = $true) & (subsetE = $true) & (subset2powerset = $true) & (exuEu = $true) & (breln1invprop = $true) & (demorgan1b = $true) & (powersetTI1 = $true) & (iffalse = $true) & (ubforcartprodlem2 = $true) & (binintersectI = $true) & (ubforcartprodlem1 = $true) & (powersetE = $true) & (singletonsuniq = $true) & (replAx = $true) & (setadjoinE = $true) & (foundationAx = $true) & (binunionLsub = $true) & (ap2apEq1 = $true) & (image1I = $true) & (emptyinPowerset = $true) & (intersectInPowersetIntersectUnions = $true) & (upairset2E = $true) & (demorgan2a2 = $true) & (ordinalMinLem1 = $true) & (emptysetAx = $true) & (setoftrueEq = $true) & (setadjoinOr = $true) & (nonemptyI = $true) & (woz2B = $true) & (symdiffE = $true) & (cartprodmempaircEq = $true) & (kpairp = $true) & (notinself = $true) & (emptyE1 = $true) & (emptysetE = $true) & (setminusIRneg = $true) & (cartprodpairsurjEq = $true) & (woz13rule1 = $true) & (binunionTILcontra = $true) & (exu__Cong = $true) & (nonemptyE1 = $true) & (upairset2IR = $true) & (symdiffI2 = $true) & (setadjoinIL = $true) & (contrasubsetT3 = $true) & (transitivesetOp2 = $true) & (binunionIR = $true) & (funcGraphProp3 = $true) & (emptyI = $true) & (cartprodmempair = $true) & (setextAx = $true) & (injFuncInInjFuncSet = $true) & (doubleComplementSub1 = $true) & (setminusELneg = $true) & (iftrueProp1 = $true) & (notequalI1 = $true) & (setminusEL = $true) & (doubleComplementE1 = $true) & (transitivesetOp1 = $true) & (woz2A = $true) & (brelnall2 = $true) & (breln1compE = $true) & (inIntersectImpInIntersectUnions = $true) & (setunionTransitive = $true) & (woz13rule0 = $true) & (dpsetconstrERa = $true) & (setunionsingleton1 = $true) & (contrasubsetT2 = $true) & (notdallE = $true) & (dpsetconstrI = $true) & (quantDeMorgan4 = $true) & (setextsub = $true) & (sepInPowerset = $true) & (subsetI1 = $true) & (setminusSubset2 = $true) & (breln1compEex = $true) & (breln1unionIL = $true) & (theeq = $true) & (eqimpsubset2 = $true) & (descrp = $true) & (binintersectEL = $true) & (setunionE = $true) & (ifp = $true) & (woz1_5 = $true) & (binintersectSubset3 = $true) & (surjFuncSetFuncIn = $true) & (cartprodsndin = $true) & (lamp = $true) & (exuI1 = $true) & (cartprodpairin = $true) & (powerset__Cong = $true) & (singletonsswitch = $true) & (funcGraphProp2 = $true) & (eqbreln = $true) & (binintersectRsub = $true) & (eta2 = $true) & (ap2apEq2 = $true) & (kfstsingleton = $true) & (demorgan1 = $true) & (upairsetE = $true) & (setminusT_lem = $true) & (exuE3e = $true) & (prop2setI = $true) & (cartprodpairmemER = $true) & (subsetE2 = $true) & (image1Equiv = $true) & (setukpairinjL = $true) & (setextT = $true) & (dsetconstrER = $true) & (binintersectSubset5 = $true) & (omega__Cong = $true) & (ex1E1 = $true) & (woz13rule4 = $true) & (binunionT_lem = $true) & (eta1 = $true) & (setunionsingleton2 = $true) & (ifSingleton = $true) & (complementTcontraSubset = $true) & (ordinalTransSet1 = $true) & (funcext2 = $true) & (setadjoinAx = $true) & (setukpairinjR11 = $true) & (emptysetOrdinal = $true) & (setminusSubset1 = $true) & (setminusERneg = $true) & (dpsetconstrSub = $true) & (disjointsetsI1 = $true) & (binunionIL = $true) & (doubleComplementSub2 = $true) & (prop2setE = $true) & (peanoSinj = $true) & (apProp = $true) & (ex1I2 = $true) & (cartprodsndpairEq = $true) & (woz1_4 = $true) & (subbreln = $true) & (subPowSU = $true) & (inCongP = $true) & (breln1invE = $true) & (binunionEcases = $true) & (binunionE = $true) & (setadjoinSub = $true) & (binintTransitive = $true) & (contraSubsetComplement = $true) & (infuncsetfunc = $true) & (funcextLem = $true) & (powersetsubset = $true) & (ubforcartprodlem3 = $true) & (quantDeMorgan1 = $true) & (complementTnotintersectT = $true) & (ex1I = $true) & (ap2p = $true) & (bs114d = $true) & (injFuncSetFuncIn = $true) & (setukpairinjL2 = $true) & (powersetE1 = $true) & (image1Ex1 = $true) & (setukpairinjR = $true) & (cartprodfstpairEq = $true) & (image1E = $true) & (breln1invI = $true) & (setOfPairsIsBReln1 = $true) & (notinemptyset = $true) & (omegaSp = $true) & (breln1SetBreln1 = $true) & (doubleComplementEq = $true) & (cartprodpairmemEL = $true) & (woz2Ex = $true) & (kfstpairEq = $true) & (secondinupair = $true) & (emptyset__Cong = $true) & (setext = $true) & (theprop = $true) & (surjCantorThm = $true) & (setadjoinSub2 = $true) & (upairsetIR = $true) & (subsetRefl = $true) & (subsetTrans = $true) & (breln1unionIR = $true) & (binunionTE = $true) & (upairinpowunion = $true) & (binintersectSubset4 = $true) & (setminusER = $true) & (contrasubsetT1 = $true) & (vacuousDall = $true) & (kpairsurjEq = $true) & (funcImageSingleton = $true) & (symdiffIneg2 = $true) & (upairequniteq = $true) & (demorgan2b = $true) & (exuE1 = $true) & (omegaSclos = $true) & (binintersectTELcontra = $true) & (iftrueProp2 = $true) & (complementT_lem = $true) & (exuI3 = $true) & (lamProp = $true) & (breln1unionprop = $true) & (doubleComplementI1 = $true) & (setukpairinjR1 = $true) & (omegaIndAx = $true) & (setukpairIL = $true) & (ksndpairEq = $true) & (complementTE1 = $true) & (powersetTE1 = $true) & (beta1 = $true) & (woz1_2 = $true) & (lam2lamEq = $true) & (demorgan2 = $true) & (omegaSAx = $true) & (dsetconstr__Cong = $true) & (iffalseProp1 = $true) & (funcinfuncset = $true) & (notsubsetI = $true) & (notinsingleton = $true) & (upairsetIL = $true) & (breln1unionEcases = $true) & (ex1E2 = $true) & (woz2W = $true) & (prop2set2propI = $true) & (woz1_1 = $true) & (singletoninpowerset = $true) & (binunionRsub = $true) & (binintersectER = $true) & (dsetconstrEL = $true) & ((((in @ sK1113 @ sK1112) = $true) & ((in @ sK1113 @ sK1113) = $true)) & ((ordinal @ sK1112) = $true)) & (omega0Ax = $true) & (breln1compI = $true) & (quantDeMorgan3 = $true) & (breln1unionI = $true) & (setunionsingleton = $true) & (dpsetconstrEL1 = $true) & (symdiffI1 = $true) & (ordinalTransSet = $true) & (symdiffIneg1 = $true) & (setunionOrdinalLem1 = $true) & (binintersectTERcontra = $true) & (powersetI1 = $true) & (subbreln1 = $true) & (iffalseProp2 = $true) & (in__Cong = $true) & (notequalI2 = $true) & (eqbreln1 = $true) & (wellorderingAx = $true) & (peano0notS = $true) & (subsetTI = $true) & (complementSubsetComplementIntersect = $true) & (setadjoin__Cong = $true) & (eqimpsubset1 = $true) & (exuE2 = $true) & (funcext = $true) & (singletoninpowunion = $true) & (demorgan2b2 = $true) & (singletonsubset = $true) & (subsetemptysetimpeq = $true) & (binintersectSubset1 = $true) & (binintersectLsub = $true) & (setunionE2 = $true) & (brelnall1 = $true) & (nonemptyI1 = $true) & (ksndsingleton = $true) & (demorgan1a = $true) & (surjFuncSetFuncSurj = $true) & (beta2 = $true) & (setunionAx = $true) & (setminusLsub = $true) & (foundation2 = $true) & (descr__Cong = $true) & (image1Ex = $true) & (binintersectT_lem = $true) & (choice2fnsingleton = $true) & (contrasubsetT = $true) & (powersetT_lem = $true) & (emptyinunitempty = $true) & (setunionI = $true) & (emptysetsubset = $true) & (complementImpComplementIntersect = $true) & (notinself2 = $true) & (setukpairIR = $true) & (woz13rule3 = $true) & (exuI2 = $true) & (setukpairinjR2 = $true) & (demorgan2a1 = $true) & (cartprodfstin = $true) & (cartprodmempair1 = $true) & (notdexE = $true) & (setukpairinjL1 = $true) & (funcGraphProp4 = $true) & (leftInvIsSurj = $true) & (breln1compprop = $true) & (iftrue = $true) & (binintersectSubset2 = $true) & (woz1_3 = $true) & (emptysetimpfalse = $true) & (upairsubunion = $true) & (powersetI = $true) & (breln1unionE = $true) & (binunionTIRcontra = $true) & (dsetconstrI = $true) & (kpairiskpair = $true) & (setunion__Cong = $true) & (injFuncSetFuncInj = $true) & (dpsetconstrER = $true) & (iftrueorfalse = $true) & (noeltsimpempty = $true) & (singletonprop = $true) & (uniqinunit = $true) & (emptyInPowerset = $true)),
% 3.09/1.81 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1112,sK1113])],[f1087,f3221,f3220])).
% 3.09/1.81 thf(f3220,plain,(
% 3.09/1.81 ? [X0 : $i] : (? [X1 : $i] : (($true = (in @ X1 @ X0)) & ($true = (in @ X1 @ X1))) & ((ordinal @ X0) = $true)) => (? [X1 : $i] : (($true = (in @ X1 @ sK1112)) & ($true = (in @ X1 @ X1))) & ((ordinal @ sK1112) = $true))),
% 3.09/1.81 introduced(definition,[],[choice_axiom])).
% 3.09/1.81 thf(f3221,plain,(
% 3.09/1.81 ? [X1 : $i] : (($true = (in @ X1 @ sK1112)) & ($true = (in @ X1 @ X1))) => (((in @ sK1113 @ sK1112) = $true) & ((in @ sK1113 @ sK1113) = $true))),
% 3.09/1.81 introduced(definition,[],[choice_axiom])).
% 3.09/1.81 thf(f1087,plain,(
% 3.09/1.81 (breln1all1 = $true) & (quantDeMorgan2 = $true) & (subsetI2 = $true) & (lam2p = $true) & (woz13rule2 = $true) & (eqinunit = $true) & (setminusILneg = $true) & (setadjoinIR = $true) & (complementTI1 = $true) & (binunionTEcontra = $true) & (inIntersectImpInUnion = $true) & (inIntersectImpInUnion2 = $true) & (complementUnionInPowersetComplement = $true) & (setOfPairsIsBReln = $true) & (setminusI = $true) & (dpsetconstrEL2 = $true) & (funcGraphProp1 = $true) & (app = $true) & (breln1unionCommutes = $true) & (inPowerset = $true) & (nonemptyImpWitness = $true) & (setukpairinjR12 = $true) & (breln1all2 = $true) & (setbeta = $true) & (demorgan2a = $true) & (complementInPowersetComplementIntersect = $true) & (powersetAx = $true) & (sepSubset = $true) & (inComplementUnionImpInComplement1 = $true) & (inComplementUnionImpNotIn1 = $true) & (exuE3u = $true) & (subsetE = $true) & (subset2powerset = $true) & (exuEu = $true) & (breln1invprop = $true) & (demorgan1b = $true) & (powersetTI1 = $true) & (iffalse = $true) & (ubforcartprodlem2 = $true) & (binintersectI = $true) & (ubforcartprodlem1 = $true) & (powersetE = $true) & (singletonsuniq = $true) & (replAx = $true) & (setadjoinE = $true) & (foundationAx = $true) & (binunionLsub = $true) & (ap2apEq1 = $true) & (image1I = $true) & (emptyinPowerset = $true) & (intersectInPowersetIntersectUnions = $true) & (upairset2E = $true) & (demorgan2a2 = $true) & (ordinalMinLem1 = $true) & (emptysetAx = $true) & (setoftrueEq = $true) & (setadjoinOr = $true) & (nonemptyI = $true) & (woz2B = $true) & (symdiffE = $true) & (cartprodmempaircEq = $true) & (kpairp = $true) & (notinself = $true) & (emptyE1 = $true) & (emptysetE = $true) & (setminusIRneg = $true) & (cartprodpairsurjEq = $true) & (woz13rule1 = $true) & (binunionTILcontra = $true) & (exu__Cong = $true) & (nonemptyE1 = $true) & (upairset2IR = $true) & (symdiffI2 = $true) & (setadjoinIL = $true) & (contrasubsetT3 = $true) & (transitivesetOp2 = $true) & (binunionIR = $true) & (funcGraphProp3 = $true) & (emptyI = $true) & (cartprodmempair = $true) & (setextAx = $true) & (injFuncInInjFuncSet = $true) & (doubleComplementSub1 = $true) & (setminusELneg = $true) & (iftrueProp1 = $true) & (notequalI1 = $true) & (setminusEL = $true) & (doubleComplementE1 = $true) & (transitivesetOp1 = $true) & (woz2A = $true) & (brelnall2 = $true) & (breln1compE = $true) & (inIntersectImpInIntersectUnions = $true) & (setunionTransitive = $true) & (woz13rule0 = $true) & (dpsetconstrERa = $true) & (setunionsingleton1 = $true) & (contrasubsetT2 = $true) & (notdallE = $true) & (dpsetconstrI = $true) & (quantDeMorgan4 = $true) & (setextsub = $true) & (sepInPowerset = $true) & (subsetI1 = $true) & (setminusSubset2 = $true) & (breln1compEex = $true) & (breln1unionIL = $true) & (theeq = $true) & (eqimpsubset2 = $true) & (descrp = $true) & (binintersectEL = $true) & (setunionE = $true) & (ifp = $true) & (woz1_5 = $true) & (binintersectSubset3 = $true) & (surjFuncSetFuncIn = $true) & (cartprodsndin = $true) & (lamp = $true) & (exuI1 = $true) & (cartprodpairin = $true) & (powerset__Cong = $true) & (singletonsswitch = $true) & (funcGraphProp2 = $true) & (eqbreln = $true) & (binintersectRsub = $true) & (eta2 = $true) & (ap2apEq2 = $true) & (kfstsingleton = $true) & (demorgan1 = $true) & (upairsetE = $true) & (setminusT_lem = $true) & (exuE3e = $true) & (prop2setI = $true) & (cartprodpairmemER = $true) & (subsetE2 = $true) & (image1Equiv = $true) & (setukpairinjL = $true) & (setextT = $true) & (dsetconstrER = $true) & (binintersectSubset5 = $true) & (omega__Cong = $true) & (ex1E1 = $true) & (woz13rule4 = $true) & (binunionT_lem = $true) & (eta1 = $true) & (setunionsingleton2 = $true) & (ifSingleton = $true) & (complementTcontraSubset = $true) & (ordinalTransSet1 = $true) & (funcext2 = $true) & (setadjoinAx = $true) & (setukpairinjR11 = $true) & (emptysetOrdinal = $true) & (setminusSubset1 = $true) & (setminusERneg = $true) & (dpsetconstrSub = $true) & (disjointsetsI1 = $true) & (binunionIL = $true) & (doubleComplementSub2 = $true) & (prop2setE = $true) & (peanoSinj = $true) & (apProp = $true) & (ex1I2 = $true) & (cartprodsndpairEq = $true) & (woz1_4 = $true) & (subbreln = $true) & (subPowSU = $true) & (inCongP = $true) & (breln1invE = $true) & (binunionEcases = $true) & (binunionE = $true) & (setadjoinSub = $true) & (binintTransitive = $true) & (contraSubsetComplement = $true) & (infuncsetfunc = $true) & (funcextLem = $true) & (powersetsubset = $true) & (ubforcartprodlem3 = $true) & (quantDeMorgan1 = $true) & (complementTnotintersectT = $true) & (ex1I = $true) & (ap2p = $true) & (bs114d = $true) & (injFuncSetFuncIn = $true) & (setukpairinjL2 = $true) & (powersetE1 = $true) & (image1Ex1 = $true) & (setukpairinjR = $true) & (cartprodfstpairEq = $true) & (image1E = $true) & (breln1invI = $true) & (setOfPairsIsBReln1 = $true) & (notinemptyset = $true) & (omegaSp = $true) & (breln1SetBreln1 = $true) & (doubleComplementEq = $true) & (cartprodpairmemEL = $true) & (woz2Ex = $true) & (kfstpairEq = $true) & (secondinupair = $true) & (emptyset__Cong = $true) & (setext = $true) & (theprop = $true) & (surjCantorThm = $true) & (setadjoinSub2 = $true) & (upairsetIR = $true) & (subsetRefl = $true) & (subsetTrans = $true) & (breln1unionIR = $true) & (binunionTE = $true) & (upairinpowunion = $true) & (binintersectSubset4 = $true) & (setminusER = $true) & (contrasubsetT1 = $true) & (vacuousDall = $true) & (kpairsurjEq = $true) & (funcImageSingleton = $true) & (symdiffIneg2 = $true) & (upairequniteq = $true) & (demorgan2b = $true) & (exuE1 = $true) & (omegaSclos = $true) & (binintersectTELcontra = $true) & (iftrueProp2 = $true) & (complementT_lem = $true) & (exuI3 = $true) & (lamProp = $true) & (breln1unionprop = $true) & (doubleComplementI1 = $true) & (setukpairinjR1 = $true) & (omegaIndAx = $true) & (setukpairIL = $true) & (ksndpairEq = $true) & (complementTE1 = $true) & (powersetTE1 = $true) & (beta1 = $true) & (woz1_2 = $true) & (lam2lamEq = $true) & (demorgan2 = $true) & (omegaSAx = $true) & (dsetconstr__Cong = $true) & (iffalseProp1 = $true) & (funcinfuncset = $true) & (notsubsetI = $true) & (notinsingleton = $true) & (upairsetIL = $true) & (breln1unionEcases = $true) & (ex1E2 = $true) & (woz2W = $true) & (prop2set2propI = $true) & (woz1_1 = $true) & (singletoninpowerset = $true) & (binunionRsub = $true) & (binintersectER = $true) & (dsetconstrEL = $true) & ? [X0 : $i] : (? [X1 : $i] : (($true = (in @ X1 @ X0)) & ($true = (in @ X1 @ X1))) & ((ordinal @ X0) = $true)) & (omega0Ax = $true) & (breln1compI = $true) & (quantDeMorgan3 = $true) & (breln1unionI = $true) & (setunionsingleton = $true) & (dpsetconstrEL1 = $true) & (symdiffI1 = $true) & (ordinalTransSet = $true) & (symdiffIneg1 = $true) & (setunionOrdinalLem1 = $true) & (binintersectTERcontra = $true) & (powersetI1 = $true) & (subbreln1 = $true) & (iffalseProp2 = $true) & (in__Cong = $true) & (notequalI2 = $true) & (eqbreln1 = $true) & (wellorderingAx = $true) & (peano0notS = $true) & (subsetTI = $true) & (complementSubsetComplementIntersect = $true) & (setadjoin__Cong = $true) & (eqimpsubset1 = $true) & (exuE2 = $true) & (funcext = $true) & (singletoninpowunion = $true) & (demorgan2b2 = $true) & (singletonsubset = $true) & (subsetemptysetimpeq = $true) & (binintersectSubset1 = $true) & (binintersectLsub = $true) & (setunionE2 = $true) & (brelnall1 = $true) & (nonemptyI1 = $true) & (ksndsingleton = $true) & (demorgan1a = $true) & (surjFuncSetFuncSurj = $true) & (beta2 = $true) & (setunionAx = $true) & (setminusLsub = $true) & (foundation2 = $true) & (descr__Cong = $true) & (image1Ex = $true) & (binintersectT_lem = $true) & (choice2fnsingleton = $true) & (contrasubsetT = $true) & (powersetT_lem = $true) & (emptyinunitempty = $true) & (setunionI = $true) & (emptysetsubset = $true) & (complementImpComplementIntersect = $true) & (notinself2 = $true) & (setukpairIR = $true) & (woz13rule3 = $true) & (exuI2 = $true) & (setukpairinjR2 = $true) & (demorgan2a1 = $true) & (cartprodfstin = $true) & (cartprodmempair1 = $true) & (notdexE = $true) & (setukpairinjL1 = $true) & (funcGraphProp4 = $true) & (leftInvIsSurj = $true) & (breln1compprop = $true) & (iftrue = $true) & (binintersectSubset2 = $true) & (woz1_3 = $true) & (emptysetimpfalse = $true) & (upairsubunion = $true) & (powersetI = $true) & (breln1unionE = $true) & (binunionTIRcontra = $true) & (dsetconstrI = $true) & (kpairiskpair = $true) & (setunion__Cong = $true) & (injFuncSetFuncInj = $true) & (dpsetconstrER = $true) & (iftrueorfalse = $true) & (noeltsimpempty = $true) & (singletonprop = $true) & (uniqinunit = $true) & (emptyInPowerset = $true)),
% 3.09/1.81 inference(flattening,[status(thm)],[f1086])).
% 3.09/1.81 thf(f1086,plain,(
% 3.09/1.81 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (? [X1 : $i] : (($true = (in @ X1 @ X0)) & ($true = (in @ X1 @ X1))) & ((ordinal @ X0) = $true)) & (emptysetOrdinal = $true)) & (setunionOrdinalLem1 = $true)) & (ordinalTransSet1 = $true)) & (ordinalTransSet = $true)) & (ordinalMinLem1 = $true)) & (setunionTransitive = $true)) & (transitivesetOp2 = $true)) & (binintTransitive = $true)) & (transitivesetOp1 = $true)) & (peanoSinj = $true)) & (peano0notS = $true)) & (omegaSclos = $true)) & (omegaSp = $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)),
% 3.09/1.81 inference(ennf_transformation,[status(thm)],[f1021])).
% 3.09/1.81 thf(f1021,plain,(
% 3.09/1.81 ~((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) => ((omegaSp = $true) => ((omegaSclos = $true) => ((peano0notS = $true) => ((peanoSinj = $true) => ((transitivesetOp1 = $true) => ((binintTransitive = $true) => ((transitivesetOp2 = $true) => ((setunionTransitive = $true) => ((ordinalMinLem1 = $true) => ((ordinalTransSet = $true) => ((ordinalTransSet1 = $true) => ((setunionOrdinalLem1 = $true) => ((emptysetOrdinal = $true) => ! [X0 : $i] : (((ordinal @ X0) = $true) => ! [X1 : $i] : (($true = (in @ X1 @ X0)) => ($true != (in @ X1 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.09/1.81 inference(flattening,[status(thm)],[f935])).
% 3.09/1.81 thf(f935,plain,(
% 3.09/1.81 ~((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) => ((omegaSp = $true) => ((omegaSclos = $true) => ((peano0notS = $true) => ((peanoSinj = $true) => ((transitivesetOp1 = $true) => ((binintTransitive = $true) => ((transitivesetOp2 = $true) => ((setunionTransitive = $true) => ((ordinalMinLem1 = $true) => ((ordinalTransSet = $true) => ((ordinalTransSet1 = $true) => ((setunionOrdinalLem1 = $true) => ((emptysetOrdinal = $true) => ! [X0 : $i] : (((ordinal @ X0) = $true) => ! [X1 : $i] : (($true = (in @ X1 @ X0)) => ~($true = (in @ X1 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.09/1.81 inference(fool_elimination,[status(thm)],[f934])).
% 3.09/1.81 thf(f934,plain,(
% 3.09/1.81 ~(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 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => (binintTransitive => (transitivesetOp2 => (setunionTransitive => (ordinalMinLem1 => (ordinalTransSet => (ordinalTransSet1 => (setunionOrdinalLem1 => (emptysetOrdinal => ! [X0 : $i] : ((ordinal @ X0) => ! [X1 : $i] : ((in @ X1 @ X0) => ~(in @ X1 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.09/1.81 inference(rectify,[status(thm)],[f343])).
% 3.09/1.81 thf(f343,negated_conjecture,(
% 3.09/1.81 ~(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 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => (binintTransitive => (transitivesetOp2 => (setunionTransitive => (ordinalMinLem1 => (ordinalTransSet => (ordinalTransSet1 => (setunionOrdinalLem1 => (emptysetOrdinal => ! [X11 : $i] : ((ordinal @ X11) => ! [X3 : $i] : ((in @ X3 @ X11) => ~(in @ X3 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.09/1.81 inference(negated_conjecture,[status(cth)],[f342])).
% 3.09/1.81 thf(f342,conjecture,(
% 3.09/1.81 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 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => (binintTransitive => (transitivesetOp2 => (setunionTransitive => (ordinalMinLem1 => (ordinalTransSet => (ordinalTransSet1 => (setunionOrdinalLem1 => (emptysetOrdinal => ! [X11 : $i] : ((ordinal @ X11) => ! [X3 : $i] : ((in @ X3 @ X11) => ~(in @ X3 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.09/1.81 file('/export/starexec/sandbox/tmp/tmp.zpLud4Dut7/DTF2THF_5045.p',ordinalIrrefl)).
% 3.09/1.81 thf(f3237,plain,(
% 3.09/1.81 ( ! [X1 : $i] : (($true != (in @ X1 @ X1)) | (notinself != $true)) )),
% 3.09/1.81 inference(cnf_transformation,[status(thm)],[f1539])).
% 3.09/1.81 thf(f1539,plain,(
% 3.09/1.81 ((notinself = $true) | ($true = (in @ sK27 @ sK27))) & (! [X1 : $i] : ($true != (in @ X1 @ X1)) | (notinself != $true))),
% 3.09/1.81 inference(skolemisation,[status(esa),new_symbols(skolem,[sK27])],[f1537,f1538])).
% 3.09/1.81 thf(f1538,plain,(
% 3.09/1.81 ? [X0 : $i] : ((in @ X0 @ X0) = $true) => ($true = (in @ sK27 @ sK27))),
% 3.09/1.81 introduced(definition,[],[choice_axiom])).
% 3.09/1.81 thf(f1537,plain,(
% 3.09/1.81 ((notinself = $true) | ? [X0 : $i] : ((in @ X0 @ X0) = $true)) & (! [X1 : $i] : ($true != (in @ X1 @ X1)) | (notinself != $true))),
% 3.09/1.81 inference(rectify,[status(thm)],[f1536])).
% 3.09/1.81 thf(f1536,plain,(
% 3.09/1.81 ((notinself = $true) | ? [X0 : $i] : ((in @ X0 @ X0) = $true)) & (! [X0 : $i] : ((in @ X0 @ X0) != $true) | (notinself != $true))),
% 3.09/1.81 inference(nnf_transformation,[status(thm)],[f1078])).
% 3.09/1.81 thf(f1078,plain,(
% 3.09/1.81 (notinself = $true) <=> ! [X0 : $i] : ((in @ X0 @ X0) != $true)),
% 3.09/1.81 inference(flattening,[status(thm)],[f1014])).
% 3.09/1.81 thf(f1014,plain,(
% 3.09/1.81 ! [X0 : $i] : ~((in @ X0 @ X0) = $true) <=> (notinself = $true)),
% 3.09/1.81 inference(fool_elimination,[status(thm)],[f1013])).
% 3.09/1.81 thf(f1013,plain,(
% 3.09/1.81 (! [X0 : $i] : ~(in @ X0 @ X0) = notinself)),
% 3.09/1.81 inference(rectify,[status(thm)],[f324])).
% 3.09/1.81 thf(f324,axiom,(
% 3.09/1.81 (! [X3 : $i] : ~(in @ X3 @ X3) = notinself)),
% 3.09/1.81 file('/export/starexec/sandbox/tmp/tmp.zpLud4Dut7/DTF2THF_5045.p',notinself)).
% 3.09/1.81 thf(f4827,plain,(
% 3.09/1.81 ((in @ sK1113 @ sK1113) = $true)),
% 3.09/1.81 inference(cnf_transformation,[status(thm)],[f3222])).
% 3.09/1.81 % SZS output end Proof for DTF2THF_5045
% 3.09/1.81 % (5165)------------------------------
% 3.09/1.81 % (5165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.81 % (5165)Termination reason: Refutation
% 3.09/1.81
% 3.09/1.81 % (5165)Memory used [KB]: 11769
% 3.09/1.81 % (5165)Time elapsed: 0.152 s
% 3.09/1.81 % (5165)Instructions burned: 489 (million)
% 3.09/1.81 % (5165)------------------------------
% 3.09/1.81 % (5165)------------------------------
% 3.09/1.81 % (5144)Success in time 0.236 s
%------------------------------------------------------------------------------