%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU736^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n020.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 09:36:46 AM UTC 2026
% Result : Theorem 2.73s 1.79s
% Output : Refutation 2.73s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 16
% Syntax : Number of formulae : 82 ( 16 unt; 0 typ; 5 def)
% Number of atoms : 5047 (1327 equ; 0 cnn)
% Maximal formula atoms : 258 ( 61 avg)
% Number of connectives : 3084 ( 172 ~; 133 |; 857 &; 841 @)
% ( 13 <=>;1068 =>; 0 <=; 0 <~>)
% Maximal formula depth : 266 ( 27 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 459 ( 456 usr; 270 con; 0-4 aty)
% Number of variables : 141 ( 0 ^; 94 !; 47 ?; 141 :)
% 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_304,type,
sP0: $i > $o > $i > $i > $o ).
thf(func_def_308,type,
sP4: $i > $i > $i > $i > $o ).
thf(func_def_310,type,
sP6: $i > $i > $o ).
thf(func_def_311,type,
sP7: $i > $o ).
thf(func_def_312,type,
sP8: $i > $i > $o ).
thf(func_def_313,type,
sP9: $i > $i > $o ).
thf(func_def_314,type,
sP10: $i > $i > $i > $o ).
thf(func_def_317,type,
sK13: $i > $i > $o ).
thf(func_def_325,type,
sK21: $i > ( $i > $o ) > $i ).
thf(func_def_326,type,
sK22: $i > $o ).
thf(func_def_344,type,
sK40: $i > $i > $i ).
thf(func_def_368,type,
sK64: $i > $i > $i > $i > $i ).
thf(func_def_369,type,
sK65: $i > $i > $i > $i > $i ).
thf(func_def_376,type,
sK72: $i > $o ).
thf(func_def_385,type,
sK81: $i > $i > $i ).
thf(func_def_397,type,
sK93: $i > $o ).
thf(func_def_398,type,
sK94: ( $i > $o ) > $i ).
thf(func_def_445,type,
sK141: $i > $o ).
thf(func_def_448,type,
sK144: $i > ( $i > $o ) > $i ).
thf(func_def_459,type,
sK155: $i > $o ).
thf(func_def_462,type,
sK158: $i > $o ).
thf(func_def_464,type,
sK160: ( $i > $o ) > $i > $i ).
thf(func_def_465,type,
sK161: ( $i > $o ) > $i > $i ).
thf(func_def_471,type,
sK167: $i > $i ).
thf(func_def_474,type,
sK170: $i > ( $i > $i ) > $i > $i ).
thf(func_def_477,type,
sK173: $o > $i > $i > $i ).
thf(func_def_489,type,
sK185: $i > $i > $i ).
thf(func_def_515,type,
sK211: $i > $i > $o ).
thf(func_def_525,type,
sK221: $i > $i > $i > $i ).
thf(func_def_526,type,
sK222: $i > $i > $i > $i ).
thf(func_def_529,type,
sK225: $i > $i > $i ).
thf(func_def_532,type,
sK228: $i > $o ).
thf(func_def_533,type,
sK229: $i > $i ).
thf(func_def_534,type,
sK230: ( $i > $o ) > $i ).
thf(func_def_567,type,
sK263: $i > $i > $o ).
thf(func_def_578,type,
sK274: $i > $i ).
thf(func_def_587,type,
sK283: $i > $i > $o ).
thf(func_def_600,type,
sK296: $i > $i > $o ).
thf(func_def_620,type,
sK316: $i > $i > $i ).
thf(func_def_629,type,
sK325: $i > $o ).
thf(func_def_636,type,
sK332: $i > $i ).
thf(func_def_638,type,
sK334: $i > $i ).
thf(func_def_650,type,
sK346: $i > $i ).
thf(func_def_660,type,
sK356: $i > $i ).
thf(func_def_662,type,
sK358: $i > ( $i > $i ) > $i > $i ).
thf(func_def_676,type,
sK372: $i > $o ).
thf(func_def_677,type,
sK373: $i > ( $i > $i ) > $i > $i ).
thf(func_def_679,type,
sK375: $i > $i ).
thf(func_def_684,type,
sK380: $i > $o ).
thf(func_def_717,type,
sK413: $i > $i > $o ).
thf(func_def_718,type,
sK414: $i > $i ).
thf(func_def_719,type,
sK415: $i > $i ).
thf(func_def_720,type,
sK416: ( $i > $i > $o ) > $i > $i ).
thf(func_def_721,type,
sK417: ( $i > $i > $o ) > $i > $i ).
thf(func_def_722,type,
sK418: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_724,type,
sK420: $i > $o ).
thf(func_def_727,type,
sK423: $i > $i > $i ).
thf(func_def_728,type,
sK424: $i > $i > $i ).
thf(func_def_734,type,
sK430: $i > $o ).
thf(func_def_737,type,
sK433: $i > ( $i > $o ) > $i ).
thf(func_def_738,type,
sK434: $i > ( $i > $o ) > $i ).
thf(func_def_748,type,
sK444: $i > $o ).
thf(func_def_749,type,
sK445: $i > $i ).
thf(func_def_750,type,
sK446: ( $i > $o ) > $i ).
thf(func_def_751,type,
sK447: $i > ( $i > $o ) > $i ).
thf(func_def_752,type,
sK448: $i > $o ).
thf(func_def_754,type,
sK450: $i > $i > $i > $i > $i ).
thf(func_def_762,type,
sK458: $i > ( $i > $o ) > $i ).
thf(func_def_763,type,
sK459: $i > $o ).
thf(func_def_765,type,
sK461: ( $i > $o ) > $i ).
thf(func_def_766,type,
sK462: ( $i > $o ) > $i ).
thf(func_def_767,type,
sK463: $i > $o ).
thf(func_def_783,type,
sK479: ( $i > $o ) > $i > $i ).
thf(func_def_785,type,
sK481: $i > $o ).
thf(func_def_789,type,
sK485: $i > $o ).
thf(func_def_792,type,
sK488: $i > $i > ( $i > $o ) > $i ).
thf(func_def_793,type,
sK489: $i > $i > $i > $i > $i ).
thf(func_def_798,type,
sK494: $i > ( $i > $o ) > $i ).
thf(func_def_799,type,
sK495: $i > $o ).
thf(func_def_806,type,
sK502: $i > $i > $i > $i > $i ).
thf(func_def_807,type,
sK503: $i > $i > $i > $i > $i ).
thf(func_def_808,type,
sK504: $i > $i > $i > $i > $i ).
thf(func_def_809,type,
sK505: $i > $i > $i > $i > $i ).
thf(func_def_813,type,
sK509: $i > $i > $o ).
thf(func_def_817,type,
sK513: $i > $i ).
thf(func_def_818,type,
sK514: ( $i > $i ) > $i > $i > $i ).
thf(func_def_831,type,
sK527: $i > $i > $i > $i ).
thf(func_def_832,type,
sK528: $i > $o ).
thf(func_def_842,type,
sK538: $i > $i > $i ).
thf(func_def_847,type,
sK543: $i > $o ).
thf(func_def_849,type,
sK545: $i > $o ).
thf(func_def_850,type,
sK546: $i > $o ).
thf(func_def_851,type,
sK547: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_852,type,
sK548: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_854,type,
sK550: $i > $o ).
thf(func_def_864,type,
sK560: $i > $i ).
thf(func_def_866,type,
sK562: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_867,type,
sK563: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_870,type,
sK566: $i > $o ).
thf(func_def_871,type,
sK567: $i > $o ).
thf(func_def_872,type,
sK568: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_873,type,
sK569: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_877,type,
sK573: $i > $o ).
thf(func_def_887,type,
sK583: $i > $i ).
thf(func_def_889,type,
sK585: $i > ( $i > $i ) > $i > $i ).
thf(func_def_898,type,
sK594: $i > $i > $i ).
thf(func_def_909,type,
sK605: $i > $i ).
thf(func_def_920,type,
sK616: $i > $o ).
thf(func_def_921,type,
sK617: $i > $o ).
thf(func_def_922,type,
sK618: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_923,type,
sK619: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_925,type,
sK621: $i > $i > $i ).
thf(func_def_926,type,
sK622: $i > $i > $i ).
thf(func_def_927,type,
sK623: $i > $i > $i ).
thf(func_def_928,type,
sK624: $i > $i > $i ).
thf(func_def_929,type,
sK625: $i > $i > $i > $i ).
thf(func_def_930,type,
sK626: $i > $i ).
thf(func_def_931,type,
sK627: $i > $i ).
thf(func_def_932,type,
sK628: $i > $i ).
thf(func_def_933,type,
sK629: $i > $i ).
thf(func_def_934,type,
sK630: $i > $i > $i ).
thf(func_def_935,type,
sK631: $i > $i > $i > $i ).
thf(func_def_936,type,
sK632: $i > $i > $i > $i ).
thf(func_def_937,type,
sK633: $i > $i > $i ).
thf(func_def_938,type,
sK634: $i > $i > $i ).
thf(func_def_939,type,
sK635: $i > $i ).
thf(func_def_940,type,
sK636: $i > $i > $i ).
thf(func_def_942,type,
sK638: $i > $i ).
thf(func_def_943,type,
sK639: $i > $i ).
thf(func_def_951,type,
sK647: $i > $o ).
thf(func_def_973,type,
sK669: $i > $o ).
thf(func_def_989,type,
sK685: $i > $i ).
thf(func_def_991,type,
sK687: $i > ( $i > $i ) > $i > $i ).
thf(func_def_995,type,
sK691: $i > ( $i > $o ) > $i ).
thf(func_def_996,type,
sK692: $i > $o ).
thf(func_def_1000,type,
sK696: $i > $i > $i > $i ).
thf(func_def_1004,type,
sK700: $i > $i > $i > $i ).
thf(func_def_1006,type,
sK702: $i > $i > $o ).
thf(func_def_1018,type,
sK714: $i > $i ).
thf(func_def_1020,type,
sK716: $i > ( $i > $o ) > $i ).
thf(func_def_1021,type,
sK717: $i > $o ).
thf(func_def_1026,type,
sK722: $i > $o ).
thf(func_def_1028,type,
sK724: $i > ( $i > $o ) > $i ).
thf(func_def_1035,type,
sK731: $i > $o ).
thf(func_def_1052,type,
sK748: $i > $i > $i > $i > $i ).
thf(func_def_1057,type,
sK753: $i > $i > $i ).
thf(func_def_1060,type,
sK756: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1061,type,
sK757: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1065,type,
sK761: $i > $o ).
thf(func_def_1072,type,
sK768: $i > $i > $i > $i ).
thf(func_def_1098,type,
sK794: $i > $o ).
thf(func_def_1104,type,
ph800:
!>[X0: $tType] : X0 ).
thf(f5341,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3982,f4021,f4082,f4144,f4198,f5095]) ).
thf(f5095,plain,
( ~ spl799_11
| ~ spl799_10
| ~ spl799_17 ),
inference(avatar_split_clause,[status(thm)],[f5094,f4015,f3954,f3958]) ).
thf(f3958,definition,
( spl799_11
<=> ( ( in @ sK165 @ sK163 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl799_11])],[avatar_definition]) ).
thf(f3954,definition,
( spl799_10
<=> ( $true
= ( in @ sK165 @ sK162 ) ) ),
introduced(definition,[new_symbols(naming,[spl799_10])],[avatar_definition]) ).
thf(f4015,definition,
( spl799_17
<=> ! [X0: $i] :
( ( ( in @ X0 @ sK164 )
!= $true )
| ( ( in @ X0 @ sK163 )
!= $true )
| ( ( in @ X0 @ sK162 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[spl799_17])],[avatar_definition]) ).
thf(f5094,plain,
( ( ( in @ sK165 @ sK163 )
!= $true )
| ~ spl799_10
| ~ spl799_17 ),
inference(trivial_inequality_removal,[status(thm)],[f5093]) ).
thf(f5093,plain,
( ( $true != $true )
| ( ( in @ sK165 @ sK163 )
!= $true )
| ~ spl799_10
| ~ spl799_17 ),
inference(forward_demodulation,[status(thm)],[f5037,f3956]) ).
thf(f3956,plain,
( ( $true
= ( in @ sK165 @ sK162 ) )
| ~ spl799_10 ),
inference(avatar_component_clause,[status(thm)],[f3954]) ).
thf(f5037,plain,
( ( $true
!= ( in @ sK165 @ sK162 ) )
| ( ( in @ sK165 @ sK163 )
!= $true )
| ~ spl799_17 ),
inference(trivial_inequality_removal,[status(thm)],[f4958]) ).
thf(f4958,plain,
( ( $true
!= ( in @ sK165 @ sK162 ) )
| ( ( in @ sK165 @ sK163 )
!= $true )
| ( $true != $true )
| ~ spl799_17 ),
inference(superposition,[status(thm)],[f4016,f2788]) ).
thf(f2788,plain,
( ( in @ sK165 @ sK164 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1362]) ).
thf(f1362,plain,
( ( cartprodpairsurjEq = $true )
& ( lamProp = $true )
& ( setminusSubset1 = $true )
& ( doubleComplementI1 = $true )
& ( iftrueProp1 = $true )
& ( setunion__Cong = $true )
& ( notsubsetI = $true )
& ( in__Cong = $true )
& ( setadjoinOr = $true )
& ( cartprodmempair1 = $true )
& ( funcGraphProp4 = $true )
& ( setminusERneg = $true )
& ( ( in @ sK162 @ ( powerset @ sK163 ) )
= $true )
& ( ( subset @ sK162 @ ( setminus @ sK163 @ sK164 ) )
= $true )
& ( ( in @ sK164 @ ( powerset @ sK163 ) )
= $true )
& ( ( in @ sK165 @ sK163 )
= $true )
& ( ( in @ sK165 @ sK164 )
= $true )
& ( ( in @ sK165 @ ( setminus @ sK163 @ sK162 ) )
!= $true )
& ( subset2powerset = $true )
& ( funcext = $true )
& ( upairset2IR = $true )
& ( iftrueProp2 = $true )
& ( singletoninpowerset = $true )
& ( setOfPairsIsBReln = $true )
& ( powerset__Cong = $true )
& ( nonemptyI = $true )
& ( lam2lamEq = $true )
& ( dpsetconstrSub = $true )
& ( cartprodmempaircEq = $true )
& ( exuE2 = $true )
& ( setadjoin__Cong = $true )
& ( setminusSubset2 = $true )
& ( nonemptyI1 = $true )
& ( emptysetimpfalse = $true )
& ( ap2apEq1 = $true )
& ( setukpairinjL1 = $true )
& ( binunionE = $true )
& ( cartprodfstpairEq = $true )
& ( setminusILneg = $true )
& ( emptyI = $true )
& ( app = $true )
& ( brelnall2 = $true )
& ( upairsubunion = $true )
& ( wellorderingAx = $true )
& ( setukpairinjR1 = $true )
& ( dsetconstrI = $true )
& ( setukpairinjR11 = $true )
& ( binunionT_lem = $true )
& ( notdexE = $true )
& ( binintersectT_lem = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( binunionRsub = $true )
& ( funcinfuncset = $true )
& ( omega0Ax = $true )
& ( sepInPowerset = $true )
& ( setminusLsub = $true )
& ( eqimpsubset2 = $true )
& ( eqbreln = $true )
& ( brelnall1 = $true )
& ( binintersectTELcontra = $true )
& ( kpairiskpair = $true )
& ( quantDeMorgan1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton2 = $true )
& ( notequalI1 = $true )
& ( singletonsuniq = $true )
& ( binintersectSubset4 = $true )
& ( apProp = $true )
& ( powersetE1 = $true )
& ( contrasubsetT1 = $true )
& ( funcImageSingleton = $true )
& ( cartprodsndin = $true )
& ( binunionIL = $true )
& ( cartprodpairmemEL = $true )
& ( dsetconstrEL = $true )
& ( setminusER = $true )
& ( cartprodmempair = $true )
& ( noeltsimpempty = $true )
& ( upairsetE = $true )
& ( contrasubsetT2 = $true )
& ( iffalseProp1 = $true )
& ( theprop = $true )
& ( cartprodpairmemER = $true )
& ( subsetRefl = $true )
& ( singletoninpowunion = $true )
& ( complementImpComplementIntersect = $true )
& ( subsetI1 = $true )
& ( binunionEcases = $true )
& ( symdiffE = $true )
& ( emptysetAx = $true )
& ( powersetTI1 = $true )
& ( cartprodpairin = $true )
& ( exuEu = $true )
& ( prop2set2propI = $true )
& ( iftrueorfalse = $true )
& ( notinemptyset = $true )
& ( lamp = $true )
& ( binintersectRsub = $true )
& ( emptyset__Cong = $true )
& ( binintersectTERcontra = $true )
& ( exu__Cong = $true )
& ( foundationAx = $true )
& ( setukpairinjL2 = $true )
& ( setunionsingleton1 = $true )
& ( emptyinPowerset = $true )
& ( dpsetconstrEL2 = $true )
& ( theeq = $true )
& ( binintersectER = $true )
& ( ubforcartprodlem2 = $true )
& ( setukpairinjR12 = $true )
& ( eta2 = $true )
& ( omegaIndAx = $true )
& ( setadjoinSub = $true )
& ( complementT_lem = $true )
& ( uniqinunit = $true )
& ( dpsetconstrI = $true )
& ( subsetTI = $true )
& ( symdiffIneg1 = $true )
& ( upairinpowunion = $true )
& ( ex1I = $true )
& ( iffalse = $true )
& ( cartprodsndpairEq = $true )
& ( beta2 = $true )
& ( nonemptyE1 = $true )
& ( prop2setE = $true )
& ( setbeta = $true )
& ( setadjoinE = $true )
& ( omega__Cong = $true )
& ( setext = $true )
& ( setadjoinIL = $true )
& ( notdallE = $true )
& ( setunionAx = $true )
& ( dsetconstrER = $true )
& ( singletonsswitch = $true )
& ( upairequniteq = $true )
& ( prop2setI = $true )
& ( binintersectLsub = $true )
& ( powersetAx = $true )
& ( sepSubset = $true )
& ( complementTE1 = $true )
& ( vacuousDall = $true )
& ( subsetE2 = $true )
& ( ksndpairEq = $true )
& ( complementTI1 = $true )
& ( quantDeMorgan4 = $true )
& ( emptyInPowerset = $true )
& ( symdiffIneg2 = $true )
& ( dpsetconstrER = $true )
& ( complementTnotintersectT = $true )
& ( binintersectSubset1 = $true )
& ( upairsetIR = $true )
& ( symdiffI1 = $true )
& ( ap2p = $true )
& ( setadjoinSub2 = $true )
& ( doubleComplementSub2 = $true )
& ( eqinunit = $true )
& ( exuE3u = $true )
& ( emptyE1 = $true )
& ( setukpairIR = $true )
& ( contrasubsetT = $true )
& ( subsetE = $true )
& ( beta1 = $true )
& ( ksndsingleton = $true )
& ( iftrue = $true )
& ( setunionsingleton = $true )
& ( descr__Cong = $true )
& ( setextsub = $true )
& ( subsetI2 = $true )
& ( omegaSAx = $true )
& ( setadjoinIR = $true )
& ( doubleComplementSub1 = $true )
& ( kfstsingleton = $true )
& ( subsetTrans = $true )
& ( powersetsubset = $true )
& ( setunionE = $true )
& ( ap2apEq2 = $true )
& ( inPowerset = $true )
& ( setextT = $true )
& ( funcGraphProp1 = $true )
& ( quantDeMorgan3 = $true )
& ( subsetemptysetimpeq = $true )
& ( replAx = $true )
& ( powersetTE1 = $true )
& ( funcGraphProp3 = $true )
& ( upairsetIL = $true )
& ( binunionLsub = $true )
& ( subbreln = $true )
& ( setminusELneg = $true )
& ( emptysetE = $true )
& ( kfstpairEq = $true )
& ( iffalseProp2 = $true )
& ( kpairsurjEq = $true )
& ( symdiffI2 = $true )
& ( inCongP = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodfstin = $true )
& ( descrp = $true )
& ( funcGraphProp2 = $true )
& ( powersetI1 = $true )
& ( exuI2 = $true )
& ( binintersectSubset2 = $true )
& ( lam2p = $true )
& ( setunionI = $true )
& ( doubleComplementEq = $true )
& ( ubforcartprodlem1 = $true )
& ( funcextLem = $true )
& ( emptyinunitempty = $true )
& ( powersetI = $true )
& ( exuE3e = $true )
& ( ex1E1 = $true )
& ( setminusEL = $true )
& ( dsetconstr__Cong = $true )
& ( powersetE = $true )
& ( notequalI2 = $true )
& ( setukpairIL = $true )
& ( ubforcartprodlem3 = $true )
& ( secondinupair = $true )
& ( dpsetconstrEL1 = $true )
& ( ex1I2 = $true )
& ( binintersectSubset3 = $true )
& ( ex1E2 = $true )
& ( setoftrueEq = $true )
& ( powersetT_lem = $true )
& ( quantDeMorgan2 = $true )
& ( subPowSU = $true )
& ( disjointsetsI1 = $true )
& ( setminusI = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( kpairp = $true )
& ( binunionIR = $true )
& ( eta1 = $true )
& ( doubleComplementE1 = $true )
& ( exuI1 = $true )
& ( infuncsetfunc = $true )
& ( exuI3 = $true )
& ( notinsingleton = $true )
& ( complementSubsetComplementIntersect = $true )
& ( contrasubsetT3 = $true )
& ( ifp = $true )
& ( exuE1 = $true )
& ( binintersectEL = $true )
& ( setadjoinAx = $true )
& ( ifSingleton = $true )
& ( setminusT_lem = $true )
& ( binintersectI = $true )
& ( dpsetconstrERa = $true )
& ( singletonsubset = $true )
& ( funcext2 = $true )
& ( emptysetsubset = $true )
& ( setukpairinjL = $true )
& ( setunionE2 = $true )
& ( setminusIRneg = $true )
& ( eqimpsubset1 = $true )
& ( bs114d = $true )
& ( setextAx = $true )
& ( upairset2E = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK162,sK163,sK164,sK165])],[f952,f1361,f1360,f1359]) ).
thf(f1359,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
= $true )
& ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ X3 @ ( setminus @ X1 @ X0 ) )
!= $true ) ) ) )
=> ( ( ( in @ sK162 @ ( powerset @ sK163 ) )
= $true )
& ? [X2: $i] :
( ( ( subset @ sK162 @ ( setminus @ sK163 @ X2 ) )
= $true )
& ( ( in @ X2 @ ( powerset @ sK163 ) )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ sK163 )
= $true )
& ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK163 @ sK162 ) )
!= $true ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1360,plain,
( ? [X2: $i] :
( ( ( subset @ sK162 @ ( setminus @ sK163 @ X2 ) )
= $true )
& ( ( in @ X2 @ ( powerset @ sK163 ) )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ sK163 )
= $true )
& ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK163 @ sK162 ) )
!= $true ) ) )
=> ( ( ( subset @ sK162 @ ( setminus @ sK163 @ sK164 ) )
= $true )
& ( ( in @ sK164 @ ( powerset @ sK163 ) )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ sK163 )
= $true )
& ( ( in @ X3 @ sK164 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK163 @ sK162 ) )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1361,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK163 )
= $true )
& ( ( in @ X3 @ sK164 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK163 @ sK162 ) )
!= $true ) )
=> ( ( ( in @ sK165 @ sK163 )
= $true )
& ( ( in @ sK165 @ sK164 )
= $true )
& ( ( in @ sK165 @ ( setminus @ sK163 @ sK162 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f952,plain,
( ( cartprodpairsurjEq = $true )
& ( lamProp = $true )
& ( setminusSubset1 = $true )
& ( doubleComplementI1 = $true )
& ( iftrueProp1 = $true )
& ( setunion__Cong = $true )
& ( notsubsetI = $true )
& ( in__Cong = $true )
& ( setadjoinOr = $true )
& ( cartprodmempair1 = $true )
& ( funcGraphProp4 = $true )
& ( setminusERneg = $true )
& ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
= $true )
& ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ X3 @ ( setminus @ X1 @ X0 ) )
!= $true ) ) ) )
& ( subset2powerset = $true )
& ( funcext = $true )
& ( upairset2IR = $true )
& ( iftrueProp2 = $true )
& ( singletoninpowerset = $true )
& ( setOfPairsIsBReln = $true )
& ( powerset__Cong = $true )
& ( nonemptyI = $true )
& ( lam2lamEq = $true )
& ( dpsetconstrSub = $true )
& ( cartprodmempaircEq = $true )
& ( exuE2 = $true )
& ( setadjoin__Cong = $true )
& ( setminusSubset2 = $true )
& ( nonemptyI1 = $true )
& ( emptysetimpfalse = $true )
& ( ap2apEq1 = $true )
& ( setukpairinjL1 = $true )
& ( binunionE = $true )
& ( cartprodfstpairEq = $true )
& ( setminusILneg = $true )
& ( emptyI = $true )
& ( app = $true )
& ( brelnall2 = $true )
& ( upairsubunion = $true )
& ( wellorderingAx = $true )
& ( setukpairinjR1 = $true )
& ( dsetconstrI = $true )
& ( setukpairinjR11 = $true )
& ( binunionT_lem = $true )
& ( notdexE = $true )
& ( binintersectT_lem = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( binunionRsub = $true )
& ( funcinfuncset = $true )
& ( omega0Ax = $true )
& ( sepInPowerset = $true )
& ( setminusLsub = $true )
& ( eqimpsubset2 = $true )
& ( eqbreln = $true )
& ( brelnall1 = $true )
& ( binintersectTELcontra = $true )
& ( kpairiskpair = $true )
& ( quantDeMorgan1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton2 = $true )
& ( notequalI1 = $true )
& ( singletonsuniq = $true )
& ( binintersectSubset4 = $true )
& ( apProp = $true )
& ( powersetE1 = $true )
& ( contrasubsetT1 = $true )
& ( funcImageSingleton = $true )
& ( cartprodsndin = $true )
& ( binunionIL = $true )
& ( cartprodpairmemEL = $true )
& ( dsetconstrEL = $true )
& ( setminusER = $true )
& ( cartprodmempair = $true )
& ( noeltsimpempty = $true )
& ( upairsetE = $true )
& ( contrasubsetT2 = $true )
& ( iffalseProp1 = $true )
& ( theprop = $true )
& ( cartprodpairmemER = $true )
& ( subsetRefl = $true )
& ( singletoninpowunion = $true )
& ( complementImpComplementIntersect = $true )
& ( subsetI1 = $true )
& ( binunionEcases = $true )
& ( symdiffE = $true )
& ( emptysetAx = $true )
& ( powersetTI1 = $true )
& ( cartprodpairin = $true )
& ( exuEu = $true )
& ( prop2set2propI = $true )
& ( iftrueorfalse = $true )
& ( notinemptyset = $true )
& ( lamp = $true )
& ( binintersectRsub = $true )
& ( emptyset__Cong = $true )
& ( binintersectTERcontra = $true )
& ( exu__Cong = $true )
& ( foundationAx = $true )
& ( setukpairinjL2 = $true )
& ( setunionsingleton1 = $true )
& ( emptyinPowerset = $true )
& ( dpsetconstrEL2 = $true )
& ( theeq = $true )
& ( binintersectER = $true )
& ( ubforcartprodlem2 = $true )
& ( setukpairinjR12 = $true )
& ( eta2 = $true )
& ( omegaIndAx = $true )
& ( setadjoinSub = $true )
& ( complementT_lem = $true )
& ( uniqinunit = $true )
& ( dpsetconstrI = $true )
& ( subsetTI = $true )
& ( symdiffIneg1 = $true )
& ( upairinpowunion = $true )
& ( ex1I = $true )
& ( iffalse = $true )
& ( cartprodsndpairEq = $true )
& ( beta2 = $true )
& ( nonemptyE1 = $true )
& ( prop2setE = $true )
& ( setbeta = $true )
& ( setadjoinE = $true )
& ( omega__Cong = $true )
& ( setext = $true )
& ( setadjoinIL = $true )
& ( notdallE = $true )
& ( setunionAx = $true )
& ( dsetconstrER = $true )
& ( singletonsswitch = $true )
& ( upairequniteq = $true )
& ( prop2setI = $true )
& ( binintersectLsub = $true )
& ( powersetAx = $true )
& ( sepSubset = $true )
& ( complementTE1 = $true )
& ( vacuousDall = $true )
& ( subsetE2 = $true )
& ( ksndpairEq = $true )
& ( complementTI1 = $true )
& ( quantDeMorgan4 = $true )
& ( emptyInPowerset = $true )
& ( symdiffIneg2 = $true )
& ( dpsetconstrER = $true )
& ( complementTnotintersectT = $true )
& ( binintersectSubset1 = $true )
& ( upairsetIR = $true )
& ( symdiffI1 = $true )
& ( ap2p = $true )
& ( setadjoinSub2 = $true )
& ( doubleComplementSub2 = $true )
& ( eqinunit = $true )
& ( exuE3u = $true )
& ( emptyE1 = $true )
& ( setukpairIR = $true )
& ( contrasubsetT = $true )
& ( subsetE = $true )
& ( beta1 = $true )
& ( ksndsingleton = $true )
& ( iftrue = $true )
& ( setunionsingleton = $true )
& ( descr__Cong = $true )
& ( setextsub = $true )
& ( subsetI2 = $true )
& ( omegaSAx = $true )
& ( setadjoinIR = $true )
& ( doubleComplementSub1 = $true )
& ( kfstsingleton = $true )
& ( subsetTrans = $true )
& ( powersetsubset = $true )
& ( setunionE = $true )
& ( ap2apEq2 = $true )
& ( inPowerset = $true )
& ( setextT = $true )
& ( funcGraphProp1 = $true )
& ( quantDeMorgan3 = $true )
& ( subsetemptysetimpeq = $true )
& ( replAx = $true )
& ( powersetTE1 = $true )
& ( funcGraphProp3 = $true )
& ( upairsetIL = $true )
& ( binunionLsub = $true )
& ( subbreln = $true )
& ( setminusELneg = $true )
& ( emptysetE = $true )
& ( kfstpairEq = $true )
& ( iffalseProp2 = $true )
& ( kpairsurjEq = $true )
& ( symdiffI2 = $true )
& ( inCongP = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodfstin = $true )
& ( descrp = $true )
& ( funcGraphProp2 = $true )
& ( powersetI1 = $true )
& ( exuI2 = $true )
& ( binintersectSubset2 = $true )
& ( lam2p = $true )
& ( setunionI = $true )
& ( doubleComplementEq = $true )
& ( ubforcartprodlem1 = $true )
& ( funcextLem = $true )
& ( emptyinunitempty = $true )
& ( powersetI = $true )
& ( exuE3e = $true )
& ( ex1E1 = $true )
& ( setminusEL = $true )
& ( dsetconstr__Cong = $true )
& ( powersetE = $true )
& ( notequalI2 = $true )
& ( setukpairIL = $true )
& ( ubforcartprodlem3 = $true )
& ( secondinupair = $true )
& ( dpsetconstrEL1 = $true )
& ( ex1I2 = $true )
& ( binintersectSubset3 = $true )
& ( ex1E2 = $true )
& ( setoftrueEq = $true )
& ( powersetT_lem = $true )
& ( quantDeMorgan2 = $true )
& ( subPowSU = $true )
& ( disjointsetsI1 = $true )
& ( setminusI = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( kpairp = $true )
& ( binunionIR = $true )
& ( eta1 = $true )
& ( doubleComplementE1 = $true )
& ( exuI1 = $true )
& ( infuncsetfunc = $true )
& ( exuI3 = $true )
& ( notinsingleton = $true )
& ( complementSubsetComplementIntersect = $true )
& ( contrasubsetT3 = $true )
& ( ifp = $true )
& ( exuE1 = $true )
& ( binintersectEL = $true )
& ( setadjoinAx = $true )
& ( ifSingleton = $true )
& ( setminusT_lem = $true )
& ( binintersectI = $true )
& ( dpsetconstrERa = $true )
& ( singletonsubset = $true )
& ( funcext2 = $true )
& ( emptysetsubset = $true )
& ( setukpairinjL = $true )
& ( setunionE2 = $true )
& ( setminusIRneg = $true )
& ( eqimpsubset1 = $true )
& ( bs114d = $true )
& ( setextAx = $true )
& ( upairset2E = $true ) ),
inference(flattening,[status(thm)],[f951]) ).
thf(f951,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( setminus @ X1 @ X0 ) )
!= $true )
& ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
= $true )
& ( ( in @ X2 @ ( powerset @ X1 ) )
= $true ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
= $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)],[f417]) ).
thf(f417,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 )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
=> ( ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( ( in @ X3 @ X2 )
= $true )
=> ( ( in @ X3 @ ( setminus @ X1 @ X0 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f416]) ).
thf(f416,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
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f258]) ).
thf(f258,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ ( setminus @ X3 @ X16 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ X16 )
=> ( in @ X1 @ ( setminus @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f257]) ).
thf(f257,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ ( setminus @ X3 @ X16 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ X16 )
=> ( in @ X1 @ ( setminus @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.1oBuXnfUb8/DTF2THF_10607.p',contraSubsetComplement) ).
thf(f4016,plain,
( ! [X0: $i] :
( ( ( in @ X0 @ sK164 )
!= $true )
| ( ( in @ X0 @ sK162 )
!= $true )
| ( ( in @ X0 @ sK163 )
!= $true ) )
| ~ spl799_17 ),
inference(avatar_component_clause,[status(thm)],[f4015]) ).
thf(f4198,plain,
spl799_18,
inference(avatar_contradiction_clause,[status(thm)],[f4197]) ).
thf(f4197,plain,
( $false
| spl799_18 ),
inference(trivial_inequality_removal,[status(thm)],[f4189]) ).
thf(f4189,plain,
( ( $true != $true )
| spl799_18 ),
inference(superposition,[status(thm)],[f4020,f2790]) ).
thf(f2790,plain,
( ( in @ sK164 @ ( powerset @ sK163 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1362]) ).
thf(f4020,plain,
( ( ( in @ sK164 @ ( powerset @ sK163 ) )
!= $true )
| spl799_18 ),
inference(avatar_component_clause,[status(thm)],[f4018]) ).
thf(f4018,definition,
( spl799_18
<=> ( ( in @ sK164 @ ( powerset @ sK163 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl799_18])],[avatar_definition]) ).
thf(f4144,plain,
spl799_12,
inference(avatar_contradiction_clause,[status(thm)],[f4143]) ).
thf(f4143,plain,
( $false
| spl799_12 ),
inference(trivial_inequality_removal,[status(thm)],[f4133]) ).
thf(f4133,plain,
( ( $true != $true )
| spl799_12 ),
inference(superposition,[status(thm)],[f3981,f2792]) ).
thf(f2792,plain,
( ( in @ sK162 @ ( powerset @ sK163 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1362]) ).
thf(f3981,plain,
( ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| spl799_12 ),
inference(avatar_component_clause,[status(thm)],[f3979]) ).
thf(f3979,definition,
( spl799_12
<=> ( ( in @ sK162 @ ( powerset @ sK163 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl799_12])],[avatar_definition]) ).
thf(f4082,plain,
spl799_11,
inference(avatar_contradiction_clause,[status(thm)],[f4081]) ).
thf(f4081,plain,
( $false
| spl799_11 ),
inference(trivial_inequality_removal,[status(thm)],[f4072]) ).
thf(f4072,plain,
( ( $true != $true )
| spl799_11 ),
inference(superposition,[status(thm)],[f3960,f2789]) ).
thf(f2789,plain,
( ( in @ sK165 @ sK163 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1362]) ).
thf(f3960,plain,
( ( ( in @ sK165 @ sK163 )
!= $true )
| spl799_11 ),
inference(avatar_component_clause,[status(thm)],[f3958]) ).
thf(f4021,plain,
( spl799_17
| ~ spl799_18 ),
inference(avatar_split_clause,[status(thm)],[f4013,f4018,f4015]) ).
thf(f4013,plain,
! [X0: $i] :
( ( ( in @ sK164 @ ( powerset @ sK163 ) )
!= $true )
| ( ( in @ X0 @ sK164 )
!= $true )
| ( ( in @ X0 @ sK162 )
!= $true )
| ( ( in @ X0 @ sK163 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4012]) ).
thf(f4012,plain,
! [X0: $i] :
( ( ( in @ X0 @ sK162 )
!= $true )
| ( $true != $true )
| ( ( in @ X0 @ sK164 )
!= $true )
| ( ( in @ X0 @ sK163 )
!= $true )
| ( ( in @ sK164 @ ( powerset @ sK163 ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4011,f2792]) ).
thf(f4011,plain,
! [X0: $i] :
( ( ( in @ sK164 @ ( powerset @ sK163 ) )
!= $true )
| ( ( in @ X0 @ sK164 )
!= $true )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| ( ( in @ X0 @ sK162 )
!= $true )
| ( ( in @ X0 @ sK163 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4010]) ).
thf(f4010,plain,
! [X0: $i] :
( ( ( in @ X0 @ sK162 )
!= $true )
| ( ( in @ X0 @ sK163 )
!= $true )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| ( ( in @ X0 @ sK164 )
!= $true )
| ( ( in @ sK164 @ ( powerset @ sK163 ) )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3986,f2643]) ).
thf(f2643,plain,
contrasubsetT = $true,
inference(cnf_transformation,[status(thm)],[f1362]) ).
thf(f3986,plain,
! [X0: $i] :
( ( ( in @ sK164 @ ( powerset @ sK163 ) )
!= $true )
| ( contrasubsetT != $true )
| ( ( in @ X0 @ sK164 )
!= $true )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| ( ( in @ X0 @ sK162 )
!= $true )
| ( ( in @ X0 @ sK163 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3984]) ).
thf(f3984,plain,
! [X0: $i] :
( ( ( in @ X0 @ sK164 )
!= $true )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| ( ( in @ X0 @ sK163 )
!= $true )
| ( ( in @ X0 @ sK162 )
!= $true )
| ( $true != $true )
| ( ( in @ sK164 @ ( powerset @ sK163 ) )
!= $true )
| ( contrasubsetT != $true ) ),
inference(superposition,[status(thm)],[f2938,f2791]) ).
thf(f2791,plain,
( ( subset @ sK162 @ ( setminus @ sK163 @ sK164 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1362]) ).
thf(f2938,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
!= $true )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true )
| ( ( in @ X3 @ X1 )
!= $true )
| ( ( in @ X2 @ ( powerset @ X1 ) )
!= $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( contrasubsetT != $true ) ),
inference(cnf_transformation,[status(thm)],[f1522]) ).
thf(f1522,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
!= $true )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( in @ X2 @ ( powerset @ X1 ) )
!= $true ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true ) )
| ( contrasubsetT != $true ) )
& ( ( contrasubsetT = $true )
| ( ( ( in @ sK272 @ sK271 )
= $true )
& ( ( subset @ sK269 @ ( setminus @ sK270 @ sK271 ) )
= $true )
& ( ( in @ sK272 @ sK269 )
= $true )
& ( ( in @ sK272 @ sK270 )
= $true )
& ( ( in @ sK271 @ ( powerset @ sK270 ) )
= $true )
& ( ( in @ sK269 @ ( powerset @ sK270 ) )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK269,sK270,sK271,sK272])],[f1518,f1521,f1520,f1519]) ).
thf(f1519,plain,
( ? [X4: $i,X5: $i] :
( ? [X6: $i] :
( ? [X7: $i] :
( ( ( in @ X7 @ X6 )
= $true )
& ( ( subset @ X4 @ ( setminus @ X5 @ X6 ) )
= $true )
& ( ( in @ X7 @ X4 )
= $true )
& ( ( in @ X7 @ X5 )
= $true ) )
& ( $true
= ( in @ X6 @ ( powerset @ X5 ) ) ) )
& ( ( in @ X4 @ ( powerset @ X5 ) )
= $true ) )
=> ( ? [X6: $i] :
( ? [X7: $i] :
( ( ( in @ X7 @ X6 )
= $true )
& ( $true
= ( subset @ sK269 @ ( setminus @ sK270 @ X6 ) ) )
& ( ( in @ X7 @ sK269 )
= $true )
& ( ( in @ X7 @ sK270 )
= $true ) )
& ( ( in @ X6 @ ( powerset @ sK270 ) )
= $true ) )
& ( ( in @ sK269 @ ( powerset @ sK270 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1520,plain,
( ? [X6: $i] :
( ? [X7: $i] :
( ( ( in @ X7 @ X6 )
= $true )
& ( $true
= ( subset @ sK269 @ ( setminus @ sK270 @ X6 ) ) )
& ( ( in @ X7 @ sK269 )
= $true )
& ( ( in @ X7 @ sK270 )
= $true ) )
& ( ( in @ X6 @ ( powerset @ sK270 ) )
= $true ) )
=> ( ? [X7: $i] :
( ( ( in @ X7 @ sK271 )
= $true )
& ( ( subset @ sK269 @ ( setminus @ sK270 @ sK271 ) )
= $true )
& ( ( in @ X7 @ sK269 )
= $true )
& ( ( in @ X7 @ sK270 )
= $true ) )
& ( ( in @ sK271 @ ( powerset @ sK270 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1521,plain,
( ? [X7: $i] :
( ( ( in @ X7 @ sK271 )
= $true )
& ( ( subset @ sK269 @ ( setminus @ sK270 @ sK271 ) )
= $true )
& ( ( in @ X7 @ sK269 )
= $true )
& ( ( in @ X7 @ sK270 )
= $true ) )
=> ( ( ( in @ sK272 @ sK271 )
= $true )
& ( ( subset @ sK269 @ ( setminus @ sK270 @ sK271 ) )
= $true )
& ( ( in @ sK272 @ sK269 )
= $true )
& ( ( in @ sK272 @ sK270 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1518,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
!= $true )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( in @ X2 @ ( powerset @ X1 ) )
!= $true ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true ) )
| ( contrasubsetT != $true ) )
& ( ( contrasubsetT = $true )
| ? [X4: $i,X5: $i] :
( ? [X6: $i] :
( ? [X7: $i] :
( ( ( in @ X7 @ X6 )
= $true )
& ( ( subset @ X4 @ ( setminus @ X5 @ X6 ) )
= $true )
& ( ( in @ X7 @ X4 )
= $true )
& ( ( in @ X7 @ X5 )
= $true ) )
& ( $true
= ( in @ X6 @ ( powerset @ X5 ) ) ) )
& ( ( in @ X4 @ ( powerset @ X5 ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1517]) ).
thf(f1517,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
!= $true )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( in @ X2 @ ( powerset @ X1 ) )
!= $true ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true ) )
| ( contrasubsetT != $true ) )
& ( ( contrasubsetT = $true )
| ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
= $true )
& ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( in @ X2 @ ( powerset @ X1 ) )
= $true ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f995]) ).
thf(f995,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
!= $true )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( in @ X2 @ ( powerset @ X1 ) )
!= $true ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true ) )
<=> ( contrasubsetT = $true ) ),
inference(flattening,[status(thm)],[f994]) ).
thf(f994,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
!= $true )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( in @ X2 @ ( powerset @ X1 ) )
!= $true ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true ) )
<=> ( contrasubsetT = $true ) ),
inference(ennf_transformation,[status(thm)],[f806]) ).
thf(f806,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
= $true )
=> ( ( ( in @ X3 @ X2 )
= $true )
=> ( ( in @ X3 @ X0 )
!= $true ) ) ) ) ) )
<=> ( contrasubsetT = $true ) ),
inference(flattening,[status(thm)],[f696]) ).
thf(f696,plain,
( ( contrasubsetT = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
= $true )
=> ( ( ( in @ X3 @ X2 )
= $true )
=> ( ( in @ X3 @ X0 )
!= $true ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f695]) ).
thf(f695,plain,
( contrasubsetT
= ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( subset @ X0 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X0 ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f244]) ).
thf(f244,axiom,
( contrasubsetT
= ( ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( subset @ X11 @ ( setminus @ X3 @ X16 ) )
=> ( ( in @ X1 @ X16 )
=> ~ ( in @ X1 @ X11 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.1oBuXnfUb8/DTF2THF_10607.p',contrasubsetT) ).
thf(f3982,plain,
( spl799_10
| ~ spl799_12 ),
inference(avatar_split_clause,[status(thm)],[f3977,f3979,f3954]) ).
thf(f3977,plain,
( ( $true
= ( in @ sK165 @ sK162 ) )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3976]) ).
thf(f3976,plain,
( ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| ( $true != $true )
| ( $true
= ( in @ sK165 @ sK162 ) ) ),
inference(forward_demodulation,[status(thm)],[f3975,f2789]) ).
thf(f3975,plain,
( ( ( in @ sK165 @ sK163 )
!= $true )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| ( $true
= ( in @ sK165 @ sK162 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3974]) ).
thf(f3974,plain,
( ( $true != $true )
| ( ( in @ sK165 @ sK163 )
!= $true )
| ( $true
= ( in @ sK165 @ sK162 ) )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3932,f2663]) ).
thf(f2663,plain,
complementTE1 = $true,
inference(cnf_transformation,[status(thm)],[f1362]) ).
thf(f3932,plain,
( ( complementTE1 != $true )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| ( ( in @ sK165 @ sK163 )
!= $true )
| ( $true
= ( in @ sK165 @ sK162 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3925]) ).
thf(f3925,plain,
( ( $true != $true )
| ( complementTE1 != $true )
| ( ( in @ sK165 @ sK163 )
!= $true )
| ( ( in @ sK162 @ ( powerset @ sK163 ) )
!= $true )
| ( $true
= ( in @ sK165 @ sK162 ) ) ),
inference(superposition,[status(thm)],[f2787,f2892]) ).
thf(f2892,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ ( setminus @ X4 @ X3 ) )
= $true )
| ( complementTE1 != $true )
| ( ( in @ X3 @ ( powerset @ X4 ) )
!= $true )
| ( ( in @ X5 @ X4 )
!= $true )
| ( ( in @ X5 @ X3 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1473]) ).
thf(f1473,plain,
( ( ( complementTE1 = $true )
| ( ( ( in @ sK237 @ ( powerset @ sK238 ) )
= $true )
& ( ( in @ sK239 @ sK238 )
= $true )
& ( ( in @ sK239 @ sK237 )
!= $true )
& ( ( in @ sK239 @ ( setminus @ sK238 @ sK237 ) )
!= $true ) ) )
& ( ! [X3: $i,X4: $i] :
( ( ( in @ X3 @ ( powerset @ X4 ) )
!= $true )
| ! [X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
| ( ( in @ X5 @ X3 )
= $true )
| ( ( in @ X5 @ ( setminus @ X4 @ X3 ) )
= $true ) ) )
| ( complementTE1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK237,sK238,sK239])],[f1470,f1472,f1471]) ).
thf(f1471,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( $true
!= ( in @ X2 @ X0 ) )
& ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
!= $true ) ) )
=> ( ( ( in @ sK237 @ ( powerset @ sK238 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ sK238 )
= $true )
& ( ( in @ X2 @ sK237 )
!= $true )
& ( ( in @ X2 @ ( setminus @ sK238 @ sK237 ) )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1472,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ sK238 )
= $true )
& ( ( in @ X2 @ sK237 )
!= $true )
& ( ( in @ X2 @ ( setminus @ sK238 @ sK237 ) )
!= $true ) )
=> ( ( ( in @ sK239 @ sK238 )
= $true )
& ( ( in @ sK239 @ sK237 )
!= $true )
& ( ( in @ sK239 @ ( setminus @ sK238 @ sK237 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1470,plain,
( ( ( complementTE1 = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( $true
!= ( in @ X2 @ X0 ) )
& ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
!= $true ) ) ) )
& ( ! [X3: $i,X4: $i] :
( ( ( in @ X3 @ ( powerset @ X4 ) )
!= $true )
| ! [X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
| ( ( in @ X5 @ X3 )
= $true )
| ( ( in @ X5 @ ( setminus @ X4 @ X3 ) )
= $true ) ) )
| ( complementTE1 != $true ) ) ),
inference(rectify,[status(thm)],[f1469]) ).
thf(f1469,plain,
( ( ( complementTE1 = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( $true
!= ( in @ X2 @ X0 ) )
& ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
!= $true ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true )
| ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X2 @ X0 ) )
| ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
= $true ) ) )
| ( complementTE1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1073]) ).
thf(f1073,plain,
( ( complementTE1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true )
| ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X2 @ X0 ) )
| ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
= $true ) ) ) ),
inference(flattening,[status(thm)],[f1072]) ).
thf(f1072,plain,
( ( complementTE1 = $true )
<=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
= $true )
| ( ( in @ X2 @ X1 )
!= $true ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f771]) ).
thf(f771,plain,
( ( complementTE1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
!= $true )
=> ( $true
= ( in @ X2 @ X0 ) ) ) ) ) ),
inference(flattening,[status(thm)],[f283]) ).
thf(f283,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
!= $true )
=> ( $true
= ( in @ X2 @ X0 ) ) ) ) )
<=> ( complementTE1 = $true ) ),
inference(fool_elimination,[status(thm)],[f282]) ).
thf(f282,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ~ ( in @ X2 @ ( setminus @ X1 @ X0 ) )
=> ( in @ X2 @ X0 ) ) ) )
= complementTE1 ),
inference(rectify,[status(thm)],[f241]) ).
thf(f241,axiom,
! [X11: $i,X3: $i] :
( ( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ ( setminus @ X3 @ X11 ) )
=> ( in @ X1 @ X11 ) ) ) )
= complementTE1 ),
file('/export/starexec/sandbox2/tmp/tmp.1oBuXnfUb8/DTF2THF_10607.p',complementTE1) ).
thf(f2787,plain,
( ( in @ sK165 @ ( setminus @ sK163 @ sK162 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1362]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU736^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n020.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 12:32:42 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : SEU736^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.45 % Domain : Set Theory
% 0.19/0.45 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.19/0.45 % subset X (setminus A Y) -> (! x:i.in x A -> in x Y ->
% 0.19/0.45 % in x (setminus A X))))
% 0.19/0.45
% 0.19/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.45 % Source : [Bro08]
% 0.19/0.45 % Names : ZFC238g [Bro08]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 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, 0.80 v4.1.0, 1.00 v3.7.0
% 0.19/0.45 % Syntax : Number of formulae : 547 ( 256 unt; 290 typ; 256 def)
% 0.19/0.45 % Number of atoms : 1476 ( 382 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 258 ( 5 avg)
% 0.19/0.45 % Number of connectives : 2954 ( 65 ~; 12 |; 44 &;2102 @)
% 0.19/0.45 % ( 14 <=>; 717 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 265 ( 2 avg)
% 0.19/0.45 % Number of types : 2 ( 0 usr)
% 0.19/0.45 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 293 ( 290 usr; 256 con; 0-4 aty)
% 0.19/0.45 % Number of variables : 860 ( 74 ^; 751 !; 35 ?; 860 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : http://mathgate.info/detsetitem.php?id=298
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 thf(in_type,type,
% 0.19/0.45 in: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu_type,type,
% 0.19/0.45 exu: ( $i > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu,definition,
% 0.19/0.45 ( exu
% 0.19/0.45 = ( ^ [Xphi: $i > $o] :
% 0.19/0.45 ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setextAx_type,type,
% 0.19/0.45 setextAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setextAx,definition,
% 0.19/0.45 ( setextAx
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 <=> ( in @ Xx @ B ) )
% 0.19/0.45 => ( A = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyset_type,type,
% 0.19/0.45 emptyset: $i ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetAx_type,type,
% 0.19/0.45 emptysetAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetAx,definition,
% 0.19/0.45 ( emptysetAx
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoin_type,type,
% 0.19/0.45 setadjoin: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinAx_type,type,
% 0.19/0.45 setadjoinAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinAx,definition,
% 0.19/0.45 ( setadjoinAx
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 <=> ( ( Xy = Xx )
% 0.19/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powerset_type,type,
% 0.19/0.45 powerset: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(powersetAx_type,type,
% 0.19/0.45 powersetAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetAx,definition,
% 0.19/0.45 ( powersetAx
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 <=> ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunion_type,type,
% 0.19/0.45 setunion: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setunionAx_type,type,
% 0.19/0.45 setunionAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionAx,definition,
% 0.19/0.45 ( setunionAx
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 <=> ? [B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omega_type,type,
% 0.19/0.45 omega: $i ).
% 0.19/0.45
% 0.19/0.45 thf(omega0Ax_type,type,
% 0.19/0.45 omega0Ax: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omega0Ax,definition,
% 0.19/0.45 ( omega0Ax
% 0.19/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omegaSAx_type,type,
% 0.19/0.45 omegaSAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omegaSAx,definition,
% 0.19/0.45 ( omegaSAx
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ omega )
% 0.19/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omegaIndAx_type,type,
% 0.19/0.45 omegaIndAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omegaIndAx,definition,
% 0.19/0.45 ( omegaIndAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( ( in @ emptyset @ A )
% 0.19/0.45 & ! [Xx: $i] :
% 0.19/0.45 ( ( ( in @ Xx @ omega )
% 0.19/0.45 & ( in @ Xx @ A ) )
% 0.19/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ omega )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(replAx_type,type,
% 0.19/0.45 replAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(replAx,definition,
% 0.19/0.45 ( replAx
% 0.19/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.45 => ? [B: $i] :
% 0.19/0.45 ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 <=> ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ A )
% 0.19/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(foundationAx_type,type,
% 0.19/0.45 foundationAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(foundationAx,definition,
% 0.19/0.45 ( foundationAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.45 => ? [B: $i] :
% 0.19/0.45 ( ( in @ B @ A )
% 0.19/0.45 & ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(wellorderingAx_type,type,
% 0.19/0.45 wellorderingAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(wellorderingAx,definition,
% 0.19/0.45 ( wellorderingAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ? [B: $i] :
% 0.19/0.45 ( ! [C: $i] :
% 0.19/0.45 ( ( in @ C @ B )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ A ) ) )
% 0.19/0.45 & ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( ( in @ Xx @ A )
% 0.19/0.45 & ( in @ Xy @ A ) )
% 0.19/0.45 => ( ! [C: $i] :
% 0.19/0.45 ( ( in @ C @ B )
% 0.19/0.45 => ( ( in @ Xx @ C )
% 0.19/0.45 <=> ( in @ Xy @ C ) ) )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 & ! [C: $i,D: $i] :
% 0.19/0.45 ( ( ( in @ C @ B )
% 0.19/0.45 & ( in @ D @ B ) )
% 0.19/0.45 => ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ D ) )
% 0.19/0.45 | ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ D )
% 0.19/0.45 => ( in @ Xx @ C ) ) ) )
% 0.19/0.45 & ! [C: $i] :
% 0.19/0.45 ( ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.45 => ? [D: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ D @ B )
% 0.19/0.45 & ( in @ Xx @ C )
% 0.19/0.45 & ~ ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ D )
% 0.19/0.45 & ( in @ Xy @ C ) )
% 0.19/0.45 & ! [E: $i] :
% 0.19/0.45 ( ( in @ E @ B )
% 0.19/0.45 => ( ! [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ E )
% 0.19/0.45 => ( in @ Xy @ D ) )
% 0.19/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(descr_type,type,
% 0.19/0.45 descr: ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(descrp_type,type,
% 0.19/0.45 descrp: $o ).
% 0.19/0.45
% 0.19/0.45 thf(descrp,definition,
% 0.19/0.45 ( descrp
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ( Xphi
% 0.19/0.45 @ ( descr
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr_type,type,
% 0.19/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrI_type,type,
% 0.19/0.45 dsetconstrI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrI,definition,
% 0.19/0.45 ( dsetconstrI
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 => ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL_type,type,
% 0.19/0.45 dsetconstrEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL,definition,
% 0.19/0.45 ( dsetconstrEL
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrER_type,type,
% 0.19/0.45 dsetconstrER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrER,definition,
% 0.19/0.45 ( dsetconstrER
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE1_type,type,
% 0.19/0.45 exuE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE1,definition,
% 0.19/0.45 ( exuE1
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set_type,type,
% 0.19/0.45 prop2set: $o > $i ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setE_type,type,
% 0.19/0.45 prop2setE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setE,definition,
% 0.19/0.45 ( prop2setE
% 0.19/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetE_type,type,
% 0.19/0.45 emptysetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetE,definition,
% 0.19/0.45 ( emptysetE
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetimpfalse_type,type,
% 0.19/0.45 emptysetimpfalse: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetimpfalse,definition,
% 0.19/0.45 ( emptysetimpfalse
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => $false ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notinemptyset_type,type,
% 0.19/0.45 notinemptyset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notinemptyset,definition,
% 0.19/0.45 ( notinemptyset
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3e_type,type,
% 0.19/0.45 exuE3e: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3e,definition,
% 0.19/0.45 ( exuE3e
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setext_type,type,
% 0.19/0.45 setext: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setext,definition,
% 0.19/0.45 ( setext
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) )
% 0.19/0.45 => ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 => ( A = B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyI_type,type,
% 0.19/0.45 emptyI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyI,definition,
% 0.19/0.45 ( emptyI
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ A )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(noeltsimpempty_type,type,
% 0.19/0.45 noeltsimpempty: $o ).
% 0.19/0.45
% 0.19/0.45 thf(noeltsimpempty,definition,
% 0.19/0.45 ( noeltsimpempty
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ A )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setbeta_type,type,
% 0.19/0.45 setbeta: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setbeta,definition,
% 0.19/0.45 ( setbeta
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonempty_type,type,
% 0.19/0.45 nonempty: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonempty,definition,
% 0.19/0.45 ( nonempty
% 0.19/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyE1_type,type,
% 0.19/0.45 nonemptyE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyE1,definition,
% 0.19/0.45 ( nonemptyE1
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( nonempty @ A )
% 0.19/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI_type,type,
% 0.19/0.45 nonemptyI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI,definition,
% 0.19/0.45 ( nonemptyI
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 => ( nonempty
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI1_type,type,
% 0.19/0.45 nonemptyI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI1,definition,
% 0.19/0.45 ( nonemptyI1
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIL_type,type,
% 0.19/0.45 setadjoinIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIL,definition,
% 0.19/0.45 ( setadjoinIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinunitempty_type,type,
% 0.19/0.45 emptyinunitempty: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinunitempty,definition,
% 0.19/0.45 ( emptyinunitempty
% 0.19/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIR_type,type,
% 0.19/0.45 setadjoinIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIR,definition,
% 0.19/0.45 ( setadjoinIR
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ A )
% 0.19/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinE_type,type,
% 0.19/0.45 setadjoinE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinE,definition,
% 0.19/0.45 ( setadjoinE
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ( ( Xy = Xx )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => ( ( ( in @ Xy @ A )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinOr_type,type,
% 0.19/0.45 setadjoinOr: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinOr,definition,
% 0.19/0.45 ( setadjoinOr
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 => ( ( Xy = Xx )
% 0.19/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setoftrueEq_type,type,
% 0.19/0.45 setoftrueEq: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setoftrueEq,definition,
% 0.19/0.45 ( setoftrueEq
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : $true )
% 0.19/0.45 = A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI_type,type,
% 0.19/0.45 powersetI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI,definition,
% 0.19/0.45 ( powersetI
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinPowerset_type,type,
% 0.19/0.45 emptyinPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinPowerset,definition,
% 0.19/0.45 ( emptyinPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyInPowerset_type,type,
% 0.19/0.45 emptyInPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyInPowerset,definition,
% 0.19/0.45 ( emptyInPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE_type,type,
% 0.19/0.45 powersetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE,definition,
% 0.19/0.45 ( powersetE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunionI_type,type,
% 0.19/0.45 setunionI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionI,definition,
% 0.19/0.45 ( setunionI
% 0.19/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( ( in @ B @ A )
% 0.19/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunionE_type,type,
% 0.19/0.45 setunionE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionE,definition,
% 0.19/0.45 ( setunionE
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ! [B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( ( in @ B @ A )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subPowSU_type,type,
% 0.19/0.45 subPowSU: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subPowSU,definition,
% 0.19/0.45 ( subPowSU
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE2_type,type,
% 0.19/0.45 exuE2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE2,definition,
% 0.19/0.45 ( exuE2
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyImpWitness_type,type,
% 0.19/0.45 nonemptyImpWitness: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyImpWitness,definition,
% 0.19/0.45 ( nonemptyImpWitness
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( nonempty @ A )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & $true ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(uniqinunit_type,type,
% 0.19/0.45 uniqinunit: $o ).
% 0.19/0.45
% 0.19/0.45 thf(uniqinunit,definition,
% 0.19/0.45 ( uniqinunit
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notinsingleton_type,type,
% 0.19/0.45 notinsingleton: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notinsingleton,definition,
% 0.19/0.45 ( notinsingleton
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx != Xy )
% 0.19/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(eqinunit_type,type,
% 0.19/0.45 eqinunit: $o ).
% 0.19/0.45
% 0.19/0.45 thf(eqinunit,definition,
% 0.19/0.45 ( eqinunit
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(singletonsswitch_type,type,
% 0.19/0.45 singletonsswitch: $o ).
% 0.19/0.45
% 0.19/0.45 thf(singletonsswitch,definition,
% 0.19/0.45 ( singletonsswitch
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetE_type,type,
% 0.19/0.45 upairsetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetE,definition,
% 0.19/0.45 ( upairsetE
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.45 => ( ( Xz = Xx )
% 0.19/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIL_type,type,
% 0.19/0.45 upairsetIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIL,definition,
% 0.19/0.45 ( upairsetIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIR_type,type,
% 0.19/0.45 upairsetIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIR,definition,
% 0.19/0.45 ( upairsetIR
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyE1_type,type,
% 0.19/0.45 emptyE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyE1,definition,
% 0.19/0.45 ( emptyE1
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ( ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 = emptyset )
% 0.19/0.45 => $false ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(vacuousDall_type,type,
% 0.19/0.45 vacuousDall: $o ).
% 0.19/0.45
% 0.19/0.45 thf(vacuousDall,definition,
% 0.19/0.45 ( vacuousDall
% 0.19/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan1_type,type,
% 0.19/0.45 quantDeMorgan1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan1,definition,
% 0.19/0.45 ( quantDeMorgan1
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan2_type,type,
% 0.19/0.45 quantDeMorgan2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan2,definition,
% 0.19/0.45 ( quantDeMorgan2
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) )
% 0.19/0.45 => ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan3_type,type,
% 0.19/0.45 quantDeMorgan3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan3,definition,
% 0.19/0.45 ( quantDeMorgan3
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan4_type,type,
% 0.19/0.45 quantDeMorgan4: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan4,definition,
% 0.19/0.45 ( quantDeMorgan4
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) )
% 0.19/0.45 => ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setI_type,type,
% 0.19/0.45 prop2setI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setI,definition,
% 0.19/0.45 ( prop2setI
% 0.19/0.45 = ( ! [Xphi: $o] :
% 0.19/0.45 ( Xphi
% 0.19/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(set2prop_type,type,
% 0.19/0.45 set2prop: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set2propI_type,type,
% 0.19/0.45 prop2set2propI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set2propI,definition,
% 0.19/0.45 ( prop2set2propI
% 0.19/0.45 = ( ! [Xphi: $o] :
% 0.19/0.45 ( Xphi
% 0.19/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notdexE_type,type,
% 0.19/0.45 notdexE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notdexE,definition,
% 0.19/0.45 ( notdexE
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notdallE_type,type,
% 0.19/0.45 notdallE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notdallE,definition,
% 0.19/0.45 ( notdallE
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI1_type,type,
% 0.19/0.45 exuI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI1,definition,
% 0.19/0.45 ( exuI1
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI3_type,type,
% 0.19/0.45 exuI3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI3,definition,
% 0.19/0.45 ( exuI3
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuI2_type,type,
% 0.19/0.46 exuI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuI2,definition,
% 0.19/0.46 ( exuI2
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 <=> ( Xy = Xx ) )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(inCongP_type,type,
% 0.19/0.46 inCongP: $o ).
% 0.19/0.46
% 0.19/0.46 thf(inCongP,definition,
% 0.19/0.46 ( inCongP
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(in__Cong_type,type,
% 0.19/0.46 in__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(in__Cong,definition,
% 0.19/0.46 ( in__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3u_type,type,
% 0.19/0.46 exuE3u: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3u,definition,
% 0.19/0.46 ( exuE3u
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exu__Cong_type,type,
% 0.19/0.46 exu__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu__Cong,definition,
% 0.19/0.46 ( exu__Cong
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 <=> ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset__Cong_type,type,
% 0.19/0.46 emptyset__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset__Cong,definition,
% 0.19/0.46 ( emptyset__Cong
% 0.19/0.46 = ( emptyset = emptyset ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin__Cong_type,type,
% 0.19/0.46 setadjoin__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin__Cong,definition,
% 0.19/0.46 ( setadjoin__Cong
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ! [Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( Xz = Xu )
% 0.19/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powerset__Cong_type,type,
% 0.19/0.46 powerset__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powerset__Cong,definition,
% 0.19/0.46 ( powerset__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( powerset @ A )
% 0.19/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunion__Cong_type,type,
% 0.19/0.46 setunion__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunion__Cong,definition,
% 0.19/0.46 ( setunion__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( setunion @ A )
% 0.19/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omega__Cong_type,type,
% 0.19/0.46 omega__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omega__Cong,definition,
% 0.19/0.46 ( omega__Cong
% 0.19/0.46 = ( omega = omega ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuEu_type,type,
% 0.19/0.46 exuEu: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuEu,definition,
% 0.19/0.46 ( exuEu
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(descr__Cong_type,type,
% 0.19/0.46 descr__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(descr__Cong,definition,
% 0.19/0.46 ( descr__Cong
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.46 => ( ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr__Cong_type,type,
% 0.19/0.46 dsetconstr__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr__Cong,definition,
% 0.19/0.46 ( dsetconstr__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.46 => ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subset_type,type,
% 0.19/0.46 subset: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(disjoint_type,type,
% 0.19/0.46 disjoint: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setsmeet_type,type,
% 0.19/0.46 setsmeet: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI1_type,type,
% 0.19/0.46 subsetI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI1,definition,
% 0.19/0.46 ( subsetI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset2_type,type,
% 0.19/0.46 eqimpsubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset2,definition,
% 0.19/0.46 ( eqimpsubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset1_type,type,
% 0.19/0.46 eqimpsubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset1,definition,
% 0.19/0.46 ( eqimpsubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI2_type,type,
% 0.19/0.46 subsetI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI2,definition,
% 0.19/0.46 ( subsetI2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetsubset_type,type,
% 0.19/0.46 emptysetsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetsubset,definition,
% 0.19/0.46 ( emptysetsubset
% 0.19/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE_type,type,
% 0.19/0.46 subsetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE,definition,
% 0.19/0.46 ( subsetE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE2_type,type,
% 0.19/0.46 subsetE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE2,definition,
% 0.19/0.46 ( subsetE2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notsubsetI_type,type,
% 0.19/0.46 notsubsetI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notsubsetI,definition,
% 0.19/0.46 ( notsubsetI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI1_type,type,
% 0.19/0.46 notequalI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI1,definition,
% 0.19/0.46 ( notequalI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ~ ( subset @ A @ B )
% 0.19/0.46 => ( A != B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI2_type,type,
% 0.19/0.46 notequalI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI2,definition,
% 0.19/0.46 ( notequalI2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( A != B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetRefl_type,type,
% 0.19/0.46 subsetRefl: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetRefl,definition,
% 0.19/0.46 ( subsetRefl
% 0.19/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTrans_type,type,
% 0.19/0.46 subsetTrans: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTrans,definition,
% 0.19/0.46 ( subsetTrans
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( subset @ B @ C )
% 0.19/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub_type,type,
% 0.19/0.46 setadjoinSub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub,definition,
% 0.19/0.46 ( setadjoinSub
% 0.19/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub2_type,type,
% 0.19/0.46 setadjoinSub2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub2,definition,
% 0.19/0.46 ( setadjoinSub2
% 0.19/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subset2powerset_type,type,
% 0.19/0.46 subset2powerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subset2powerset,definition,
% 0.19/0.46 ( subset2powerset
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextsub_type,type,
% 0.19/0.46 setextsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextsub,definition,
% 0.19/0.46 ( setextsub
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( subset @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetemptysetimpeq_type,type,
% 0.19/0.46 subsetemptysetimpeq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetemptysetimpeq,definition,
% 0.19/0.46 ( subsetemptysetimpeq
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( subset @ A @ emptyset )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI1_type,type,
% 0.19/0.46 powersetI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI1,definition,
% 0.19/0.46 ( powersetI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ B @ A )
% 0.19/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE1_type,type,
% 0.19/0.46 powersetE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE1,definition,
% 0.19/0.46 ( powersetE1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(inPowerset_type,type,
% 0.19/0.46 inPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(inPowerset,definition,
% 0.19/0.46 ( inPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetsubset_type,type,
% 0.19/0.46 powersetsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetsubset,definition,
% 0.19/0.46 ( powersetsubset
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sepInPowerset_type,type,
% 0.19/0.46 sepInPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sepInPowerset,definition,
% 0.19/0.46 ( sepInPowerset
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( in
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sepSubset_type,type,
% 0.19/0.46 sepSubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sepSubset,definition,
% 0.19/0.46 ( sepSubset
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( subset
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunion_type,type,
% 0.19/0.46 binunion: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIL_type,type,
% 0.19/0.46 binunionIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIL,definition,
% 0.19/0.46 ( binunionIL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2IR_type,type,
% 0.19/0.46 upairset2IR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2IR,definition,
% 0.19/0.46 ( upairset2IR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIR_type,type,
% 0.19/0.46 binunionIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIR,definition,
% 0.19/0.46 ( binunionIR
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionEcases_type,type,
% 0.19/0.46 binunionEcases: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionEcases,definition,
% 0.19/0.46 ( binunionEcases
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.46 => ( ( ( in @ Xx @ A )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => ( ( ( in @ Xx @ B )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionE_type,type,
% 0.19/0.46 binunionE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionE,definition,
% 0.19/0.46 ( binunionE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionLsub_type,type,
% 0.19/0.46 binunionLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionLsub,definition,
% 0.19/0.46 ( binunionLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionRsub_type,type,
% 0.19/0.46 binunionRsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionRsub,definition,
% 0.19/0.46 ( binunionRsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersect_type,type,
% 0.19/0.46 binintersect: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectI_type,type,
% 0.19/0.46 binintersectI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectI,definition,
% 0.19/0.46 ( binintersectI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset5_type,type,
% 0.19/0.46 binintersectSubset5: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset5,definition,
% 0.19/0.46 ( binintersectSubset5
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( subset @ C @ A )
% 0.19/0.46 => ( ( subset @ C @ B )
% 0.19/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectEL_type,type,
% 0.19/0.46 binintersectEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectEL,definition,
% 0.19/0.46 ( binintersectEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectLsub_type,type,
% 0.19/0.46 binintersectLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectLsub,definition,
% 0.19/0.46 ( binintersectLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset2_type,type,
% 0.19/0.46 binintersectSubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset2,definition,
% 0.19/0.46 ( binintersectSubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset3_type,type,
% 0.19/0.46 binintersectSubset3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset3,definition,
% 0.19/0.46 ( binintersectSubset3
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( binintersect @ A @ B )
% 0.19/0.46 = B )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectER_type,type,
% 0.19/0.46 binintersectER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectER,definition,
% 0.19/0.46 ( binintersectER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(disjointsetsI1_type,type,
% 0.19/0.46 disjointsetsI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(disjointsetsI1,definition,
% 0.19/0.46 ( disjointsetsI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( in @ Xx @ B ) )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectRsub_type,type,
% 0.19/0.46 binintersectRsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectRsub,definition,
% 0.19/0.46 ( binintersectRsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset4_type,type,
% 0.19/0.46 binintersectSubset4: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset4,definition,
% 0.19/0.46 ( binintersectSubset4
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ B @ A )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset1_type,type,
% 0.19/0.46 binintersectSubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset1,definition,
% 0.19/0.46 ( binintersectSubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( binintersect @ A @ B )
% 0.19/0.46 = A )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(bs114d_type,type,
% 0.19/0.46 bs114d: $o ).
% 0.19/0.46
% 0.19/0.46 thf(bs114d,definition,
% 0.19/0.46 ( bs114d
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(regular_type,type,
% 0.19/0.46 regular: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminus_type,type,
% 0.19/0.46 setminus: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setminusI_type,type,
% 0.19/0.46 setminusI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusI,definition,
% 0.19/0.46 ( setminusI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusEL_type,type,
% 0.19/0.46 setminusEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusEL,definition,
% 0.19/0.46 ( setminusEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusER_type,type,
% 0.19/0.46 setminusER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusER,definition,
% 0.19/0.46 ( setminusER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset2_type,type,
% 0.19/0.46 setminusSubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset2,definition,
% 0.19/0.46 ( setminusSubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( setminus @ A @ B )
% 0.19/0.46 = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusERneg_type,type,
% 0.19/0.46 setminusERneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusERneg,definition,
% 0.19/0.46 ( setminusERneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusELneg_type,type,
% 0.19/0.46 setminusELneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusELneg,definition,
% 0.19/0.46 ( setminusELneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusILneg_type,type,
% 0.19/0.46 setminusILneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusILneg,definition,
% 0.19/0.46 ( setminusILneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusIRneg_type,type,
% 0.19/0.46 setminusIRneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusIRneg,definition,
% 0.19/0.46 ( setminusIRneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusLsub_type,type,
% 0.19/0.46 setminusLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusLsub,definition,
% 0.19/0.46 ( setminusLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset1_type,type,
% 0.19/0.46 setminusSubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset1,definition,
% 0.19/0.46 ( setminusSubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( setminus @ A @ B )
% 0.19/0.46 = emptyset )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiff_type,type,
% 0.19/0.46 symdiff: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffE_type,type,
% 0.19/0.46 symdiffE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffE,definition,
% 0.19/0.46 ( symdiffE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI1_type,type,
% 0.19/0.46 symdiffI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI1,definition,
% 0.19/0.46 ( symdiffI1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI2_type,type,
% 0.19/0.46 symdiffI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI2,definition,
% 0.19/0.46 ( symdiffI2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg1_type,type,
% 0.19/0.46 symdiffIneg1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg1,definition,
% 0.19/0.46 ( symdiffIneg1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg2_type,type,
% 0.19/0.46 symdiffIneg2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg2,definition,
% 0.19/0.46 ( symdiffIneg2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iskpair_type,type,
% 0.19/0.46 iskpair: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(secondinupair_type,type,
% 0.19/0.46 secondinupair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(secondinupair,definition,
% 0.19/0.46 ( secondinupair
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIL_type,type,
% 0.19/0.46 setukpairIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIL,definition,
% 0.19/0.46 ( setukpairIL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIR_type,type,
% 0.19/0.46 setukpairIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIR,definition,
% 0.19/0.46 ( setukpairIR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpairiskpair_type,type,
% 0.19/0.46 kpairiskpair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairiskpair,definition,
% 0.19/0.46 ( kpairiskpair
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpair_type,type,
% 0.19/0.46 kpair: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(kpairp_type,type,
% 0.19/0.46 kpairp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairp,definition,
% 0.19/0.46 ( kpairp
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprod_type,type,
% 0.19/0.46 cartprod: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsubset_type,type,
% 0.19/0.46 singletonsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsubset,definition,
% 0.19/0.46 ( singletonsubset
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowerset_type,type,
% 0.19/0.46 singletoninpowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowerset,definition,
% 0.19/0.46 ( singletoninpowerset
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowunion_type,type,
% 0.19/0.46 singletoninpowunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowunion,definition,
% 0.19/0.46 ( singletoninpowunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2E_type,type,
% 0.19/0.46 upairset2E: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2E,definition,
% 0.19/0.46 ( upairset2E
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xx )
% 0.19/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsubunion_type,type,
% 0.19/0.46 upairsubunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsubunion,definition,
% 0.19/0.46 ( upairsubunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairinpowunion_type,type,
% 0.19/0.46 upairinpowunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairinpowunion,definition,
% 0.19/0.46 ( upairinpowunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem1_type,type,
% 0.19/0.46 ubforcartprodlem1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem1,definition,
% 0.19/0.46 ( ubforcartprodlem1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem2_type,type,
% 0.19/0.46 ubforcartprodlem2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem2,definition,
% 0.19/0.46 ( ubforcartprodlem2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem3_type,type,
% 0.19/0.46 ubforcartprodlem3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem3,definition,
% 0.19/0.46 ( ubforcartprodlem3
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairin_type,type,
% 0.19/0.46 cartprodpairin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairin,definition,
% 0.19/0.46 ( cartprodpairin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair1_type,type,
% 0.19/0.46 cartprodmempair1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair1,definition,
% 0.19/0.46 ( cartprodmempair1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 & ( Xu
% 0.19/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair_type,type,
% 0.19/0.46 cartprodmempair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair,definition,
% 0.19/0.46 ( cartprodmempair
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE2_type,type,
% 0.19/0.46 setunionE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE2,definition,
% 0.19/0.46 ( setunionE2
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 => ? [X: $i] :
% 0.19/0.46 ( ( in @ X @ A )
% 0.19/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton1_type,type,
% 0.19/0.46 setunionsingleton1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton1,definition,
% 0.19/0.46 ( setunionsingleton1
% 0.19/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton2_type,type,
% 0.19/0.46 setunionsingleton2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton2,definition,
% 0.19/0.46 ( setunionsingleton2
% 0.19/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton_type,type,
% 0.19/0.46 setunionsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton,definition,
% 0.19/0.46 ( setunionsingleton
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.19/0.46 = Xx ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singleton_type,type,
% 0.19/0.46 singleton: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonprop_type,type,
% 0.19/0.46 singletonprop: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonprop,definition,
% 0.19/0.46 ( singletonprop
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) )
% 0.19/0.46 => ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1_type,type,
% 0.19/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1,definition,
% 0.19/0.46 ( ex1
% 0.19/0.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( singleton
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E1_type,type,
% 0.19/0.46 ex1E1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E1,definition,
% 0.19/0.46 ( ex1E1
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ( ex1 @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I_type,type,
% 0.19/0.46 ex1I: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I,definition,
% 0.19/0.46 ( ex1I
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xy = Xx ) ) )
% 0.19/0.46 => ( ex1 @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I2_type,type,
% 0.19/0.46 ex1I2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I2,definition,
% 0.19/0.46 ( ex1I2
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) )
% 0.19/0.46 => ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ( ex1 @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsuniq_type,type,
% 0.19/0.46 singletonsuniq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsuniq,definition,
% 0.19/0.46 ( singletonsuniq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.19/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(atmost1p_type,type,
% 0.19/0.46 atmost1p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(atleast2p_type,type,
% 0.19/0.46 atleast2p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(atmost2p_type,type,
% 0.19/0.46 atmost2p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetp_type,type,
% 0.19/0.46 upairsetp: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL1_type,type,
% 0.19/0.46 setukpairinjL1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL1,definition,
% 0.19/0.46 ( setukpairinjL1
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kfstsingleton_type,type,
% 0.19/0.46 kfstsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kfstsingleton,definition,
% 0.19/0.46 ( kfstsingleton
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(theprop_type,type,
% 0.19/0.46 theprop: $o ).
% 0.19/0.46
% 0.19/0.46 thf(theprop,definition,
% 0.19/0.46 ( theprop
% 0.19/0.46 = ( ! [X: $i] :
% 0.19/0.46 ( ( singleton @ X )
% 0.19/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kfst_type,type,
% 0.19/0.46 kfst: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(kfstpairEq_type,type,
% 0.19/0.46 kfstpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kfstpairEq,definition,
% 0.19/0.46 ( kfstpairEq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xx ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstin_type,type,
% 0.19/0.46 cartprodfstin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstin,definition,
% 0.19/0.46 ( cartprodfstin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL2_type,type,
% 0.19/0.46 setukpairinjL2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL2,definition,
% 0.19/0.46 ( setukpairinjL2
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL_type,type,
% 0.19/0.46 setukpairinjL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL,definition,
% 0.19/0.46 ( setukpairinjL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xz @ Xu ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR11_type,type,
% 0.19/0.46 setukpairinjR11: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR11,definition,
% 0.19/0.46 ( setukpairinjR11
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR12_type,type,
% 0.19/0.46 setukpairinjR12: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR12,definition,
% 0.19/0.46 ( setukpairinjR12
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR1_type,type,
% 0.19/0.46 setukpairinjR1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR1,definition,
% 0.19/0.46 ( setukpairinjR1
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xu )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairequniteq_type,type,
% 0.19/0.46 upairequniteq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairequniteq,definition,
% 0.19/0.46 ( upairequniteq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR2_type,type,
% 0.19/0.46 setukpairinjR2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR2,definition,
% 0.19/0.46 ( setukpairinjR2
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR_type,type,
% 0.19/0.46 setukpairinjR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR,definition,
% 0.19/0.46 ( setukpairinjR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xz @ Xu ) )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ksndsingleton_type,type,
% 0.19/0.46 ksndsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ksndsingleton,definition,
% 0.19/0.46 ( ksndsingleton
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.46 @ ^ [Xx: $i] :
% 0.19/0.46 ( Xu
% 0.19/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ksnd_type,type,
% 0.19/0.46 ksnd: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(ksndpairEq_type,type,
% 0.19/0.46 ksndpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ksndpairEq,definition,
% 0.19/0.46 ( ksndpairEq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xy ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpairsurjEq_type,type,
% 0.19/0.46 kpairsurjEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairsurjEq,definition,
% 0.19/0.46 ( kpairsurjEq
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.46 = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndin_type,type,
% 0.19/0.46 cartprodsndin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndin,definition,
% 0.19/0.46 ( cartprodsndin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemEL_type,type,
% 0.19/0.46 cartprodpairmemEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemEL,definition,
% 0.19/0.46 ( cartprodpairmemEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemER_type,type,
% 0.19/0.46 cartprodpairmemER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemER,definition,
% 0.19/0.46 ( cartprodpairmemER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempaircEq_type,type,
% 0.19/0.46 cartprodmempaircEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempaircEq,definition,
% 0.19/0.46 ( cartprodmempaircEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstpairEq_type,type,
% 0.19/0.46 cartprodfstpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstpairEq,definition,
% 0.19/0.46 ( cartprodfstpairEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndpairEq_type,type,
% 0.19/0.46 cartprodsndpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndpairEq,definition,
% 0.19/0.46 ( cartprodsndpairEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairsurjEq_type,type,
% 0.19/0.46 cartprodpairsurjEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairsurjEq,definition,
% 0.19/0.46 ( cartprodpairsurjEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.46 = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(breln_type,type,
% 0.19/0.46 breln: $i > $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(breln,definition,
% 0.19/0.46 ( breln
% 0.19/0.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstr_type,type,
% 0.19/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrI_type,type,
% 0.19/0.46 dpsetconstrI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrI,definition,
% 0.19/0.46 ( dpsetconstrI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrSub_type,type,
% 0.19/0.46 dpsetconstrSub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrSub,definition,
% 0.19/0.46 ( dpsetconstrSub
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.46 ( subset
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.19/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setOfPairsIsBReln_type,type,
% 0.19/0.46 setOfPairsIsBReln: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setOfPairsIsBReln,definition,
% 0.19/0.46 ( setOfPairsIsBReln
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.46 ( breln @ A @ B
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrERa_type,type,
% 0.19/0.46 dpsetconstrERa: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrERa,definition,
% 0.19/0.46 ( dpsetconstrERa
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL1_type,type,
% 0.19/0.46 dpsetconstrEL1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL1,definition,
% 0.19/0.46 ( dpsetconstrEL1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL2_type,type,
% 0.19/0.46 dpsetconstrEL2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL2,definition,
% 0.19/0.46 ( dpsetconstrEL2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrER_type,type,
% 0.19/0.46 dpsetconstrER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrER,definition,
% 0.19/0.46 ( dpsetconstrER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(func_type,type,
% 0.19/0.46 func: $i > $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcSet_type,type,
% 0.19/0.46 funcSet: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(funcImageSingleton_type,type,
% 0.19/0.46 funcImageSingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcImageSingleton,definition,
% 0.19/0.46 ( funcImageSingleton
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(apProp_type,type,
% 0.19/0.46 apProp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(apProp,definition,
% 0.19/0.46 ( apProp
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in
% 0.19/0.46 @ ( setunion
% 0.19/0.46 @ ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.19/0.46 @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ap_type,type,
% 0.19/0.46 ap: $i > $i > $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(app_type,type,
% 0.19/0.46 app: $o ).
% 0.19/0.46
% 0.19/0.46 thf(app,definition,
% 0.19/0.46 ( app
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(infuncsetfunc_type,type,
% 0.19/0.46 infuncsetfunc: $o ).
% 0.19/0.46
% 0.19/0.46 thf(infuncsetfunc,definition,
% 0.19/0.46 ( infuncsetfunc
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ap2p_type,type,
% 0.19/0.46 ap2p: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ap2p,definition,
% 0.19/0.46 ( ap2p
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcinfuncset_type,type,
% 0.19/0.46 funcinfuncset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcinfuncset,definition,
% 0.19/0.46 ( funcinfuncset
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(lamProp_type,type,
% 0.19/0.46 lamProp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(lamProp,definition,
% 0.19/0.46 ( lamProp
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ( func @ A @ B
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xf @ Xx )
% 0.19/0.46 = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(lam_type,type,
% 0.19/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(lamp_type,type,
% 0.19/0.46 lamp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(lamp,definition,
% 0.19/0.46 ( lamp
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ( func @ A @ B
% 0.19/0.46 @ ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(lam2p_type,type,
% 0.19/0.46 lam2p: $o ).
% 0.19/0.46
% 0.19/0.46 thf(lam2p,definition,
% 0.19/0.46 ( lam2p
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ( in
% 0.19/0.46 @ ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(brelnall1_type,type,
% 0.19/0.46 brelnall1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(brelnall1,definition,
% 0.19/0.46 ( brelnall1
% 0.19/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 => ! [Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ R )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(brelnall2_type,type,
% 0.19/0.46 brelnall2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(brelnall2,definition,
% 0.19/0.46 ( brelnall2
% 0.19/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 => ! [Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ R )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E2_type,type,
% 0.19/0.46 ex1E2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E2,definition,
% 0.19/0.46 ( ex1E2
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ( ex1 @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp1_type,type,
% 0.19/0.46 funcGraphProp1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp1,definition,
% 0.19/0.46 ( funcGraphProp1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp3_type,type,
% 0.19/0.46 funcGraphProp3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp3,definition,
% 0.19/0.46 ( funcGraphProp3
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp2_type,type,
% 0.19/0.46 funcGraphProp2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp2,definition,
% 0.19/0.46 ( funcGraphProp2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcextLem_type,type,
% 0.19/0.46 funcextLem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcextLem,definition,
% 0.19/0.46 ( funcextLem
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xg: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xg )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp4_type,type,
% 0.19/0.46 funcGraphProp4: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp4,definition,
% 0.19/0.46 ( funcGraphProp4
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subbreln_type,type,
% 0.19/0.46 subbreln: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subbreln,definition,
% 0.19/0.46 ( subbreln
% 0.19/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 => ! [S: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ S )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqbreln_type,type,
% 0.19/0.46 eqbreln: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqbreln,definition,
% 0.19/0.46 ( eqbreln
% 0.19/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 => ! [S: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ S )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.19/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcext_type,type,
% 0.19/0.46 funcext: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcext,definition,
% 0.19/0.46 ( funcext
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xg: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xg )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcext2_type,type,
% 0.19/0.46 funcext2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcext2,definition,
% 0.19/0.46 ( funcext2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xg: $i] :
% 0.19/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ap2apEq1_type,type,
% 0.19/0.46 ap2apEq1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ap2apEq1,definition,
% 0.19/0.46 ( ap2apEq1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ap2apEq2_type,type,
% 0.19/0.46 ap2apEq2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ap2apEq2,definition,
% 0.19/0.46 ( ap2apEq2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(beta1_type,type,
% 0.19/0.46 beta1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(beta1,definition,
% 0.19/0.46 ( beta1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B
% 0.19/0.46 @ ( lam @ A @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.46 @ Xx )
% 0.19/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eta1_type,type,
% 0.19/0.46 eta1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eta1,definition,
% 0.19/0.46 ( eta1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ( ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.46 = Xf ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(lam2lamEq_type,type,
% 0.19/0.46 lam2lamEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(lam2lamEq,definition,
% 0.19/0.46 ( lam2lamEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ( ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.46 = ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(beta2_type,type,
% 0.19/0.46 beta2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(beta2,definition,
% 0.19/0.46 ( beta2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B
% 0.19/0.46 @ ( lam @ A @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.46 @ Xx )
% 0.19/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eta2_type,type,
% 0.19/0.46 eta2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eta2,definition,
% 0.19/0.46 ( eta2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ( ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.46 = Xf ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iffalseProp1_type,type,
% 0.19/0.46 iffalseProp1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iffalseProp1,definition,
% 0.19/0.46 ( iffalseProp1
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ~ Xphi
% 0.19/0.46 => ( in @ Xy
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iffalseProp2_type,type,
% 0.19/0.46 iffalseProp2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iffalseProp2,definition,
% 0.19/0.46 ( iffalseProp2
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ~ Xphi
% 0.19/0.46 => ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) )
% 0.19/0.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueProp1_type,type,
% 0.19/0.46 iftrueProp1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueProp1,definition,
% 0.19/0.46 ( iftrueProp1
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 => ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueProp2_type,type,
% 0.19/0.46 iftrueProp2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueProp2,definition,
% 0.19/0.46 ( iftrueProp2
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 => ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) )
% 0.19/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ifSingleton_type,type,
% 0.19/0.46 ifSingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ifSingleton,definition,
% 0.19/0.46 ( ifSingleton
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(if_type,type,
% 0.19/0.46 if: $i > $o > $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(ifp_type,type,
% 0.19/0.46 ifp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ifp,definition,
% 0.19/0.46 ( ifp
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(theeq_type,type,
% 0.19/0.46 theeq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(theeq,definition,
% 0.19/0.46 ( theeq
% 0.19/0.46 = ( ! [X: $i] :
% 0.19/0.46 ( ( singleton @ X )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ X )
% 0.19/0.46 => ( ( setunion @ X )
% 0.19/0.46 = Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iftrue_type,type,
% 0.19/0.46 iftrue: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iftrue,definition,
% 0.19/0.46 ( iftrue
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.46 = Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iffalse_type,type,
% 0.19/0.46 iffalse: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iffalse,definition,
% 0.19/0.46 ( iffalse
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ~ Xphi
% 0.19/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.46 = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueorfalse_type,type,
% 0.19/0.46 iftrueorfalse: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueorfalse,definition,
% 0.19/0.46 ( iftrueorfalse
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectT_lem_type,type,
% 0.19/0.46 binintersectT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectT_lem,definition,
% 0.19/0.46 ( binintersectT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionT_lem_type,type,
% 0.19/0.46 binunionT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionT_lem,definition,
% 0.19/0.46 ( binunionT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetT_lem_type,type,
% 0.19/0.46 powersetT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetT_lem,definition,
% 0.19/0.46 ( powersetT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusT_lem_type,type,
% 0.19/0.46 setminusT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusT_lem,definition,
% 0.19/0.46 ( setminusT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(complementT_lem_type,type,
% 0.19/0.46 complementT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(complementT_lem,definition,
% 0.19/0.46 ( complementT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextT_type,type,
% 0.19/0.46 setextT: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextT,definition,
% 0.19/0.46 ( setextT
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ( in @ Xx @ Y ) ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ Y )
% 0.19/0.46 => ( in @ Xx @ X ) ) )
% 0.19/0.46 => ( X = Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTI_type,type,
% 0.19/0.46 subsetTI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTI,definition,
% 0.19/0.46 ( subsetTI
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ( in @ Xx @ Y ) ) )
% 0.19/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetTI1_type,type,
% 0.19/0.46 powersetTI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetTI1,definition,
% 0.19/0.46 ( powersetTI1
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ( in @ Xx @ Y ) ) )
% 0.19/0.46 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetTE1_type,type,
% 0.19/0.46 powersetTE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetTE1,definition,
% 0.19/0.46 ( powersetTE1
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ X @ ( powerset @ Y ) )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(complementTI1_type,type,
% 0.19/0.46 complementTI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(complementTI1,definition,
% 0.19/0.46 ( complementTI1
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(complementTE1_type,type,
% 0.19/0.46 complementTE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(complementTE1,definition,
% 0.19/0.46 ( complementTE1
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectTELcontra_type,type,
% 0.19/0.46 binintersectTELcontra: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectTELcontra,definition,
% 0.19/0.46 ( binintersectTELcontra
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ X )
% 0.19/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectTERcontra_type,type,
% 0.19/0.46 binintersectTERcontra: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectTERcontra,definition,
% 0.19/0.46 ( binintersectTERcontra
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ Y )
% 0.19/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(contrasubsetT_type,type,
% 0.19/0.46 contrasubsetT: $o ).
% 0.19/0.46
% 0.19/0.46 thf(contrasubsetT,definition,
% 0.19/0.46 ( contrasubsetT
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.19/0.46 => ( ( in @ Xx @ Y )
% 0.19/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(contrasubsetT1_type,type,
% 0.19/0.46 contrasubsetT1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(contrasubsetT1,definition,
% 0.19/0.46 ( contrasubsetT1
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( subset @ X @ Y )
% 0.19/0.46 => ( ~ ( in @ Xx @ Y )
% 0.19/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(contrasubsetT2_type,type,
% 0.19/0.46 contrasubsetT2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(contrasubsetT2,definition,
% 0.19/0.46 ( contrasubsetT2
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( ( subset @ X @ Y )
% 0.19/0.46 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(contrasubsetT3_type,type,
% 0.19/0.46 contrasubsetT3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(contrasubsetT3,definition,
% 0.19/0.46 ( contrasubsetT3
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.19/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementI1_type,type,
% 0.19/0.46 doubleComplementI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementI1,definition,
% 0.19/0.46 ( doubleComplementI1
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementE1_type,type,
% 0.19/0.46 doubleComplementE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementE1,definition,
% 0.19/0.46 ( doubleComplementE1
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.19/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementSub1_type,type,
% 0.19/0.46 doubleComplementSub1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementSub1,definition,
% 0.19/0.46 ( doubleComplementSub1
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementSub2_type,type,
% 0.19/0.46 doubleComplementSub2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementSub2,definition,
% 0.19/0.46 ( doubleComplementSub2
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementEq_type,type,
% 0.19/0.46 doubleComplementEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(doubleComplementEq,definition,
% 0.19/0.46 ( doubleComplementEq
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ( X
% 0.19/0.46 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(complementTnotintersectT_type,type,
% 0.19/0.46 complementTnotintersectT: $o ).
% 0.19/0.46
% 0.19/0.46 thf(complementTnotintersectT,definition,
% 0.19/0.46 ( complementTnotintersectT
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(complementImpComplementIntersect_type,type,
% 0.19/0.46 complementImpComplementIntersect: $o ).
% 0.19/0.46
% 0.19/0.46 thf(complementImpComplementIntersect,definition,
% 0.19/0.46 ( complementImpComplementIntersect
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(complementSubsetComplementIntersect_type,type,
% 0.19/0.46 complementSubsetComplementIntersect: $o ).
% 0.19/0.46
% 0.19/0.46 thf(complementSubsetComplementIntersect,definition,
% 0.19/0.46 ( complementSubsetComplementIntersect
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(complementInPowersetComplementIntersect_type,type,
% 0.19/0.46 complementInPowersetComplementIntersect: $o ).
% 0.19/0.46
% 0.19/0.46 thf(complementInPowersetComplementIntersect,definition,
% 0.19/0.46 ( complementInPowersetComplementIntersect
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(contraSubsetComplement,conjecture,
% 0.19/0.46 ( setextAx
% 0.19/0.46 => ( emptysetAx
% 0.19/0.46 => ( setadjoinAx
% 0.19/0.46 => ( powersetAx
% 0.19/0.46 => ( setunionAx
% 0.19/0.46 => ( omega0Ax
% 0.19/0.46 => ( omegaSAx
% 0.19/0.46 => ( omegaIndAx
% 0.19/0.46 => ( replAx
% 0.19/0.46 => ( foundationAx
% 0.19/0.46 => ( wellorderingAx
% 0.19/0.46 => ( descrp
% 0.19/0.46 => ( dsetconstrI
% 0.19/0.46 => ( dsetconstrEL
% 0.19/0.46 => ( dsetconstrER
% 0.19/0.46 => ( exuE1
% 0.19/0.46 => ( prop2setE
% 0.19/0.46 => ( emptysetE
% 0.19/0.46 => ( emptysetimpfalse
% 0.19/0.46 => ( notinemptyset
% 0.19/0.46 => ( exuE3e
% 0.19/0.46 => ( setext
% 0.19/0.46 => ( emptyI
% 0.19/0.46 => ( noeltsimpempty
% 0.19/0.46 => ( setbeta
% 0.19/0.46 => ( nonemptyE1
% 0.19/0.46 => ( nonemptyI
% 0.19/0.46 => ( nonemptyI1
% 0.19/0.46 => ( setadjoinIL
% 0.19/0.46 => ( emptyinunitempty
% 0.19/0.46 => ( setadjoinIR
% 0.19/0.46 => ( setadjoinE
% 0.19/0.46 => ( setadjoinOr
% 0.19/0.46 => ( setoftrueEq
% 0.19/0.46 => ( powersetI
% 0.19/0.46 => ( emptyinPowerset
% 0.19/0.46 => ( emptyInPowerset
% 0.19/0.46 => ( powersetE
% 0.19/0.46 => ( setunionI
% 0.19/0.46 => ( setunionE
% 0.19/0.46 => ( subPowSU
% 0.19/0.46 => ( exuE2
% 0.19/0.46 => ( nonemptyImpWitness
% 0.19/0.46 => ( uniqinunit
% 0.19/0.46 => ( notinsingleton
% 0.19/0.46 => ( eqinunit
% 0.19/0.46 => ( singletonsswitch
% 0.19/0.46 => ( upairsetE
% 0.19/0.46 => ( upairsetIL
% 0.19/0.46 => ( upairsetIR
% 0.19/0.46 => ( emptyE1
% 0.19/0.46 => ( vacuousDall
% 0.19/0.46 => ( quantDeMorgan1
% 0.19/0.46 => ( quantDeMorgan2
% 0.19/0.46 => ( quantDeMorgan3
% 0.19/0.46 => ( quantDeMorgan4
% 0.19/0.46 => ( prop2setI
% 0.19/0.46 => ( prop2set2propI
% 0.19/0.46 => ( notdexE
% 0.19/0.46 => ( notdallE
% 0.19/0.46 => ( exuI1
% 0.19/0.46 => ( exuI3
% 0.19/0.46 => ( exuI2
% 0.19/0.46 => ( inCongP
% 0.19/0.46 => ( in__Cong
% 0.19/0.46 => ( exuE3u
% 0.19/0.46 => ( exu__Cong
% 0.19/0.46 => ( emptyset__Cong
% 0.19/0.46 => ( setadjoin__Cong
% 0.19/0.46 => ( powerset__Cong
% 0.19/0.46 => ( setunion__Cong
% 0.19/0.46 => ( omega__Cong
% 0.19/0.46 => ( exuEu
% 0.19/0.46 => ( descr__Cong
% 0.19/0.46 => ( dsetconstr__Cong
% 0.19/0.46 => ( subsetI1
% 0.19/0.46 => ( eqimpsubset2
% 0.19/0.46 => ( eqimpsubset1
% 0.19/0.46 => ( subsetI2
% 0.19/0.46 => ( emptysetsubset
% 0.19/0.46 => ( subsetE
% 0.19/0.46 => ( subsetE2
% 0.19/0.46 => ( notsubsetI
% 0.19/0.46 => ( notequalI1
% 0.19/0.46 => ( notequalI2
% 0.19/0.46 => ( subsetRefl
% 0.19/0.46 => ( subsetTrans
% 0.19/0.46 => ( setadjoinSub
% 0.19/0.46 => ( setadjoinSub2
% 0.19/0.46 => ( subset2powerset
% 0.19/0.46 => ( setextsub
% 0.19/0.46 => ( subsetemptysetimpeq
% 0.19/0.46 => ( powersetI1
% 0.19/0.46 => ( powersetE1
% 0.19/0.46 => ( inPowerset
% 0.19/0.46 => ( powersetsubset
% 0.19/0.46 => ( sepInPowerset
% 0.19/0.46 => ( sepSubset
% 0.19/0.46 => ( binunionIL
% 0.19/0.46 => ( upairset2IR
% 0.19/0.46 => ( binunionIR
% 0.19/0.46 => ( binunionEcases
% 0.19/0.46 => ( binunionE
% 0.19/0.46 => ( binunionLsub
% 0.19/0.46 => ( binunionRsub
% 0.19/0.46 => ( binintersectI
% 0.19/0.46 => ( binintersectSubset5
% 0.19/0.46 => ( binintersectEL
% 0.19/0.46 => ( binintersectLsub
% 0.19/0.46 => ( binintersectSubset2
% 0.19/0.46 => ( binintersectSubset3
% 0.19/0.46 => ( binintersectER
% 0.19/0.46 => ( disjointsetsI1
% 0.19/0.46 => ( binintersectRsub
% 0.19/0.46 => ( binintersectSubset4
% 0.19/0.46 => ( binintersectSubset1
% 0.19/0.46 => ( bs114d
% 0.19/0.46 => ( setminusI
% 0.19/0.46 => ( setminusEL
% 0.19/0.46 => ( setminusER
% 0.19/0.46 => ( setminusSubset2
% 0.19/0.46 => ( setminusERneg
% 0.19/0.46 => ( setminusELneg
% 0.19/0.46 => ( setminusILneg
% 0.19/0.46 => ( setminusIRneg
% 0.19/0.46 => ( setminusLsub
% 0.19/0.46 => ( setminusSubset1
% 0.19/0.46 => ( symdiffE
% 0.19/0.46 => ( symdiffI1
% 0.19/0.46 => ( symdiffI2
% 0.19/0.46 => ( symdiffIneg1
% 0.19/0.46 => ( symdiffIneg2
% 0.19/0.46 => ( secondinupair
% 0.19/0.46 => ( setukpairIL
% 0.19/0.46 => ( setukpairIR
% 0.19/0.46 => ( kpairiskpair
% 0.19/0.46 => ( kpairp
% 0.19/0.46 => ( singletonsubset
% 0.19/0.46 => ( singletoninpowerset
% 0.19/0.46 => ( singletoninpowunion
% 0.19/0.46 => ( upairset2E
% 0.19/0.46 => ( upairsubunion
% 0.19/0.46 => ( upairinpowunion
% 0.19/0.46 => ( ubforcartprodlem1
% 0.19/0.46 => ( ubforcartprodlem2
% 0.19/0.46 => ( ubforcartprodlem3
% 0.19/0.46 => ( cartprodpairin
% 0.19/0.46 => ( cartprodmempair1
% 0.19/0.46 => ( cartprodmempair
% 0.19/0.46 => ( setunionE2
% 0.19/0.46 => ( setunionsingleton1
% 0.19/0.46 => ( setunionsingleton2
% 0.19/0.46 => ( setunionsingleton
% 0.19/0.46 => ( singletonprop
% 0.19/0.46 => ( ex1E1
% 0.19/0.46 => ( ex1I
% 0.19/0.46 => ( ex1I2
% 0.19/0.46 => ( singletonsuniq
% 0.19/0.46 => ( setukpairinjL1
% 0.19/0.46 => ( kfstsingleton
% 0.19/0.46 => ( theprop
% 0.19/0.46 => ( kfstpairEq
% 0.19/0.46 => ( cartprodfstin
% 0.19/0.46 => ( setukpairinjL2
% 0.19/0.46 => ( setukpairinjL
% 0.19/0.46 => ( setukpairinjR11
% 0.19/0.46 => ( setukpairinjR12
% 0.19/0.46 => ( setukpairinjR1
% 0.19/0.46 => ( upairequniteq
% 0.19/0.46 => ( setukpairinjR2
% 0.19/0.46 => ( setukpairinjR
% 0.19/0.47 => ( ksndsingleton
% 0.19/0.47 => ( ksndpairEq
% 0.19/0.47 => ( kpairsurjEq
% 0.19/0.47 => ( cartprodsndin
% 0.19/0.47 => ( cartprodpairmemEL
% 0.19/0.47 => ( cartprodpairmemER
% 0.19/0.47 => ( cartprodmempaircEq
% 0.19/0.47 => ( cartprodfstpairEq
% 0.19/0.47 => ( cartprodsndpairEq
% 0.19/0.47 => ( cartprodpairsurjEq
% 0.19/0.47 => ( dpsetconstrI
% 0.19/0.47 => ( dpsetconstrSub
% 0.19/0.47 => ( setOfPairsIsBReln
% 0.19/0.47 => ( dpsetconstrERa
% 0.19/0.47 => ( dpsetconstrEL1
% 0.19/0.47 => ( dpsetconstrEL2
% 0.19/0.47 => ( dpsetconstrER
% 0.19/0.47 => ( funcImageSingleton
% 0.19/0.47 => ( apProp
% 0.19/0.47 => ( app
% 0.19/0.47 => ( infuncsetfunc
% 0.30/0.47 => ( ap2p
% 0.30/0.47 => ( funcinfuncset
% 0.30/0.47 => ( lamProp
% 0.30/0.47 => ( lamp
% 0.30/0.47 => ( lam2p
% 0.30/0.47 => ( brelnall1
% 0.30/0.47 => ( brelnall2
% 0.30/0.47 => ( ex1E2
% 0.30/0.47 => ( funcGraphProp1
% 0.30/0.47 => ( funcGraphProp3
% 0.30/0.47 => ( funcGraphProp2
% 0.30/0.47 => ( funcextLem
% 0.30/0.47 => ( funcGraphProp4
% 0.30/0.47 => ( subbreln
% 0.30/0.47 => ( eqbreln
% 0.30/0.47 => ( funcext
% 0.30/0.47 => ( funcext2
% 0.30/0.47 => ( ap2apEq1
% 0.30/0.47 => ( ap2apEq2
% 0.30/0.47 => ( beta1
% 0.30/0.47 => ( eta1
% 0.30/0.47 => ( lam2lamEq
% 0.30/0.47 => ( beta2
% 0.30/0.47 => ( eta2
% 0.30/0.47 => ( iffalseProp1
% 0.30/0.47 => ( iffalseProp2
% 0.30/0.47 => ( iftrueProp1
% 0.30/0.47 => ( iftrueProp2
% 0.30/0.47 => ( ifSingleton
% 0.30/0.47 => ( ifp
% 0.30/0.47 => ( theeq
% 0.30/0.47 => ( iftrue
% 0.30/0.47 => ( iffalse
% 0.30/0.47 => ( iftrueorfalse
% 0.30/0.47 => ( binintersectT_lem
% 0.30/0.47 => ( binunionT_lem
% 0.30/0.47 => ( powersetT_lem
% 0.30/0.47 => ( setminusT_lem
% 0.30/0.47 => ( complementT_lem
% 0.30/0.47 => ( setextT
% 0.30/0.47 => ( subsetTI
% 0.30/0.47 => ( powersetTI1
% 0.30/0.47 => ( powersetTE1
% 0.30/0.47 => ( complementTI1
% 0.30/0.47 => ( complementTE1
% 0.30/0.47 => ( binintersectTELcontra
% 0.30/0.47 => ( binintersectTERcontra
% 0.30/0.47 => ( contrasubsetT
% 0.30/0.47 => ( contrasubsetT1
% 0.30/0.47 => ( contrasubsetT2
% 0.30/0.47 => ( contrasubsetT3
% 0.30/0.47 => ( doubleComplementI1
% 0.30/0.47 => ( doubleComplementE1
% 0.30/0.47 => ( doubleComplementSub1
% 0.30/0.47 => ( doubleComplementSub2
% 0.30/0.47 => ( doubleComplementEq
% 0.30/0.47 => ( complementTnotintersectT
% 0.30/0.47 => ( complementImpComplementIntersect
% 0.30/0.47 => ( complementSubsetComplementIntersect
% 0.30/0.47 => ( complementInPowersetComplementIntersect
% 0.30/0.47 => ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.45/1.47 ---- Embedded in THF ---
% 2.45/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.45/1.47 %%% Generated on Tue Feb 24 12:32:43 EST 2026
% 2.45/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.45/1.47 %%% Logic specification used:
% 2.45/1.47 %%% thf(spec, logic, $$dhol).
% 2.45/1.47
% 2.45/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.1oBuXnfUb8/DTF2DTF10607.p
% See solution above
% 2.45/1.47 ------------------------
% 2.68/1.52 ---- Cleaned THF ---
% 2.68/1.53 thf(in_type,type,
% 2.68/1.53 in: $i > $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(exu_type,type,
% 2.68/1.53 exu: ( $i > $o ) > $o ).
% 2.68/1.53
% 2.68/1.53 thf(exu,definition,
% 2.68/1.53 ( exu
% 2.68/1.53 = ( ^ [Xphi: $i > $o] :
% 2.68/1.53 ? [Xx: $i] :
% 2.68/1.53 ( ( Xphi @ Xx )
% 2.68/1.53 & ! [Xy: $i] :
% 2.68/1.53 ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setextAx_type,type,
% 2.68/1.53 setextAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setextAx,definition,
% 2.68/1.53 ( setextAx
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 <=> ( in @ Xx @ B ) )
% 2.68/1.53 => ( A = B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptyset_type,type,
% 2.68/1.53 emptyset: $i ).
% 2.68/1.53
% 2.68/1.53 thf(emptysetAx_type,type,
% 2.68/1.53 emptysetAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptysetAx,definition,
% 2.68/1.53 ( emptysetAx
% 2.68/1.53 = ( ! [Xx: $i] :
% 2.68/1.53 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoin_type,type,
% 2.68/1.53 setadjoin: $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinAx_type,type,
% 2.68/1.53 setadjoinAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinAx,definition,
% 2.68/1.53 ( setadjoinAx
% 2.68/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.68/1.53 <=> ( ( Xy = Xx )
% 2.68/1.53 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powerset_type,type,
% 2.68/1.53 powerset: $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(powersetAx_type,type,
% 2.68/1.53 powersetAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetAx,definition,
% 2.68/1.53 ( powersetAx
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.68/1.53 <=> ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setunion_type,type,
% 2.68/1.53 setunion: $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(setunionAx_type,type,
% 2.68/1.53 setunionAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setunionAx,definition,
% 2.68/1.53 ( setunionAx
% 2.68/1.53 = ( ! [A: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.68/1.53 <=> ? [B: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 & ( in @ B @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(omega_type,type,
% 2.68/1.53 omega: $i ).
% 2.68/1.53
% 2.68/1.53 thf(omega0Ax_type,type,
% 2.68/1.53 omega0Ax: $o ).
% 2.68/1.53
% 2.68/1.53 thf(omega0Ax,definition,
% 2.68/1.53 ( omega0Ax
% 2.68/1.53 = ( in @ emptyset @ omega ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(omegaSAx_type,type,
% 2.68/1.53 omegaSAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(omegaSAx,definition,
% 2.68/1.53 ( omegaSAx
% 2.68/1.53 = ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ omega )
% 2.68/1.53 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(omegaIndAx_type,type,
% 2.68/1.53 omegaIndAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(omegaIndAx,definition,
% 2.68/1.53 ( omegaIndAx
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ( ( in @ emptyset @ A )
% 2.68/1.53 & ! [Xx: $i] :
% 2.68/1.53 ( ( ( in @ Xx @ omega )
% 2.68/1.53 & ( in @ Xx @ A ) )
% 2.68/1.53 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ omega )
% 2.68/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(replAx_type,type,
% 2.68/1.53 replAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(replAx,definition,
% 2.68/1.53 ( replAx
% 2.68/1.53 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( exu
% 2.68/1.53 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.68/1.53 => ? [B: $i] :
% 2.68/1.53 ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 <=> ? [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(foundationAx_type,type,
% 2.68/1.53 foundationAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(foundationAx,definition,
% 2.68/1.53 ( foundationAx
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.68/1.53 => ? [B: $i] :
% 2.68/1.53 ( ( in @ B @ A )
% 2.68/1.53 & ~ ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(wellorderingAx_type,type,
% 2.68/1.53 wellorderingAx: $o ).
% 2.68/1.53
% 2.68/1.53 thf(wellorderingAx,definition,
% 2.68/1.53 ( wellorderingAx
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ? [B: $i] :
% 2.68/1.53 ( ! [C: $i] :
% 2.68/1.53 ( ( in @ C @ B )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ C )
% 2.68/1.53 => ( in @ Xx @ A ) ) )
% 2.68/1.53 & ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( ( in @ Xx @ A )
% 2.68/1.53 & ( in @ Xy @ A ) )
% 2.68/1.53 => ( ! [C: $i] :
% 2.68/1.53 ( ( in @ C @ B )
% 2.68/1.53 => ( ( in @ Xx @ C )
% 2.68/1.53 <=> ( in @ Xy @ C ) ) )
% 2.68/1.53 => ( Xx = Xy ) ) )
% 2.68/1.53 & ! [C: $i,D: $i] :
% 2.68/1.53 ( ( ( in @ C @ B )
% 2.68/1.53 & ( in @ D @ B ) )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ C )
% 2.68/1.53 => ( in @ Xx @ D ) )
% 2.68/1.53 | ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ D )
% 2.68/1.53 => ( in @ Xx @ C ) ) ) )
% 2.68/1.53 & ! [C: $i] :
% 2.68/1.53 ( ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ C )
% 2.68/1.53 => ( in @ Xx @ A ) )
% 2.68/1.53 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.68/1.53 => ? [D: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ D @ B )
% 2.68/1.53 & ( in @ Xx @ C )
% 2.68/1.53 & ~ ? [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ D )
% 2.68/1.53 & ( in @ Xy @ C ) )
% 2.68/1.53 & ! [E: $i] :
% 2.68/1.53 ( ( in @ E @ B )
% 2.68/1.53 => ( ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ E )
% 2.68/1.53 => ( in @ Xy @ D ) )
% 2.68/1.53 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(descr_type,type,
% 2.68/1.53 descr: ( $i > $o ) > $i ).
% 2.68/1.53
% 2.68/1.53 thf(descrp_type,type,
% 2.68/1.53 descrp: $o ).
% 2.68/1.53
% 2.68/1.53 thf(descrp,definition,
% 2.68/1.53 ( descrp
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ( Xphi
% 2.68/1.53 @ ( descr
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstr_type,type,
% 2.68/1.53 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstrI_type,type,
% 2.68/1.53 dsetconstrI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstrI,definition,
% 2.68/1.53 ( dsetconstrI
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 => ( in @ Xx
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstrEL_type,type,
% 2.68/1.53 dsetconstrEL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstrEL,definition,
% 2.68/1.53 ( dsetconstrEL
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.68/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstrER_type,type,
% 2.68/1.53 dsetconstrER: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstrER,definition,
% 2.68/1.53 ( dsetconstrER
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.68/1.53 => ( Xphi @ Xx ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exuE1_type,type,
% 2.68/1.53 exuE1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exuE1,definition,
% 2.68/1.53 ( exuE1
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ? [Xx: $i] :
% 2.68/1.53 ( ( Xphi @ Xx )
% 2.68/1.53 & ! [Xy: $i] :
% 2.68/1.53 ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(prop2set_type,type,
% 2.68/1.53 prop2set: $o > $i ).
% 2.68/1.53
% 2.68/1.53 thf(prop2setE_type,type,
% 2.68/1.53 prop2setE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(prop2setE,definition,
% 2.68/1.53 ( prop2setE
% 2.68/1.53 = ( ! [Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.68/1.53 => Xphi ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptysetE_type,type,
% 2.68/1.53 emptysetE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptysetE,definition,
% 2.68/1.53 ( emptysetE
% 2.68/1.53 = ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ emptyset )
% 2.68/1.53 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptysetimpfalse_type,type,
% 2.68/1.53 emptysetimpfalse: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptysetimpfalse,definition,
% 2.68/1.53 ( emptysetimpfalse
% 2.68/1.53 = ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ emptyset )
% 2.68/1.53 => $false ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(notinemptyset_type,type,
% 2.68/1.53 notinemptyset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(notinemptyset,definition,
% 2.68/1.53 ( notinemptyset
% 2.68/1.53 = ( ! [Xx: $i] :
% 2.68/1.53 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exuE3e_type,type,
% 2.68/1.53 exuE3e: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exuE3e,definition,
% 2.68/1.53 ( exuE3e
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setext_type,type,
% 2.68/1.53 setext: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setext,definition,
% 2.68/1.53 ( setext
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ Xx @ B ) )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ A ) )
% 2.68/1.53 => ( A = B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptyI_type,type,
% 2.68/1.53 emptyI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptyI,definition,
% 2.68/1.53 ( emptyI
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ~ ( in @ Xx @ A )
% 2.68/1.53 => ( A = emptyset ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(noeltsimpempty_type,type,
% 2.68/1.53 noeltsimpempty: $o ).
% 2.68/1.53
% 2.68/1.53 thf(noeltsimpempty,definition,
% 2.68/1.53 ( noeltsimpempty
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ~ ( in @ Xx @ A )
% 2.68/1.53 => ( A = emptyset ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setbeta_type,type,
% 2.68/1.53 setbeta: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setbeta,definition,
% 2.68/1.53 ( setbeta
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.68/1.53 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(nonempty_type,type,
% 2.68/1.53 nonempty: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(nonempty,definition,
% 2.68/1.53 ( nonempty
% 2.68/1.53 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(nonemptyE1_type,type,
% 2.68/1.53 nonemptyE1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(nonemptyE1,definition,
% 2.68/1.53 ( nonemptyE1
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ( nonempty @ A )
% 2.68/1.53 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(nonemptyI_type,type,
% 2.68/1.53 nonemptyI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(nonemptyI,definition,
% 2.68/1.53 ( nonemptyI
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 => ( nonempty
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(nonemptyI1_type,type,
% 2.68/1.53 nonemptyI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(nonemptyI1,definition,
% 2.68/1.53 ( nonemptyI1
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.68/1.53 => ( nonempty @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinIL_type,type,
% 2.68/1.53 setadjoinIL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinIL,definition,
% 2.68/1.53 ( setadjoinIL
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptyinunitempty_type,type,
% 2.68/1.53 emptyinunitempty: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptyinunitempty,definition,
% 2.68/1.53 ( emptyinunitempty
% 2.68/1.53 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinIR_type,type,
% 2.68/1.53 setadjoinIR: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinIR,definition,
% 2.68/1.53 ( setadjoinIR
% 2.68/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinE_type,type,
% 2.68/1.53 setadjoinE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinE,definition,
% 2.68/1.53 ( setadjoinE
% 2.68/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.68/1.53 => ! [Xphi: $o] :
% 2.68/1.53 ( ( ( Xy = Xx )
% 2.68/1.53 => Xphi )
% 2.68/1.53 => ( ( ( in @ Xy @ A )
% 2.68/1.53 => Xphi )
% 2.68/1.53 => Xphi ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinOr_type,type,
% 2.68/1.53 setadjoinOr: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinOr,definition,
% 2.68/1.53 ( setadjoinOr
% 2.68/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.68/1.53 => ( ( Xy = Xx )
% 2.68/1.53 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setoftrueEq_type,type,
% 2.68/1.53 setoftrueEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setoftrueEq,definition,
% 2.68/1.53 ( setoftrueEq
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xx: $i] : $true )
% 2.68/1.53 = A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powersetI_type,type,
% 2.68/1.53 powersetI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetI,definition,
% 2.68/1.53 ( powersetI
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ A ) )
% 2.68/1.53 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptyinPowerset_type,type,
% 2.68/1.53 emptyinPowerset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptyinPowerset,definition,
% 2.68/1.53 ( emptyinPowerset
% 2.68/1.53 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptyInPowerset_type,type,
% 2.68/1.53 emptyInPowerset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptyInPowerset,definition,
% 2.68/1.53 ( emptyInPowerset
% 2.68/1.53 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powersetE_type,type,
% 2.68/1.53 powersetE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetE,definition,
% 2.68/1.53 ( powersetE
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.68/1.53 => ( ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setunionI_type,type,
% 2.68/1.53 setunionI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setunionI,definition,
% 2.68/1.53 ( setunionI
% 2.68/1.53 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 => ( ( in @ B @ A )
% 2.68/1.53 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setunionE_type,type,
% 2.68/1.53 setunionE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setunionE,definition,
% 2.68/1.53 ( setunionE
% 2.68/1.53 = ( ! [A: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.68/1.53 => ! [Xphi: $o] :
% 2.68/1.53 ( ! [B: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 => ( ( in @ B @ A )
% 2.68/1.53 => Xphi ) )
% 2.68/1.53 => Xphi ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subPowSU_type,type,
% 2.68/1.53 subPowSU: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subPowSU,definition,
% 2.68/1.53 ( subPowSU
% 2.68/1.53 = ( ! [A: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exuE2_type,type,
% 2.68/1.53 exuE2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exuE2,definition,
% 2.68/1.53 ( exuE2
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ? [Xx: $i] :
% 2.68/1.53 ! [Xy: $i] :
% 2.68/1.53 ( ( Xphi @ Xy )
% 2.68/1.53 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(nonemptyImpWitness_type,type,
% 2.68/1.53 nonemptyImpWitness: $o ).
% 2.68/1.53
% 2.68/1.53 thf(nonemptyImpWitness,definition,
% 2.68/1.53 ( nonemptyImpWitness
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ( nonempty @ A )
% 2.68/1.53 => ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & $true ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(uniqinunit_type,type,
% 2.68/1.53 uniqinunit: $o ).
% 2.68/1.53
% 2.68/1.53 thf(uniqinunit,definition,
% 2.68/1.53 ( uniqinunit
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(notinsingleton_type,type,
% 2.68/1.53 notinsingleton: $o ).
% 2.68/1.53
% 2.68/1.53 thf(notinsingleton,definition,
% 2.68/1.53 ( notinsingleton
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx != Xy )
% 2.68/1.53 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(eqinunit_type,type,
% 2.68/1.53 eqinunit: $o ).
% 2.68/1.53
% 2.68/1.53 thf(eqinunit,definition,
% 2.68/1.53 ( eqinunit
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx = Xy )
% 2.68/1.53 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(singletonsswitch_type,type,
% 2.68/1.53 singletonsswitch: $o ).
% 2.68/1.53
% 2.68/1.53 thf(singletonsswitch,definition,
% 2.68/1.53 ( singletonsswitch
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.53 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(upairsetE_type,type,
% 2.68/1.53 upairsetE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairsetE,definition,
% 2.68/1.53 ( upairsetE
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.68/1.53 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.68/1.53 => ( ( Xz = Xx )
% 2.68/1.53 | ( Xz = Xy ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(upairsetIL_type,type,
% 2.68/1.53 upairsetIL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairsetIL,definition,
% 2.68/1.53 ( upairsetIL
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(upairsetIR_type,type,
% 2.68/1.53 upairsetIR: $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairsetIR,definition,
% 2.68/1.53 ( upairsetIR
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptyE1_type,type,
% 2.68/1.53 emptyE1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptyE1,definition,
% 2.68/1.53 ( emptyE1
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ( Xphi @ Xx ) )
% 2.68/1.53 => ( ( ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 = emptyset )
% 2.68/1.53 => $false ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(vacuousDall_type,type,
% 2.68/1.53 vacuousDall: $o ).
% 2.68/1.53
% 2.68/1.53 thf(vacuousDall,definition,
% 2.68/1.53 ( vacuousDall
% 2.68/1.53 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ emptyset )
% 2.68/1.53 => ( Xphi @ Xx ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(quantDeMorgan1_type,type,
% 2.68/1.53 quantDeMorgan1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(quantDeMorgan1,definition,
% 2.68/1.53 ( quantDeMorgan1
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ~ ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( Xphi @ Xx ) )
% 2.68/1.53 => ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(quantDeMorgan2_type,type,
% 2.68/1.53 quantDeMorgan2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(quantDeMorgan2,definition,
% 2.68/1.53 ( quantDeMorgan2
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ~ ( Xphi @ Xx ) )
% 2.68/1.53 => ~ ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(quantDeMorgan3_type,type,
% 2.68/1.53 quantDeMorgan3: $o ).
% 2.68/1.53
% 2.68/1.53 thf(quantDeMorgan3,definition,
% 2.68/1.53 ( quantDeMorgan3
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ~ ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ( Xphi @ Xx ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(quantDeMorgan4_type,type,
% 2.68/1.53 quantDeMorgan4: $o ).
% 2.68/1.53
% 2.68/1.53 thf(quantDeMorgan4,definition,
% 2.68/1.53 ( quantDeMorgan4
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ~ ( Xphi @ Xx ) )
% 2.68/1.53 => ~ ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(prop2setI_type,type,
% 2.68/1.53 prop2setI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(prop2setI,definition,
% 2.68/1.53 ( prop2setI
% 2.68/1.53 = ( ! [Xphi: $o] :
% 2.68/1.53 ( Xphi
% 2.68/1.53 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(set2prop_type,type,
% 2.68/1.53 set2prop: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(prop2set2propI_type,type,
% 2.68/1.53 prop2set2propI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(prop2set2propI,definition,
% 2.68/1.53 ( prop2set2propI
% 2.68/1.53 = ( ! [Xphi: $o] :
% 2.68/1.53 ( Xphi
% 2.68/1.53 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(notdexE_type,type,
% 2.68/1.53 notdexE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(notdexE,definition,
% 2.68/1.53 ( notdexE
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ~ ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ( Xphi @ Xx ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(notdallE_type,type,
% 2.68/1.53 notdallE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(notdallE,definition,
% 2.68/1.53 ( notdallE
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ~ ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( Xphi @ Xx ) )
% 2.68/1.53 => ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exuI1_type,type,
% 2.68/1.53 exuI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exuI1,definition,
% 2.68/1.53 ( exuI1
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ? [Xx: $i] :
% 2.68/1.53 ( ( Xphi @ Xx )
% 2.68/1.53 & ! [Xy: $i] :
% 2.68/1.53 ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) )
% 2.68/1.53 => ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exuI3_type,type,
% 2.68/1.53 exuI3: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exuI3,definition,
% 2.68/1.53 ( exuI3
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.68/1.53 => ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xphi @ Xx )
% 2.68/1.53 => ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) )
% 2.68/1.53 => ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exuI2_type,type,
% 2.68/1.53 exuI2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exuI2,definition,
% 2.68/1.53 ( exuI2
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ? [Xx: $i] :
% 2.68/1.53 ! [Xy: $i] :
% 2.68/1.53 ( ( Xphi @ Xy )
% 2.68/1.53 <=> ( Xy = Xx ) )
% 2.68/1.53 => ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(inCongP_type,type,
% 2.68/1.53 inCongP: $o ).
% 2.68/1.53
% 2.68/1.53 thf(inCongP,definition,
% 2.68/1.53 ( inCongP
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( A = B )
% 2.68/1.53 => ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx = Xy )
% 2.68/1.53 => ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(in__Cong_type,type,
% 2.68/1.53 in__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(in__Cong,definition,
% 2.68/1.53 ( in__Cong
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( A = B )
% 2.68/1.53 => ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx = Xy )
% 2.68/1.53 => ( ( in @ Xx @ A )
% 2.68/1.53 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exuE3u_type,type,
% 2.68/1.53 exuE3u: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exuE3u,definition,
% 2.68/1.53 ( exuE3u
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xphi @ Xx )
% 2.68/1.53 => ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exu__Cong_type,type,
% 2.68/1.53 exu__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exu__Cong,definition,
% 2.68/1.53 ( exu__Cong
% 2.68/1.53 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.68/1.53 ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx = Xy )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 <=> ( Xpsi @ Xy ) ) )
% 2.68/1.53 => ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 <=> ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptyset__Cong_type,type,
% 2.68/1.53 emptyset__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptyset__Cong,definition,
% 2.68/1.53 ( emptyset__Cong
% 2.68/1.53 = ( emptyset = emptyset ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoin__Cong_type,type,
% 2.68/1.53 setadjoin__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoin__Cong,definition,
% 2.68/1.53 ( setadjoin__Cong
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx = Xy )
% 2.68/1.53 => ! [Xz: $i,Xu: $i] :
% 2.68/1.53 ( ( Xz = Xu )
% 2.68/1.53 => ( ( setadjoin @ Xx @ Xz )
% 2.68/1.53 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powerset__Cong_type,type,
% 2.68/1.53 powerset__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powerset__Cong,definition,
% 2.68/1.53 ( powerset__Cong
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( A = B )
% 2.68/1.53 => ( ( powerset @ A )
% 2.68/1.53 = ( powerset @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setunion__Cong_type,type,
% 2.68/1.53 setunion__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setunion__Cong,definition,
% 2.68/1.53 ( setunion__Cong
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( A = B )
% 2.68/1.53 => ( ( setunion @ A )
% 2.68/1.53 = ( setunion @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(omega__Cong_type,type,
% 2.68/1.53 omega__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(omega__Cong,definition,
% 2.68/1.53 ( omega__Cong
% 2.68/1.53 = ( omega = omega ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(exuEu_type,type,
% 2.68/1.53 exuEu: $o ).
% 2.68/1.53
% 2.68/1.53 thf(exuEu,definition,
% 2.68/1.53 ( exuEu
% 2.68/1.53 = ( ! [Xphi: $i > $o] :
% 2.68/1.53 ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xphi @ Xx )
% 2.68/1.53 => ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(descr__Cong_type,type,
% 2.68/1.53 descr__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(descr__Cong,definition,
% 2.68/1.53 ( descr__Cong
% 2.68/1.53 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.68/1.53 ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx = Xy )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 <=> ( Xpsi @ Xy ) ) )
% 2.68/1.53 => ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ( ( exu
% 2.68/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.68/1.53 => ( ( descr
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 = ( descr
% 2.68/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstr__Cong_type,type,
% 2.68/1.53 dsetconstr__Cong: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dsetconstr__Cong,definition,
% 2.68/1.53 ( dsetconstr__Cong
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( A = B )
% 2.68/1.53 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( Xx = Xy )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.68/1.53 => ( ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 = ( dsetconstr @ B
% 2.68/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subset_type,type,
% 2.68/1.53 subset: $i > $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(disjoint_type,type,
% 2.68/1.53 disjoint: $i > $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(setsmeet_type,type,
% 2.68/1.53 setsmeet: $i > $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetI1_type,type,
% 2.68/1.53 subsetI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetI1,definition,
% 2.68/1.53 ( subsetI1
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ Xx @ B ) )
% 2.68/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(eqimpsubset2_type,type,
% 2.68/1.53 eqimpsubset2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(eqimpsubset2,definition,
% 2.68/1.53 ( eqimpsubset2
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( A = B )
% 2.68/1.53 => ( subset @ B @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(eqimpsubset1_type,type,
% 2.68/1.53 eqimpsubset1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(eqimpsubset1,definition,
% 2.68/1.53 ( eqimpsubset1
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( A = B )
% 2.68/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subsetI2_type,type,
% 2.68/1.53 subsetI2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetI2,definition,
% 2.68/1.53 ( subsetI2
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ Xx @ B ) )
% 2.68/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(emptysetsubset_type,type,
% 2.68/1.53 emptysetsubset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(emptysetsubset,definition,
% 2.68/1.53 ( emptysetsubset
% 2.68/1.53 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subsetE_type,type,
% 2.68/1.53 subsetE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetE,definition,
% 2.68/1.53 ( subsetE
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subsetE2_type,type,
% 2.68/1.53 subsetE2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetE2,definition,
% 2.68/1.53 ( subsetE2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( ~ ( in @ Xx @ B )
% 2.68/1.53 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(notsubsetI_type,type,
% 2.68/1.53 notsubsetI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(notsubsetI,definition,
% 2.68/1.53 ( notsubsetI
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ B )
% 2.68/1.53 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(notequalI1_type,type,
% 2.68/1.53 notequalI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(notequalI1,definition,
% 2.68/1.53 ( notequalI1
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ~ ( subset @ A @ B )
% 2.68/1.53 => ( A != B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(notequalI2_type,type,
% 2.68/1.53 notequalI2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(notequalI2,definition,
% 2.68/1.53 ( notequalI2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ B )
% 2.68/1.53 => ( A != B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subsetRefl_type,type,
% 2.68/1.53 subsetRefl: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetRefl,definition,
% 2.68/1.53 ( subsetRefl
% 2.68/1.53 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subsetTrans_type,type,
% 2.68/1.53 subsetTrans: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetTrans,definition,
% 2.68/1.53 ( subsetTrans
% 2.68/1.53 = ( ! [A: $i,B: $i,C: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( ( subset @ B @ C )
% 2.68/1.53 => ( subset @ A @ C ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinSub_type,type,
% 2.68/1.53 setadjoinSub: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinSub,definition,
% 2.68/1.53 ( setadjoinSub
% 2.68/1.53 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinSub2_type,type,
% 2.68/1.53 setadjoinSub2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setadjoinSub2,definition,
% 2.68/1.53 ( setadjoinSub2
% 2.68/1.53 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subset2powerset_type,type,
% 2.68/1.53 subset2powerset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subset2powerset,definition,
% 2.68/1.53 ( subset2powerset
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setextsub_type,type,
% 2.68/1.53 setextsub: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setextsub,definition,
% 2.68/1.53 ( setextsub
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( ( subset @ B @ A )
% 2.68/1.53 => ( A = B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subsetemptysetimpeq_type,type,
% 2.68/1.53 subsetemptysetimpeq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetemptysetimpeq,definition,
% 2.68/1.53 ( subsetemptysetimpeq
% 2.68/1.53 = ( ! [A: $i] :
% 2.68/1.53 ( ( subset @ A @ emptyset )
% 2.68/1.53 => ( A = emptyset ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powersetI1_type,type,
% 2.68/1.53 powersetI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetI1,definition,
% 2.68/1.53 ( powersetI1
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( subset @ B @ A )
% 2.68/1.53 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powersetE1_type,type,
% 2.68/1.53 powersetE1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetE1,definition,
% 2.68/1.53 ( powersetE1
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.68/1.53 => ( subset @ B @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(inPowerset_type,type,
% 2.68/1.53 inPowerset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(inPowerset,definition,
% 2.68/1.53 ( inPowerset
% 2.68/1.53 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powersetsubset_type,type,
% 2.68/1.53 powersetsubset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetsubset,definition,
% 2.68/1.53 ( powersetsubset
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(sepInPowerset_type,type,
% 2.68/1.53 sepInPowerset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(sepInPowerset,definition,
% 2.68/1.53 ( sepInPowerset
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( in
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 @ ( powerset @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(sepSubset_type,type,
% 2.68/1.53 sepSubset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(sepSubset,definition,
% 2.68/1.53 ( sepSubset
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( subset
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binunion_type,type,
% 2.68/1.53 binunion: $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(binunionIL_type,type,
% 2.68/1.53 binunionIL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binunionIL,definition,
% 2.68/1.53 ( binunionIL
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(upairset2IR_type,type,
% 2.68/1.53 upairset2IR: $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairset2IR,definition,
% 2.68/1.53 ( upairset2IR
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binunionIR_type,type,
% 2.68/1.53 binunionIR: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binunionIR,definition,
% 2.68/1.53 ( binunionIR
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binunionEcases_type,type,
% 2.68/1.53 binunionEcases: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binunionEcases,definition,
% 2.68/1.53 ( binunionEcases
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.68/1.53 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.68/1.53 => ( ( ( in @ Xx @ A )
% 2.68/1.53 => Xphi )
% 2.68/1.53 => ( ( ( in @ Xx @ B )
% 2.68/1.53 => Xphi )
% 2.68/1.53 => Xphi ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binunionE_type,type,
% 2.68/1.53 binunionE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binunionE,definition,
% 2.68/1.53 ( binunionE
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.68/1.53 => ( ( in @ Xx @ A )
% 2.68/1.53 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binunionLsub_type,type,
% 2.68/1.53 binunionLsub: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binunionLsub,definition,
% 2.68/1.53 ( binunionLsub
% 2.68/1.53 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binunionRsub_type,type,
% 2.68/1.53 binunionRsub: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binunionRsub,definition,
% 2.68/1.53 ( binunionRsub
% 2.68/1.53 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersect_type,type,
% 2.68/1.53 binintersect: $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectI_type,type,
% 2.68/1.53 binintersectI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectI,definition,
% 2.68/1.53 ( binintersectI
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset5_type,type,
% 2.68/1.53 binintersectSubset5: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset5,definition,
% 2.68/1.53 ( binintersectSubset5
% 2.68/1.53 = ( ! [A: $i,B: $i,C: $i] :
% 2.68/1.53 ( ( subset @ C @ A )
% 2.68/1.53 => ( ( subset @ C @ B )
% 2.68/1.53 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectEL_type,type,
% 2.68/1.53 binintersectEL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectEL,definition,
% 2.68/1.53 ( binintersectEL
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.68/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectLsub_type,type,
% 2.68/1.53 binintersectLsub: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectLsub,definition,
% 2.68/1.53 ( binintersectLsub
% 2.68/1.53 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset2_type,type,
% 2.68/1.53 binintersectSubset2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset2,definition,
% 2.68/1.53 ( binintersectSubset2
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( ( binintersect @ A @ B )
% 2.68/1.53 = A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset3_type,type,
% 2.68/1.53 binintersectSubset3: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset3,definition,
% 2.68/1.53 ( binintersectSubset3
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( ( binintersect @ A @ B )
% 2.68/1.53 = B )
% 2.68/1.53 => ( subset @ B @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectER_type,type,
% 2.68/1.53 binintersectER: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectER,definition,
% 2.68/1.53 ( binintersectER
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.68/1.53 => ( in @ Xx @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(disjointsetsI1_type,type,
% 2.68/1.53 disjointsetsI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(disjointsetsI1,definition,
% 2.68/1.53 ( disjointsetsI1
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ~ ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ( in @ Xx @ B ) )
% 2.68/1.53 => ( ( binintersect @ A @ B )
% 2.68/1.53 = emptyset ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectRsub_type,type,
% 2.68/1.53 binintersectRsub: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectRsub,definition,
% 2.68/1.53 ( binintersectRsub
% 2.68/1.53 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset4_type,type,
% 2.68/1.53 binintersectSubset4: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset4,definition,
% 2.68/1.53 ( binintersectSubset4
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( subset @ B @ A )
% 2.68/1.53 => ( ( binintersect @ A @ B )
% 2.68/1.53 = B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset1_type,type,
% 2.68/1.53 binintersectSubset1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectSubset1,definition,
% 2.68/1.53 ( binintersectSubset1
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( ( binintersect @ A @ B )
% 2.68/1.53 = A )
% 2.68/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(bs114d_type,type,
% 2.68/1.53 bs114d: $o ).
% 2.68/1.53
% 2.68/1.53 thf(bs114d,definition,
% 2.68/1.53 ( bs114d
% 2.68/1.53 = ( ! [A: $i,B: $i,C: $i] :
% 2.68/1.53 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.68/1.53 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(regular_type,type,
% 2.68/1.53 regular: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminus_type,type,
% 2.68/1.53 setminus: $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(setminusI_type,type,
% 2.68/1.53 setminusI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusI,definition,
% 2.68/1.53 ( setminusI
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusEL_type,type,
% 2.68/1.53 setminusEL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusEL,definition,
% 2.68/1.53 ( setminusEL
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.68/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusER_type,type,
% 2.68/1.53 setminusER: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusER,definition,
% 2.68/1.53 ( setminusER
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.68/1.53 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusSubset2_type,type,
% 2.68/1.53 setminusSubset2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusSubset2,definition,
% 2.68/1.53 ( setminusSubset2
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( subset @ A @ B )
% 2.68/1.53 => ( ( setminus @ A @ B )
% 2.68/1.53 = emptyset ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusERneg_type,type,
% 2.68/1.53 setminusERneg: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusERneg,definition,
% 2.68/1.53 ( setminusERneg
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.68/1.53 => ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusELneg_type,type,
% 2.68/1.53 setminusELneg: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusELneg,definition,
% 2.68/1.53 ( setminusELneg
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.68/1.53 => ( ~ ( in @ Xx @ B )
% 2.68/1.53 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusILneg_type,type,
% 2.68/1.53 setminusILneg: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusILneg,definition,
% 2.68/1.53 ( setminusILneg
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ~ ( in @ Xx @ A )
% 2.68/1.53 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusIRneg_type,type,
% 2.68/1.53 setminusIRneg: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusIRneg,definition,
% 2.68/1.53 ( setminusIRneg
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ B )
% 2.68/1.53 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusLsub_type,type,
% 2.68/1.53 setminusLsub: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusLsub,definition,
% 2.68/1.53 ( setminusLsub
% 2.68/1.53 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusSubset1_type,type,
% 2.68/1.53 setminusSubset1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusSubset1,definition,
% 2.68/1.53 ( setminusSubset1
% 2.68/1.53 = ( ! [A: $i,B: $i] :
% 2.68/1.53 ( ( ( setminus @ A @ B )
% 2.68/1.53 = emptyset )
% 2.68/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(symdiff_type,type,
% 2.68/1.53 symdiff: $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffE_type,type,
% 2.68/1.53 symdiffE: $o ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffE,definition,
% 2.68/1.53 ( symdiffE
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.68/1.53 => ! [Xphi: $o] :
% 2.68/1.53 ( ( ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ B )
% 2.68/1.53 => Xphi ) )
% 2.68/1.53 => ( ( ~ ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ B )
% 2.68/1.53 => Xphi ) )
% 2.68/1.53 => Xphi ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffI1_type,type,
% 2.68/1.53 symdiffI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffI1,definition,
% 2.68/1.53 ( symdiffI1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffI2_type,type,
% 2.68/1.53 symdiffI2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffI2,definition,
% 2.68/1.53 ( symdiffI2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ~ ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ B )
% 2.68/1.53 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffIneg1_type,type,
% 2.68/1.53 symdiffIneg1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffIneg1,definition,
% 2.68/1.53 ( symdiffIneg1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ B )
% 2.68/1.53 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffIneg2_type,type,
% 2.68/1.53 symdiffIneg2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(symdiffIneg2,definition,
% 2.68/1.53 ( symdiffIneg2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ~ ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ B )
% 2.68/1.53 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(iskpair_type,type,
% 2.68/1.53 iskpair: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(secondinupair_type,type,
% 2.68/1.53 secondinupair: $o ).
% 2.68/1.53
% 2.68/1.53 thf(secondinupair,definition,
% 2.68/1.53 ( secondinupair
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairIL_type,type,
% 2.68/1.53 setukpairIL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairIL,definition,
% 2.68/1.53 ( setukpairIL
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairIR_type,type,
% 2.68/1.53 setukpairIR: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairIR,definition,
% 2.68/1.53 ( setukpairIR
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(kpairiskpair_type,type,
% 2.68/1.53 kpairiskpair: $o ).
% 2.68/1.53
% 2.68/1.53 thf(kpairiskpair,definition,
% 2.68/1.53 ( kpairiskpair
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(kpair_type,type,
% 2.68/1.53 kpair: $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(kpairp_type,type,
% 2.68/1.53 kpairp: $o ).
% 2.68/1.53
% 2.68/1.53 thf(kpairp,definition,
% 2.68/1.53 ( kpairp
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprod_type,type,
% 2.68/1.53 cartprod: $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(singletonsubset_type,type,
% 2.68/1.53 singletonsubset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(singletonsubset,definition,
% 2.68/1.53 ( singletonsubset
% 2.68/1.53 = ( ! [A: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(singletoninpowerset_type,type,
% 2.68/1.53 singletoninpowerset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(singletoninpowerset,definition,
% 2.68/1.53 ( singletoninpowerset
% 2.68/1.53 = ( ! [A: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(singletoninpowunion_type,type,
% 2.68/1.53 singletoninpowunion: $o ).
% 2.68/1.53
% 2.68/1.53 thf(singletoninpowunion,definition,
% 2.68/1.53 ( singletoninpowunion
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(upairset2E_type,type,
% 2.68/1.53 upairset2E: $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairset2E,definition,
% 2.68/1.53 ( upairset2E
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.68/1.53 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.68/1.53 => ( ( Xz = Xx )
% 2.68/1.53 | ( Xz = Xy ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(upairsubunion_type,type,
% 2.68/1.53 upairsubunion: $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairsubunion,definition,
% 2.68/1.53 ( upairsubunion
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(upairinpowunion_type,type,
% 2.68/1.53 upairinpowunion: $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairinpowunion,definition,
% 2.68/1.53 ( upairinpowunion
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ubforcartprodlem1_type,type,
% 2.68/1.53 ubforcartprodlem1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ubforcartprodlem1,definition,
% 2.68/1.53 ( ubforcartprodlem1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ubforcartprodlem2_type,type,
% 2.68/1.53 ubforcartprodlem2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ubforcartprodlem2,definition,
% 2.68/1.53 ( ubforcartprodlem2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ubforcartprodlem3_type,type,
% 2.68/1.53 ubforcartprodlem3: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ubforcartprodlem3,definition,
% 2.68/1.53 ( ubforcartprodlem3
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodpairin_type,type,
% 2.68/1.53 cartprodpairin: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodpairin,definition,
% 2.68/1.53 ( cartprodpairin
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodmempair1_type,type,
% 2.68/1.53 cartprodmempair1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodmempair1,definition,
% 2.68/1.53 ( cartprodmempair1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.53 => ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ? [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 & ( Xu
% 2.68/1.53 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodmempair_type,type,
% 2.68/1.53 cartprodmempair: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodmempair,definition,
% 2.68/1.53 ( cartprodmempair
% 2.68/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.53 => ( iskpair @ Xu ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setunionE2_type,type,
% 2.68/1.53 setunionE2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setunionE2,definition,
% 2.68/1.53 ( setunionE2
% 2.68/1.53 = ( ! [A: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.68/1.53 => ? [X: $i] :
% 2.68/1.53 ( ( in @ X @ A )
% 2.68/1.53 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setunionsingleton1_type,type,
% 2.68/1.53 setunionsingleton1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setunionsingleton1,definition,
% 2.68/1.53 ( setunionsingleton1
% 2.68/1.53 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setunionsingleton2_type,type,
% 2.68/1.53 setunionsingleton2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setunionsingleton2,definition,
% 2.68/1.53 ( setunionsingleton2
% 2.68/1.53 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setunionsingleton_type,type,
% 2.68/1.53 setunionsingleton: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setunionsingleton,definition,
% 2.68/1.53 ( setunionsingleton
% 2.68/1.53 = ( ! [Xx: $i] :
% 2.68/1.53 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.68/1.53 = Xx ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(singleton_type,type,
% 2.68/1.53 singleton: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(singletonprop_type,type,
% 2.68/1.53 singletonprop: $o ).
% 2.68/1.53
% 2.68/1.53 thf(singletonprop,definition,
% 2.68/1.53 ( singletonprop
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 => ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) )
% 2.68/1.53 => ( ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ( Xphi @ Xx ) )
% 2.68/1.53 => ( singleton
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ex1_type,type,
% 2.68/1.53 ex1: $i > ( $i > $o ) > $o ).
% 2.68/1.53
% 2.68/1.53 thf(ex1,definition,
% 2.68/1.53 ( ex1
% 2.68/1.53 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( singleton
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ex1E1_type,type,
% 2.68/1.53 ex1E1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ex1E1,definition,
% 2.68/1.53 ( ex1E1
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ( ex1 @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ex1I_type,type,
% 2.68/1.53 ex1I: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ex1I,definition,
% 2.68/1.53 ( ex1I
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 => ( ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xy = Xx ) ) )
% 2.68/1.53 => ( ex1 @ A
% 2.68/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ex1I2_type,type,
% 2.68/1.53 ex1I2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ex1I2,definition,
% 2.68/1.53 ( ex1I2
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 => ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) )
% 2.68/1.53 => ( ? [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 & ( Xphi @ Xx ) )
% 2.68/1.53 => ( ex1 @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(singletonsuniq_type,type,
% 2.68/1.53 singletonsuniq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(singletonsuniq,definition,
% 2.68/1.53 ( singletonsuniq
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( ( setadjoin @ Xx @ emptyset )
% 2.68/1.53 = ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(atmost1p_type,type,
% 2.68/1.53 atmost1p: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(atleast2p_type,type,
% 2.68/1.53 atleast2p: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(atmost2p_type,type,
% 2.68/1.53 atmost2p: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairsetp_type,type,
% 2.68/1.53 upairsetp: $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjL1_type,type,
% 2.68/1.53 setukpairinjL1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjL1,definition,
% 2.68/1.53 ( setukpairinjL1
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.68/1.53 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.68/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(kfstsingleton_type,type,
% 2.68/1.53 kfstsingleton: $o ).
% 2.68/1.53
% 2.68/1.53 thf(kfstsingleton,definition,
% 2.68/1.53 ( kfstsingleton
% 2.68/1.53 = ( ! [Xu: $i] :
% 2.68/1.53 ( ( iskpair @ Xu )
% 2.68/1.53 => ( singleton
% 2.68/1.53 @ ( dsetconstr @ ( setunion @ Xu )
% 2.68/1.53 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(theprop_type,type,
% 2.68/1.53 theprop: $o ).
% 2.68/1.53
% 2.68/1.53 thf(theprop,definition,
% 2.68/1.53 ( theprop
% 2.68/1.53 = ( ! [X: $i] :
% 2.68/1.53 ( ( singleton @ X )
% 2.68/1.53 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(kfst_type,type,
% 2.68/1.53 kfst: $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(kfstpairEq_type,type,
% 2.68/1.53 kfstpairEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(kfstpairEq,definition,
% 2.68/1.53 ( kfstpairEq
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.68/1.53 = Xx ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodfstin_type,type,
% 2.68/1.53 cartprodfstin: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodfstin,definition,
% 2.68/1.53 ( cartprodfstin
% 2.68/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.53 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjL2_type,type,
% 2.68/1.53 setukpairinjL2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjL2,definition,
% 2.68/1.53 ( setukpairinjL2
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.68/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.68/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjL_type,type,
% 2.68/1.53 setukpairinjL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjL,definition,
% 2.68/1.53 ( setukpairinjL
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.53 ( ( ( kpair @ Xx @ Xy )
% 2.68/1.53 = ( kpair @ Xz @ Xu ) )
% 2.68/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR11_type,type,
% 2.68/1.53 setukpairinjR11: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR11,definition,
% 2.68/1.53 ( setukpairinjR11
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx = Xy )
% 2.68/1.53 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.53 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR12_type,type,
% 2.68/1.53 setukpairinjR12: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR12,definition,
% 2.68/1.53 ( setukpairinjR12
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xx = Xy )
% 2.68/1.53 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.68/1.53 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR1_type,type,
% 2.68/1.53 setukpairinjR1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR1,definition,
% 2.68/1.53 ( setukpairinjR1
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.68/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.68/1.53 => ( ( Xz = Xu )
% 2.68/1.53 => ( Xy = Xu ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(upairequniteq_type,type,
% 2.68/1.53 upairequniteq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(upairequniteq,definition,
% 2.68/1.53 ( upairequniteq
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.68/1.53 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.53 = ( setadjoin @ Xz @ emptyset ) )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR2_type,type,
% 2.68/1.53 setukpairinjR2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR2,definition,
% 2.68/1.53 ( setukpairinjR2
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.68/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.68/1.53 => ( Xy = Xu ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR_type,type,
% 2.68/1.53 setukpairinjR: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setukpairinjR,definition,
% 2.68/1.53 ( setukpairinjR
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.53 ( ( ( kpair @ Xx @ Xy )
% 2.68/1.53 = ( kpair @ Xz @ Xu ) )
% 2.68/1.53 => ( Xy = Xu ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ksndsingleton_type,type,
% 2.68/1.53 ksndsingleton: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ksndsingleton,definition,
% 2.68/1.53 ( ksndsingleton
% 2.68/1.53 = ( ! [Xu: $i] :
% 2.68/1.53 ( ( iskpair @ Xu )
% 2.68/1.53 => ( singleton
% 2.68/1.53 @ ( dsetconstr @ ( setunion @ Xu )
% 2.68/1.53 @ ^ [Xx: $i] :
% 2.68/1.53 ( Xu
% 2.68/1.53 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ksnd_type,type,
% 2.68/1.53 ksnd: $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(ksndpairEq_type,type,
% 2.68/1.53 ksndpairEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ksndpairEq,definition,
% 2.68/1.53 ( ksndpairEq
% 2.68/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.68/1.53 = Xy ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(kpairsurjEq_type,type,
% 2.68/1.53 kpairsurjEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(kpairsurjEq,definition,
% 2.68/1.53 ( kpairsurjEq
% 2.68/1.53 = ( ! [Xu: $i] :
% 2.68/1.53 ( ( iskpair @ Xu )
% 2.68/1.53 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.68/1.53 = Xu ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodsndin_type,type,
% 2.68/1.53 cartprodsndin: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodsndin,definition,
% 2.68/1.53 ( cartprodsndin
% 2.68/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.53 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodpairmemEL_type,type,
% 2.68/1.53 cartprodpairmemEL: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodpairmemEL,definition,
% 2.68/1.53 ( cartprodpairmemEL
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.68/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodpairmemER_type,type,
% 2.68/1.53 cartprodpairmemER: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodpairmemER,definition,
% 2.68/1.53 ( cartprodpairmemER
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.68/1.53 => ( in @ Xy @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodmempaircEq_type,type,
% 2.68/1.53 cartprodmempaircEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodmempaircEq,definition,
% 2.68/1.53 ( cartprodmempaircEq
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( kpair @ Xx @ Xy )
% 2.68/1.53 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodfstpairEq_type,type,
% 2.68/1.53 cartprodfstpairEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodfstpairEq,definition,
% 2.68/1.53 ( cartprodfstpairEq
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.68/1.53 = Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodsndpairEq_type,type,
% 2.68/1.53 cartprodsndpairEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodsndpairEq,definition,
% 2.68/1.53 ( cartprodsndpairEq
% 2.68/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.68/1.53 = Xy ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodpairsurjEq_type,type,
% 2.68/1.53 cartprodpairsurjEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(cartprodpairsurjEq,definition,
% 2.68/1.53 ( cartprodpairsurjEq
% 2.68/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.53 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.68/1.53 = Xu ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(breln_type,type,
% 2.68/1.53 breln: $i > $i > $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(breln,definition,
% 2.68/1.53 ( breln
% 2.68/1.53 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstr_type,type,
% 2.68/1.53 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrI_type,type,
% 2.68/1.53 dpsetconstrI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrI,definition,
% 2.68/1.53 ( dpsetconstrI
% 2.68/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( Xphi @ Xx @ Xy )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.53 @ ( dpsetconstr @ A @ B
% 2.68/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrSub_type,type,
% 2.68/1.53 dpsetconstrSub: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrSub,definition,
% 2.68/1.53 ( dpsetconstrSub
% 2.68/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.68/1.53 ( subset
% 2.68/1.53 @ ( dpsetconstr @ A @ B
% 2.68/1.53 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.68/1.53 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setOfPairsIsBReln_type,type,
% 2.68/1.53 setOfPairsIsBReln: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setOfPairsIsBReln,definition,
% 2.68/1.53 ( setOfPairsIsBReln
% 2.68/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.68/1.53 ( breln @ A @ B
% 2.68/1.53 @ ( dpsetconstr @ A @ B
% 2.68/1.53 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrERa_type,type,
% 2.68/1.53 dpsetconstrERa: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrERa,definition,
% 2.68/1.53 ( dpsetconstrERa
% 2.68/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.53 @ ( dpsetconstr @ A @ B
% 2.68/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.68/1.53 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrEL1_type,type,
% 2.68/1.53 dpsetconstrEL1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrEL1,definition,
% 2.68/1.53 ( dpsetconstrEL1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.53 @ ( dpsetconstr @ A @ B
% 2.68/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.68/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrEL2_type,type,
% 2.68/1.53 dpsetconstrEL2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrEL2,definition,
% 2.68/1.53 ( dpsetconstrEL2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.53 @ ( dpsetconstr @ A @ B
% 2.68/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.68/1.53 => ( in @ Xy @ B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrER_type,type,
% 2.68/1.53 dpsetconstrER: $o ).
% 2.68/1.53
% 2.68/1.53 thf(dpsetconstrER,definition,
% 2.68/1.53 ( dpsetconstrER
% 2.68/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.53 @ ( dpsetconstr @ A @ B
% 2.68/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.68/1.53 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(func_type,type,
% 2.68/1.53 func: $i > $i > $i > $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcSet_type,type,
% 2.68/1.53 funcSet: $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(funcImageSingleton_type,type,
% 2.68/1.53 funcImageSingleton: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcImageSingleton,definition,
% 2.68/1.53 ( funcImageSingleton
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( singleton
% 2.68/1.53 @ ( dsetconstr @ B
% 2.68/1.53 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(apProp_type,type,
% 2.68/1.53 apProp: $o ).
% 2.68/1.53
% 2.68/1.53 thf(apProp,definition,
% 2.68/1.53 ( apProp
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in
% 2.68/1.53 @ ( setunion
% 2.68/1.53 @ ( dsetconstr @ B
% 2.68/1.53 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.68/1.53 @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ap_type,type,
% 2.68/1.53 ap: $i > $i > $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(app_type,type,
% 2.68/1.53 app: $o ).
% 2.68/1.53
% 2.68/1.53 thf(app,definition,
% 2.68/1.53 ( app
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(infuncsetfunc_type,type,
% 2.68/1.53 infuncsetfunc: $o ).
% 2.68/1.53
% 2.68/1.53 thf(infuncsetfunc,definition,
% 2.68/1.53 ( infuncsetfunc
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.53 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ap2p_type,type,
% 2.68/1.53 ap2p: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ap2p,definition,
% 2.68/1.53 ( ap2p
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(funcinfuncset_type,type,
% 2.68/1.53 funcinfuncset: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcinfuncset,definition,
% 2.68/1.53 ( funcinfuncset
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(lamProp_type,type,
% 2.68/1.53 lamProp: $o ).
% 2.68/1.53
% 2.68/1.53 thf(lamProp,definition,
% 2.68/1.53 ( lamProp
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.53 => ( func @ A @ B
% 2.68/1.53 @ ( dpsetconstr @ A @ B
% 2.68/1.53 @ ^ [Xx: $i,Xy: $i] :
% 2.68/1.53 ( ( Xf @ Xx )
% 2.68/1.53 = Xy ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(lam_type,type,
% 2.68/1.53 lam: $i > $i > ( $i > $i ) > $i ).
% 2.68/1.53
% 2.68/1.53 thf(lamp_type,type,
% 2.68/1.53 lamp: $o ).
% 2.68/1.53
% 2.68/1.53 thf(lamp,definition,
% 2.68/1.53 ( lamp
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.53 => ( func @ A @ B
% 2.68/1.53 @ ( lam @ A @ B
% 2.68/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(lam2p_type,type,
% 2.68/1.53 lam2p: $o ).
% 2.68/1.53
% 2.68/1.53 thf(lam2p,definition,
% 2.68/1.53 ( lam2p
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.53 => ( in
% 2.68/1.53 @ ( lam @ A @ B
% 2.68/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.68/1.53 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(brelnall1_type,type,
% 2.68/1.53 brelnall1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(brelnall1,definition,
% 2.68/1.53 ( brelnall1
% 2.68/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.68/1.53 ( ( breln @ A @ B @ R )
% 2.68/1.53 => ! [Xphi: $i > $o] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.68/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ R )
% 2.68/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(brelnall2_type,type,
% 2.68/1.53 brelnall2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(brelnall2,definition,
% 2.68/1.53 ( brelnall2
% 2.68/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.68/1.53 ( ( breln @ A @ B @ R )
% 2.68/1.53 => ! [Xphi: $i > $o] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.68/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ R )
% 2.68/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ex1E2_type,type,
% 2.68/1.53 ex1E2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ex1E2,definition,
% 2.68/1.53 ( ex1E2
% 2.68/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.53 ( ( ex1 @ A
% 2.68/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( ( Xphi @ Xx )
% 2.68/1.53 => ( ( Xphi @ Xy )
% 2.68/1.53 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(funcGraphProp1_type,type,
% 2.68/1.53 funcGraphProp1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcGraphProp1,definition,
% 2.68/1.53 ( funcGraphProp1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(funcGraphProp3_type,type,
% 2.68/1.53 funcGraphProp3: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcGraphProp3,definition,
% 2.68/1.53 ( funcGraphProp3
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(funcGraphProp2_type,type,
% 2.68/1.53 funcGraphProp2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcGraphProp2,definition,
% 2.68/1.53 ( funcGraphProp2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.68/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.53 = Xy ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(funcextLem_type,type,
% 2.68/1.53 funcextLem: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcextLem,definition,
% 2.68/1.53 ( funcextLem
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ! [Xg: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xg )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(funcGraphProp4_type,type,
% 2.68/1.53 funcGraphProp4: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcGraphProp4,definition,
% 2.68/1.53 ( funcGraphProp4
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.68/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.53 = Xy ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subbreln_type,type,
% 2.68/1.53 subbreln: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subbreln,definition,
% 2.68/1.53 ( subbreln
% 2.68/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.68/1.53 ( ( breln @ A @ B @ R )
% 2.68/1.53 => ! [S: $i] :
% 2.68/1.53 ( ( breln @ A @ B @ S )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.68/1.53 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(eqbreln_type,type,
% 2.68/1.53 eqbreln: $o ).
% 2.68/1.53
% 2.68/1.53 thf(eqbreln,definition,
% 2.68/1.53 ( eqbreln
% 2.68/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.68/1.53 ( ( breln @ A @ B @ R )
% 2.68/1.53 => ! [S: $i] :
% 2.68/1.53 ( ( breln @ A @ B @ S )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ B )
% 2.68/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.68/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.68/1.53 => ( R = S ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(funcext_type,type,
% 2.68/1.53 funcext: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcext,definition,
% 2.68/1.53 ( funcext
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ! [Xg: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xg )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.68/1.53 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(funcext2_type,type,
% 2.68/1.53 funcext2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(funcext2,definition,
% 2.68/1.53 ( funcext2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.53 => ! [Xg: $i] :
% 2.68/1.53 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.68/1.53 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ap2apEq1_type,type,
% 2.68/1.53 ap2apEq1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ap2apEq1,definition,
% 2.68/1.53 ( ap2apEq1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.53 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ap2apEq2_type,type,
% 2.68/1.53 ap2apEq2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ap2apEq2,definition,
% 2.68/1.53 ( ap2apEq2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.53 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(beta1_type,type,
% 2.68/1.53 beta1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(beta1,definition,
% 2.68/1.53 ( beta1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( ap @ A @ B
% 2.68/1.53 @ ( lam @ A @ B
% 2.68/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.68/1.53 @ Xx )
% 2.68/1.53 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(eta1_type,type,
% 2.68/1.53 eta1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(eta1,definition,
% 2.68/1.53 ( eta1
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( func @ A @ B @ Xf )
% 2.68/1.53 => ( ( lam @ A @ B
% 2.68/1.53 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.68/1.53 = Xf ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(lam2lamEq_type,type,
% 2.68/1.53 lam2lamEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(lam2lamEq,definition,
% 2.68/1.53 ( lam2lamEq
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.53 => ( ( lam @ A @ B
% 2.68/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.68/1.53 = ( lam @ A @ B
% 2.68/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(beta2_type,type,
% 2.68/1.53 beta2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(beta2,definition,
% 2.68/1.53 ( beta2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.53 ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( ap @ A @ B
% 2.68/1.53 @ ( lam @ A @ B
% 2.68/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.68/1.53 @ Xx )
% 2.68/1.53 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(eta2_type,type,
% 2.68/1.53 eta2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(eta2,definition,
% 2.68/1.53 ( eta2
% 2.68/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.53 => ( ( lam @ A @ B
% 2.68/1.53 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.68/1.53 = Xf ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(iffalseProp1_type,type,
% 2.68/1.53 iffalseProp1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(iffalseProp1,definition,
% 2.68/1.53 ( iffalseProp1
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( ~ Xphi
% 2.68/1.53 => ( in @ Xy
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xz: $i] :
% 2.68/1.53 ( ( Xphi
% 2.68/1.53 & ( Xz = Xx ) )
% 2.68/1.53 | ( ~ Xphi
% 2.68/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(iffalseProp2_type,type,
% 2.68/1.53 iffalseProp2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(iffalseProp2,definition,
% 2.68/1.53 ( iffalseProp2
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( ~ Xphi
% 2.68/1.53 => ( ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xz: $i] :
% 2.68/1.53 ( ( Xphi
% 2.68/1.53 & ( Xz = Xx ) )
% 2.68/1.53 | ( ~ Xphi
% 2.68/1.53 & ( Xz = Xy ) ) ) )
% 2.68/1.53 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(iftrueProp1_type,type,
% 2.68/1.53 iftrueProp1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(iftrueProp1,definition,
% 2.68/1.53 ( iftrueProp1
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( Xphi
% 2.68/1.53 => ( in @ Xx
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xz: $i] :
% 2.68/1.53 ( ( Xphi
% 2.68/1.53 & ( Xz = Xx ) )
% 2.68/1.53 | ( ~ Xphi
% 2.68/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(iftrueProp2_type,type,
% 2.68/1.53 iftrueProp2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(iftrueProp2,definition,
% 2.68/1.53 ( iftrueProp2
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( Xphi
% 2.68/1.53 => ( ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xz: $i] :
% 2.68/1.53 ( ( Xphi
% 2.68/1.53 & ( Xz = Xx ) )
% 2.68/1.53 | ( ~ Xphi
% 2.68/1.53 & ( Xz = Xy ) ) ) )
% 2.68/1.53 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(ifSingleton_type,type,
% 2.68/1.53 ifSingleton: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ifSingleton,definition,
% 2.68/1.53 ( ifSingleton
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( singleton
% 2.68/1.53 @ ( dsetconstr @ A
% 2.68/1.53 @ ^ [Xz: $i] :
% 2.68/1.53 ( ( Xphi
% 2.68/1.53 & ( Xz = Xx ) )
% 2.68/1.53 | ( ~ Xphi
% 2.68/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(if_type,type,
% 2.68/1.53 if: $i > $o > $i > $i > $i ).
% 2.68/1.53
% 2.68/1.53 thf(ifp_type,type,
% 2.68/1.53 ifp: $o ).
% 2.68/1.53
% 2.68/1.53 thf(ifp,definition,
% 2.68/1.53 ( ifp
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(theeq_type,type,
% 2.68/1.53 theeq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(theeq,definition,
% 2.68/1.53 ( theeq
% 2.68/1.53 = ( ! [X: $i] :
% 2.68/1.53 ( ( singleton @ X )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ X )
% 2.68/1.53 => ( ( setunion @ X )
% 2.68/1.53 = Xx ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(iftrue_type,type,
% 2.68/1.53 iftrue: $o ).
% 2.68/1.53
% 2.68/1.53 thf(iftrue,definition,
% 2.68/1.53 ( iftrue
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( Xphi
% 2.68/1.53 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.68/1.53 = Xx ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(iffalse_type,type,
% 2.68/1.53 iffalse: $o ).
% 2.68/1.53
% 2.68/1.53 thf(iffalse,definition,
% 2.68/1.53 ( iffalse
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( ~ Xphi
% 2.68/1.53 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.68/1.53 = Xy ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(iftrueorfalse_type,type,
% 2.68/1.53 iftrueorfalse: $o ).
% 2.68/1.53
% 2.68/1.53 thf(iftrueorfalse,definition,
% 2.68/1.53 ( iftrueorfalse
% 2.68/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ! [Xy: $i] :
% 2.68/1.53 ( ( in @ Xy @ A )
% 2.68/1.53 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectT_lem_type,type,
% 2.68/1.53 binintersectT_lem: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectT_lem,definition,
% 2.68/1.53 ( binintersectT_lem
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binunionT_lem_type,type,
% 2.68/1.53 binunionT_lem: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binunionT_lem,definition,
% 2.68/1.53 ( binunionT_lem
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powersetT_lem_type,type,
% 2.68/1.53 powersetT_lem: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetT_lem,definition,
% 2.68/1.53 ( powersetT_lem
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setminusT_lem_type,type,
% 2.68/1.53 setminusT_lem: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setminusT_lem,definition,
% 2.68/1.53 ( setminusT_lem
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(complementT_lem_type,type,
% 2.68/1.53 complementT_lem: $o ).
% 2.68/1.53
% 2.68/1.53 thf(complementT_lem,definition,
% 2.68/1.53 ( complementT_lem
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(setextT_type,type,
% 2.68/1.53 setextT: $o ).
% 2.68/1.53
% 2.68/1.53 thf(setextT,definition,
% 2.68/1.53 ( setextT
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ X )
% 2.68/1.53 => ( in @ Xx @ Y ) ) )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ Y )
% 2.68/1.53 => ( in @ Xx @ X ) ) )
% 2.68/1.53 => ( X = Y ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(subsetTI_type,type,
% 2.68/1.53 subsetTI: $o ).
% 2.68/1.53
% 2.68/1.53 thf(subsetTI,definition,
% 2.68/1.53 ( subsetTI
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ X )
% 2.68/1.53 => ( in @ Xx @ Y ) ) )
% 2.68/1.53 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powersetTI1_type,type,
% 2.68/1.53 powersetTI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetTI1,definition,
% 2.68/1.53 ( powersetTI1
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ X )
% 2.68/1.53 => ( in @ Xx @ Y ) ) )
% 2.68/1.53 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(powersetTE1_type,type,
% 2.68/1.53 powersetTE1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(powersetTE1,definition,
% 2.68/1.53 ( powersetTE1
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ X @ ( powerset @ Y ) )
% 2.68/1.53 => ( ( in @ Xx @ X )
% 2.68/1.53 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(complementTI1_type,type,
% 2.68/1.53 complementTI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(complementTI1,definition,
% 2.68/1.53 ( complementTI1
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ X )
% 2.68/1.53 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(complementTE1_type,type,
% 2.68/1.53 complementTE1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(complementTE1,definition,
% 2.68/1.53 ( complementTE1
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.68/1.53 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectTELcontra_type,type,
% 2.68/1.53 binintersectTELcontra: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectTELcontra,definition,
% 2.68/1.53 ( binintersectTELcontra
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ X )
% 2.68/1.53 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectTERcontra_type,type,
% 2.68/1.53 binintersectTERcontra: $o ).
% 2.68/1.53
% 2.68/1.53 thf(binintersectTERcontra,definition,
% 2.68/1.53 ( binintersectTERcontra
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ~ ( in @ Xx @ Y )
% 2.68/1.53 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(contrasubsetT_type,type,
% 2.68/1.53 contrasubsetT: $o ).
% 2.68/1.53
% 2.68/1.53 thf(contrasubsetT,definition,
% 2.68/1.53 ( contrasubsetT
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.68/1.53 => ( ( in @ Xx @ Y )
% 2.68/1.53 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(contrasubsetT1_type,type,
% 2.68/1.53 contrasubsetT1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(contrasubsetT1,definition,
% 2.68/1.53 ( contrasubsetT1
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( subset @ X @ Y )
% 2.68/1.53 => ( ~ ( in @ Xx @ Y )
% 2.68/1.53 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(contrasubsetT2_type,type,
% 2.68/1.53 contrasubsetT2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(contrasubsetT2,definition,
% 2.68/1.53 ( contrasubsetT2
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( ( subset @ X @ Y )
% 2.68/1.53 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(contrasubsetT3_type,type,
% 2.68/1.53 contrasubsetT3: $o ).
% 2.68/1.53
% 2.68/1.53 thf(contrasubsetT3,definition,
% 2.68/1.53 ( contrasubsetT3
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.68/1.53 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementI1_type,type,
% 2.68/1.53 doubleComplementI1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementI1,definition,
% 2.68/1.53 ( doubleComplementI1
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ X )
% 2.68/1.53 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementE1_type,type,
% 2.68/1.53 doubleComplementE1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementE1,definition,
% 2.68/1.53 ( doubleComplementE1
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.68/1.53 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementSub1_type,type,
% 2.68/1.53 doubleComplementSub1: $o ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementSub1,definition,
% 2.68/1.53 ( doubleComplementSub1
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementSub2_type,type,
% 2.68/1.53 doubleComplementSub2: $o ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementSub2,definition,
% 2.68/1.53 ( doubleComplementSub2
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementEq_type,type,
% 2.68/1.53 doubleComplementEq: $o ).
% 2.68/1.53
% 2.68/1.53 thf(doubleComplementEq,definition,
% 2.68/1.53 ( doubleComplementEq
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ( X
% 2.68/1.53 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(complementTnotintersectT_type,type,
% 2.68/1.53 complementTnotintersectT: $o ).
% 2.68/1.53
% 2.68/1.53 thf(complementTnotintersectT,definition,
% 2.68/1.53 ( complementTnotintersectT
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.68/1.53 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(complementImpComplementIntersect_type,type,
% 2.68/1.53 complementImpComplementIntersect: $o ).
% 2.68/1.53
% 2.68/1.53 thf(complementImpComplementIntersect,definition,
% 2.68/1.53 ( complementImpComplementIntersect
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.68/1.53 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(complementSubsetComplementIntersect_type,type,
% 2.68/1.53 complementSubsetComplementIntersect: $o ).
% 2.68/1.53
% 2.68/1.53 thf(complementSubsetComplementIntersect,definition,
% 2.68/1.53 ( complementSubsetComplementIntersect
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(complementInPowersetComplementIntersect_type,type,
% 2.68/1.53 complementInPowersetComplementIntersect: $o ).
% 2.68/1.53
% 2.68/1.53 thf(complementInPowersetComplementIntersect,definition,
% 2.68/1.53 ( complementInPowersetComplementIntersect
% 2.68/1.53 = ( ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(contraSubsetComplement,conjecture,
% 2.68/1.53 ( setextAx
% 2.68/1.53 => ( emptysetAx
% 2.68/1.53 => ( setadjoinAx
% 2.68/1.53 => ( powersetAx
% 2.68/1.53 => ( setunionAx
% 2.68/1.53 => ( omega0Ax
% 2.68/1.53 => ( omegaSAx
% 2.68/1.53 => ( omegaIndAx
% 2.68/1.53 => ( replAx
% 2.68/1.53 => ( foundationAx
% 2.68/1.53 => ( wellorderingAx
% 2.68/1.53 => ( descrp
% 2.68/1.53 => ( dsetconstrI
% 2.68/1.53 => ( dsetconstrEL
% 2.68/1.53 => ( dsetconstrER
% 2.68/1.53 => ( exuE1
% 2.68/1.53 => ( prop2setE
% 2.68/1.53 => ( emptysetE
% 2.68/1.53 => ( emptysetimpfalse
% 2.68/1.53 => ( notinemptyset
% 2.68/1.53 => ( exuE3e
% 2.68/1.53 => ( setext
% 2.68/1.53 => ( emptyI
% 2.68/1.53 => ( noeltsimpempty
% 2.68/1.53 => ( setbeta
% 2.68/1.53 => ( nonemptyE1
% 2.68/1.53 => ( nonemptyI
% 2.68/1.53 => ( nonemptyI1
% 2.68/1.53 => ( setadjoinIL
% 2.68/1.53 => ( emptyinunitempty
% 2.68/1.53 => ( setadjoinIR
% 2.68/1.53 => ( setadjoinE
% 2.68/1.53 => ( setadjoinOr
% 2.68/1.53 => ( setoftrueEq
% 2.68/1.53 => ( powersetI
% 2.68/1.53 => ( emptyinPowerset
% 2.68/1.53 => ( emptyInPowerset
% 2.68/1.53 => ( powersetE
% 2.68/1.53 => ( setunionI
% 2.68/1.53 => ( setunionE
% 2.68/1.53 => ( subPowSU
% 2.68/1.53 => ( exuE2
% 2.68/1.53 => ( nonemptyImpWitness
% 2.68/1.53 => ( uniqinunit
% 2.68/1.53 => ( notinsingleton
% 2.68/1.53 => ( eqinunit
% 2.68/1.53 => ( singletonsswitch
% 2.68/1.53 => ( upairsetE
% 2.68/1.53 => ( upairsetIL
% 2.68/1.53 => ( upairsetIR
% 2.68/1.53 => ( emptyE1
% 2.68/1.53 => ( vacuousDall
% 2.68/1.53 => ( quantDeMorgan1
% 2.68/1.53 => ( quantDeMorgan2
% 2.68/1.53 => ( quantDeMorgan3
% 2.68/1.53 => ( quantDeMorgan4
% 2.68/1.53 => ( prop2setI
% 2.68/1.53 => ( prop2set2propI
% 2.68/1.53 => ( notdexE
% 2.68/1.53 => ( notdallE
% 2.68/1.53 => ( exuI1
% 2.68/1.53 => ( exuI3
% 2.68/1.53 => ( exuI2
% 2.68/1.53 => ( inCongP
% 2.68/1.53 => ( in__Cong
% 2.68/1.53 => ( exuE3u
% 2.68/1.53 => ( exu__Cong
% 2.68/1.53 => ( emptyset__Cong
% 2.68/1.53 => ( setadjoin__Cong
% 2.68/1.53 => ( powerset__Cong
% 2.68/1.53 => ( setunion__Cong
% 2.68/1.53 => ( omega__Cong
% 2.68/1.53 => ( exuEu
% 2.68/1.53 => ( descr__Cong
% 2.68/1.53 => ( dsetconstr__Cong
% 2.68/1.53 => ( subsetI1
% 2.68/1.53 => ( eqimpsubset2
% 2.68/1.53 => ( eqimpsubset1
% 2.68/1.53 => ( subsetI2
% 2.68/1.53 => ( emptysetsubset
% 2.68/1.53 => ( subsetE
% 2.68/1.53 => ( subsetE2
% 2.68/1.53 => ( notsubsetI
% 2.68/1.53 => ( notequalI1
% 2.68/1.53 => ( notequalI2
% 2.68/1.53 => ( subsetRefl
% 2.68/1.53 => ( subsetTrans
% 2.68/1.53 => ( setadjoinSub
% 2.68/1.53 => ( setadjoinSub2
% 2.68/1.53 => ( subset2powerset
% 2.68/1.53 => ( setextsub
% 2.68/1.53 => ( subsetemptysetimpeq
% 2.68/1.53 => ( powersetI1
% 2.68/1.53 => ( powersetE1
% 2.68/1.53 => ( inPowerset
% 2.68/1.53 => ( powersetsubset
% 2.68/1.53 => ( sepInPowerset
% 2.68/1.53 => ( sepSubset
% 2.68/1.53 => ( binunionIL
% 2.68/1.53 => ( upairset2IR
% 2.68/1.53 => ( binunionIR
% 2.68/1.53 => ( binunionEcases
% 2.68/1.53 => ( binunionE
% 2.68/1.53 => ( binunionLsub
% 2.68/1.53 => ( binunionRsub
% 2.68/1.53 => ( binintersectI
% 2.68/1.53 => ( binintersectSubset5
% 2.68/1.53 => ( binintersectEL
% 2.68/1.53 => ( binintersectLsub
% 2.68/1.53 => ( binintersectSubset2
% 2.68/1.53 => ( binintersectSubset3
% 2.68/1.53 => ( binintersectER
% 2.68/1.53 => ( disjointsetsI1
% 2.68/1.53 => ( binintersectRsub
% 2.68/1.53 => ( binintersectSubset4
% 2.68/1.53 => ( binintersectSubset1
% 2.68/1.53 => ( bs114d
% 2.68/1.53 => ( setminusI
% 2.68/1.53 => ( setminusEL
% 2.68/1.53 => ( setminusER
% 2.68/1.53 => ( setminusSubset2
% 2.68/1.53 => ( setminusERneg
% 2.68/1.53 => ( setminusELneg
% 2.68/1.53 => ( setminusILneg
% 2.68/1.53 => ( setminusIRneg
% 2.68/1.53 => ( setminusLsub
% 2.68/1.53 => ( setminusSubset1
% 2.68/1.53 => ( symdiffE
% 2.68/1.53 => ( symdiffI1
% 2.68/1.53 => ( symdiffI2
% 2.68/1.53 => ( symdiffIneg1
% 2.68/1.53 => ( symdiffIneg2
% 2.68/1.53 => ( secondinupair
% 2.68/1.53 => ( setukpairIL
% 2.68/1.53 => ( setukpairIR
% 2.68/1.53 => ( kpairiskpair
% 2.68/1.53 => ( kpairp
% 2.68/1.53 => ( singletonsubset
% 2.68/1.53 => ( singletoninpowerset
% 2.68/1.53 => ( singletoninpowunion
% 2.68/1.53 => ( upairset2E
% 2.68/1.53 => ( upairsubunion
% 2.68/1.53 => ( upairinpowunion
% 2.68/1.53 => ( ubforcartprodlem1
% 2.68/1.53 => ( ubforcartprodlem2
% 2.68/1.53 => ( ubforcartprodlem3
% 2.68/1.53 => ( cartprodpairin
% 2.68/1.53 => ( cartprodmempair1
% 2.68/1.53 => ( cartprodmempair
% 2.68/1.53 => ( setunionE2
% 2.68/1.53 => ( setunionsingleton1
% 2.68/1.53 => ( setunionsingleton2
% 2.68/1.53 => ( setunionsingleton
% 2.68/1.53 => ( singletonprop
% 2.68/1.53 => ( ex1E1
% 2.68/1.53 => ( ex1I
% 2.68/1.53 => ( ex1I2
% 2.68/1.53 => ( singletonsuniq
% 2.68/1.53 => ( setukpairinjL1
% 2.68/1.53 => ( kfstsingleton
% 2.68/1.53 => ( theprop
% 2.68/1.53 => ( kfstpairEq
% 2.68/1.53 => ( cartprodfstin
% 2.68/1.53 => ( setukpairinjL2
% 2.68/1.53 => ( setukpairinjL
% 2.68/1.53 => ( setukpairinjR11
% 2.68/1.53 => ( setukpairinjR12
% 2.68/1.53 => ( setukpairinjR1
% 2.68/1.53 => ( upairequniteq
% 2.68/1.53 => ( setukpairinjR2
% 2.68/1.53 => ( setukpairinjR
% 2.68/1.53 => ( ksndsingleton
% 2.68/1.53 => ( ksndpairEq
% 2.68/1.53 => ( kpairsurjEq
% 2.68/1.53 => ( cartprodsndin
% 2.68/1.53 => ( cartprodpairmemEL
% 2.68/1.53 => ( cartprodpairmemER
% 2.68/1.53 => ( cartprodmempaircEq
% 2.68/1.53 => ( cartprodfstpairEq
% 2.68/1.53 => ( cartprodsndpairEq
% 2.68/1.53 => ( cartprodpairsurjEq
% 2.68/1.53 => ( dpsetconstrI
% 2.68/1.53 => ( dpsetconstrSub
% 2.68/1.53 => ( setOfPairsIsBReln
% 2.68/1.53 => ( dpsetconstrERa
% 2.68/1.53 => ( dpsetconstrEL1
% 2.68/1.53 => ( dpsetconstrEL2
% 2.68/1.53 => ( dpsetconstrER
% 2.68/1.53 => ( funcImageSingleton
% 2.68/1.53 => ( apProp
% 2.68/1.53 => ( app
% 2.68/1.53 => ( infuncsetfunc
% 2.68/1.53 => ( ap2p
% 2.68/1.53 => ( funcinfuncset
% 2.68/1.53 => ( lamProp
% 2.68/1.53 => ( lamp
% 2.68/1.53 => ( lam2p
% 2.68/1.53 => ( brelnall1
% 2.68/1.53 => ( brelnall2
% 2.68/1.53 => ( ex1E2
% 2.68/1.53 => ( funcGraphProp1
% 2.68/1.53 => ( funcGraphProp3
% 2.68/1.53 => ( funcGraphProp2
% 2.68/1.53 => ( funcextLem
% 2.68/1.53 => ( funcGraphProp4
% 2.68/1.53 => ( subbreln
% 2.68/1.53 => ( eqbreln
% 2.68/1.53 => ( funcext
% 2.68/1.53 => ( funcext2
% 2.68/1.53 => ( ap2apEq1
% 2.68/1.53 => ( ap2apEq2
% 2.68/1.53 => ( beta1
% 2.68/1.53 => ( eta1
% 2.68/1.53 => ( lam2lamEq
% 2.68/1.53 => ( beta2
% 2.68/1.53 => ( eta2
% 2.68/1.53 => ( iffalseProp1
% 2.68/1.53 => ( iffalseProp2
% 2.68/1.53 => ( iftrueProp1
% 2.68/1.53 => ( iftrueProp2
% 2.68/1.53 => ( ifSingleton
% 2.68/1.53 => ( ifp
% 2.68/1.53 => ( theeq
% 2.68/1.53 => ( iftrue
% 2.68/1.53 => ( iffalse
% 2.68/1.53 => ( iftrueorfalse
% 2.68/1.53 => ( binintersectT_lem
% 2.68/1.53 => ( binunionT_lem
% 2.68/1.53 => ( powersetT_lem
% 2.68/1.53 => ( setminusT_lem
% 2.68/1.53 => ( complementT_lem
% 2.68/1.53 => ( setextT
% 2.68/1.53 => ( subsetTI
% 2.68/1.53 => ( powersetTI1
% 2.68/1.53 => ( powersetTE1
% 2.68/1.53 => ( complementTI1
% 2.68/1.53 => ( complementTE1
% 2.68/1.53 => ( binintersectTELcontra
% 2.68/1.53 => ( binintersectTERcontra
% 2.68/1.53 => ( contrasubsetT
% 2.68/1.53 => ( contrasubsetT1
% 2.68/1.53 => ( contrasubsetT2
% 2.68/1.53 => ( contrasubsetT3
% 2.68/1.53 => ( doubleComplementI1
% 2.68/1.53 => ( doubleComplementE1
% 2.68/1.53 => ( doubleComplementSub1
% 2.68/1.53 => ( doubleComplementSub2
% 2.68/1.53 => ( doubleComplementEq
% 2.68/1.53 => ( complementTnotintersectT
% 2.68/1.53 => ( complementImpComplementIntersect
% 2.68/1.53 => ( complementSubsetComplementIntersect
% 2.68/1.53 => ( complementInPowersetComplementIntersect
% 2.68/1.53 => ! [A: $i,X: $i] :
% 2.68/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.68/1.53 => ! [Y: $i] :
% 2.68/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.68/1.53 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.68/1.53 => ! [Xx: $i] :
% 2.68/1.53 ( ( in @ Xx @ A )
% 2.68/1.53 => ( ( in @ Xx @ Y )
% 2.68/1.53 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.68/1.53 ------------------------
% 2.68/1.57 % (10869)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_10607 for (2999ds/27Mi)
% 2.68/1.57 % (10870)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/2Mi)
% 2.68/1.57 % (10870)Instruction limit reached!
% 2.68/1.57 % (10870)------------------------------
% 2.68/1.57 % (10870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (10870)Termination reason: Unknown
% 2.68/1.57 % (10870)Termination phase: shuffling
% 2.68/1.57
% 2.68/1.57 % (10870)Memory used [KB]: 1535
% 2.68/1.57 % (10870)Time elapsed: 0.003 s
% 2.68/1.57 % (10870)Instructions burned: 4 (million)
% 2.68/1.57 % (10870)------------------------------
% 2.68/1.57 % (10870)------------------------------
% 2.68/1.57 % (10871)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_10607 for (2999ds/2Mi)
% 2.68/1.57 % (10871)Instruction limit reached!
% 2.68/1.57 % (10871)------------------------------
% 2.68/1.57 % (10871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (10871)Termination reason: Unknown
% 2.68/1.57 % (10871)Termination phase: shuffling
% 2.68/1.57
% 2.68/1.57 % (10871)Memory used [KB]: 1535
% 2.68/1.57 % (10871)Time elapsed: 0.003 s
% 2.68/1.57 % (10871)Instructions burned: 5 (million)
% 2.68/1.57 % (10871)------------------------------
% 2.68/1.57 % (10871)------------------------------
% 2.68/1.57 % (10869)Instruction limit reached!
% 2.68/1.57 % (10869)------------------------------
% 2.68/1.57 % (10869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (10869)Termination reason: Unknown
% 2.68/1.57 % (10869)Termination phase: shuffling
% 2.68/1.57
% 2.68/1.57 % (10869)Memory used [KB]: 1918
% 2.68/1.57 % (10869)Time elapsed: 0.009 s
% 2.68/1.57 % (10869)Instructions burned: 27 (million)
% 2.68/1.57 % (10869)------------------------------
% 2.68/1.57 % (10869)------------------------------
% 2.68/1.57 % (10868)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_10607 for (2999ds/4Mi)
% 2.68/1.57 % (10867)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/183Mi)
% 2.68/1.57 % (10872)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_10607 for (2999ds/275Mi)
% 2.68/1.57 % (10873)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/18Mi)
% 2.68/1.58 % (10868)Instruction limit reached!
% 2.68/1.58 % (10868)------------------------------
% 2.68/1.58 % (10868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (10868)Termination reason: Unknown
% 2.68/1.58 % (10868)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (10868)Memory used [KB]: 1535
% 2.68/1.58 % (10868)Time elapsed: 0.004 s
% 2.68/1.58 % (10868)Instructions burned: 5 (million)
% 2.68/1.58 % (10868)------------------------------
% 2.68/1.58 % (10868)------------------------------
% 2.68/1.58 % (10880)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/3Mi)
% 2.68/1.58 % (10880)Instruction limit reached!
% 2.68/1.58 % (10880)------------------------------
% 2.68/1.58 % (10880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (10880)Termination reason: Unknown
% 2.68/1.58 % (10880)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (10880)Memory used [KB]: 1535
% 2.68/1.58 % (10880)Time elapsed: 0.003 s
% 2.68/1.58 % (10880)Instructions burned: 5 (million)
% 2.68/1.58 % (10880)------------------------------
% 2.68/1.58 % (10880)------------------------------
% 2.68/1.58 % (10881)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_10607 for (2999ds/37Mi)
% 2.68/1.58 % (10882)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_10607 for (2999ds/15Mi)
% 2.68/1.59 % (10873)Instruction limit reached!
% 2.68/1.59 % (10873)------------------------------
% 2.68/1.59 % (10873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.59 % (10873)Termination reason: Unknown
% 2.68/1.59 % (10873)Termination phase: shuffling
% 2.68/1.59
% 2.68/1.59 % (10873)Memory used [KB]: 1791
% 2.68/1.59 % (10873)Time elapsed: 0.013 s
% 2.68/1.59 % (10873)Instructions burned: 19 (million)
% 2.68/1.59 % (10873)------------------------------
% 2.68/1.59 % (10873)------------------------------
% 2.68/1.59 % (10882)Instruction limit reached!
% 2.68/1.59 % (10882)------------------------------
% 2.68/1.59 % (10882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.59 % (10882)Termination reason: Unknown
% 2.68/1.59 % (10882)Termination phase: shuffling
% 2.68/1.59
% 2.68/1.59 % (10882)Memory used [KB]: 1791
% 2.68/1.59 % (10882)Time elapsed: 0.007 s
% 2.68/1.59 % (10882)Instructions burned: 17 (million)
% 2.68/1.59 % (10882)------------------------------
% 2.68/1.59 % (10882)------------------------------
% 2.68/1.59 % (10884)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_10607 for (2999ds/1041Mi)
% 2.68/1.59 % (10881)Instruction limit reached!
% 2.68/1.59 % (10881)------------------------------
% 2.68/1.59 % (10881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.59 % (10881)Termination reason: Unknown
% 2.68/1.59 % (10881)Termination phase: shuffling
% 2.68/1.59
% 2.68/1.59 % (10881)Memory used [KB]: 2302
% 2.68/1.59 % (10881)Time elapsed: 0.012 s
% 2.68/1.59 % (10881)Instructions burned: 39 (million)
% 2.68/1.59 % (10881)------------------------------
% 2.68/1.59 % (10881)------------------------------
% 2.68/1.59 % (10883)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_10607 for (2999ds/3Mi)
% 2.68/1.60 % (10883)Instruction limit reached!
% 2.68/1.60 % (10883)------------------------------
% 2.68/1.60 % (10883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.60 % (10883)Termination reason: Unknown
% 2.68/1.60 % (10883)Termination phase: shuffling
% 2.68/1.60
% 2.68/1.60 % (10883)Memory used [KB]: 1535
% 2.68/1.60 % (10883)Time elapsed: 0.004 s
% 2.68/1.60 % (10883)Instructions burned: 3 (million)
% 2.68/1.60 % (10883)------------------------------
% 2.68/1.60 % (10883)------------------------------
% 2.73/1.60 % (10894)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/16Mi)
% 2.73/1.60 % (10897)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_10607 for (2999ds/3Mi)
% 2.73/1.60 % (10897)Instruction limit reached!
% 2.73/1.60 % (10897)------------------------------
% 2.73/1.60 % (10897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (10897)Termination reason: Unknown
% 2.73/1.60 % (10897)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (10897)Memory used [KB]: 1535
% 2.73/1.60 % (10897)Time elapsed: 0.003 s
% 2.73/1.60 % (10897)Instructions burned: 6 (million)
% 2.73/1.60 % (10897)------------------------------
% 2.73/1.60 % (10897)------------------------------
% 2.73/1.60 % (10894)Instruction limit reached!
% 2.73/1.60 % (10894)------------------------------
% 2.73/1.60 % (10894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (10894)Termination reason: Unknown
% 2.73/1.60 % (10894)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (10894)Memory used [KB]: 1791
% 2.73/1.60 % (10894)Time elapsed: 0.006 s
% 2.73/1.60 % (10894)Instructions burned: 17 (million)
% 2.73/1.60 % (10894)------------------------------
% 2.73/1.60 % (10894)------------------------------
% 2.73/1.60 % (10890)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/7Mi)
% 2.73/1.61 % (10890)Instruction limit reached!
% 2.73/1.61 % (10890)------------------------------
% 2.73/1.61 % (10890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (10890)Termination reason: Unknown
% 2.73/1.61 % (10890)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (10890)Memory used [KB]: 1663
% 2.73/1.61 % (10890)Time elapsed: 0.006 s
% 2.73/1.61 % (10890)Instructions burned: 7 (million)
% 2.73/1.61 % (10890)------------------------------
% 2.73/1.61 % (10890)------------------------------
% 2.73/1.61 % (10904)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/7Mi)
% 2.73/1.61 % (10901)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_10607 for (2999ds/3Mi)
% 2.73/1.61 % (10905)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/3Mi)
% 2.73/1.61 % (10905)Instruction limit reached!
% 2.73/1.61 % (10905)------------------------------
% 2.73/1.61 % (10905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (10905)Termination reason: Unknown
% 2.73/1.61 % (10905)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (10905)Memory used [KB]: 1407
% 2.73/1.61 % (10901)Instruction limit reached!
% 2.73/1.61 % (10901)------------------------------
% 2.73/1.61 % (10901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (10905)Time elapsed: 0.002 s
% 2.73/1.61 % (10905)Instructions burned: 3 (million)
% 2.73/1.61 % (10901)Termination reason: Unknown
% 2.73/1.61 % (10905)------------------------------
% 2.73/1.61 % (10905)------------------------------
% 2.73/1.61 % (10901)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (10901)Memory used [KB]: 1535
% 2.73/1.61 % (10901)Time elapsed: 0.003 s
% 2.73/1.61 % (10901)Instructions burned: 3 (million)
% 2.73/1.61 % (10901)------------------------------
% 2.73/1.61 % (10901)------------------------------
% 2.73/1.61 % (10904)Instruction limit reached!
% 2.73/1.61 % (10904)------------------------------
% 2.73/1.61 % (10904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (10904)Termination reason: Unknown
% 2.73/1.61 % (10904)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (10904)Memory used [KB]: 1535
% 2.73/1.61 % (10904)Time elapsed: 0.003 s
% 2.73/1.61 % (10904)Instructions burned: 7 (million)
% 2.73/1.61 % (10904)------------------------------
% 2.73/1.61 % (10904)------------------------------
% 2.73/1.62 % (10909)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/18Mi)
% 2.73/1.62 % (10912)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/6Mi)
% 2.73/1.62 % (10910)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/710Mi)
% 2.73/1.62 % (10912)Instruction limit reached!
% 2.73/1.62 % (10912)------------------------------
% 2.73/1.62 % (10912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.62 % (10912)Termination reason: Unknown
% 2.73/1.62 % (10912)Termination phase: shuffling
% 2.73/1.62
% 2.73/1.62 % (10912)Memory used [KB]: 1535
% 2.73/1.62 % (10912)Time elapsed: 0.003 s
% 2.73/1.62 % (10912)Instructions burned: 6 (million)
% 2.73/1.62 % (10912)------------------------------
% 2.73/1.62 % (10912)------------------------------
% 2.73/1.62 % (10906)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/4Mi)
% 2.73/1.63 % (10909)Instruction limit reached!
% 2.73/1.63 % (10909)------------------------------
% 2.73/1.63 % (10909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.63 % (10909)Termination reason: Unknown
% 2.73/1.63 % (10909)Termination phase: shuffling
% 2.73/1.63
% 2.73/1.63 % (10909)Memory used [KB]: 1791
% 2.73/1.63 % (10909)Time elapsed: 0.006 s
% 2.73/1.63 % (10909)Instructions burned: 18 (million)
% 2.73/1.63 % (10909)------------------------------
% 2.73/1.63 % (10909)------------------------------
% 2.73/1.63 % (10906)Instruction limit reached!
% 2.73/1.63 % (10906)------------------------------
% 2.73/1.63 % (10906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.63 % (10906)Termination reason: Unknown
% 2.73/1.63 % (10906)Termination phase: shuffling
% 2.73/1.63
% 2.73/1.63 % (10906)Memory used [KB]: 1535
% 2.73/1.63 % (10906)Time elapsed: 0.005 s
% 2.73/1.63 % (10906)Instructions burned: 5 (million)
% 2.73/1.63 % (10906)------------------------------
% 2.73/1.63 % (10906)------------------------------
% 2.73/1.63 % (10918)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_10607 for (2999ds/902Mi)
% 2.73/1.64 % (10920)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_10607 for (2999ds/21Mi)
% 2.73/1.64 % (10920)Instruction limit reached!
% 2.73/1.64 % (10920)------------------------------
% 2.73/1.64 % (10920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.64 % (10920)Termination reason: Unknown
% 2.73/1.64 % (10920)Termination phase: shuffling
% 2.73/1.64
% 2.73/1.64 % (10920)Memory used [KB]: 1918
% 2.73/1.64 % (10920)Time elapsed: 0.007 s
% 2.73/1.64 % (10920)Instructions burned: 21 (million)
% 2.73/1.64 % (10920)------------------------------
% 2.73/1.64 % (10920)------------------------------
% 2.73/1.65 % (10922)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_10607 for (2999ds/5Mi)
% 2.73/1.65 % (10922)Instruction limit reached!
% 2.73/1.65 % (10922)------------------------------
% 2.73/1.65 % (10922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.65 % (10922)Termination reason: Unknown
% 2.73/1.65 % (10922)Termination phase: shuffling
% 2.73/1.65
% 2.73/1.65 % (10922)Memory used [KB]: 1535
% 2.73/1.65 % (10922)Time elapsed: 0.005 s
% 2.73/1.65 % (10922)Instructions burned: 5 (million)
% 2.73/1.65 % (10922)------------------------------
% 2.73/1.65 % (10922)------------------------------
% 2.73/1.65 % (10929)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_10607 for (2999ds/6Mi)
% 2.73/1.65 % (10929)Instruction limit reached!
% 2.73/1.65 % (10929)------------------------------
% 2.73/1.65 % (10929)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.65 % (10929)Termination reason: Unknown
% 2.73/1.65 % (10929)Termination phase: shuffling
% 2.73/1.65
% 2.73/1.65 % (10929)Memory used [KB]: 1535
% 2.73/1.65 % (10929)Time elapsed: 0.003 s
% 2.73/1.65 % (10929)Instructions burned: 7 (million)
% 2.73/1.65 % (10929)------------------------------
% 2.73/1.65 % (10929)------------------------------
% 2.73/1.66 % (10867)Instruction limit reached!
% 2.73/1.66 % (10867)------------------------------
% 2.73/1.66 % (10867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.66 % (10867)Termination reason: Unknown
% 2.73/1.66 % (10867)Termination phase: Saturation
% 2.73/1.66
% 2.73/1.66 % (10867)Memory used [KB]: 7675
% 2.73/1.66 % (10867)Time elapsed: 0.085 s
% 2.73/1.66 % (10867)Instructions burned: 184 (million)
% 2.73/1.66 % (10867)------------------------------
% 2.73/1.66 % (10867)------------------------------
% 2.73/1.66 % (10934)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_10607 for (2998ds/779Mi)
% 2.73/1.67 % (10930)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_10607 for (2998ds/377Mi)
% 2.73/1.67 % (10936)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_10607 for (2998ds/19Mi)
% 2.73/1.67 % (10936)Instruction limit reached!
% 2.73/1.67 % (10936)------------------------------
% 2.73/1.67 % (10936)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.67 % (10936)Termination reason: Unknown
% 2.73/1.67 % (10936)Termination phase: shuffling
% 2.73/1.67
% 2.73/1.67 % (10936)Memory used [KB]: 1791
% 2.73/1.67 % (10936)Time elapsed: 0.007 s
% 2.73/1.67 % (10936)Instructions burned: 20 (million)
% 2.73/1.67 % (10936)------------------------------
% 2.73/1.67 % (10936)------------------------------
% 2.73/1.68 % (10941)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_10607 for (2998ds/879Mi)
% 2.73/1.70 % (10872)Instruction limit reached!
% 2.73/1.70 % (10872)------------------------------
% 2.73/1.70 % (10872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.70 % (10872)Termination reason: Unknown
% 2.73/1.70 % (10872)Termination phase: Saturation
% 2.73/1.70
% 2.73/1.70 % (10872)Memory used [KB]: 9594
% 2.73/1.70 % (10872)Time elapsed: 0.128 s
% 2.73/1.70 % (10872)Instructions burned: 277 (million)
% 2.73/1.70 % (10872)------------------------------
% 2.73/1.70 % (10872)------------------------------
% 2.73/1.71 % (10952)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_10607 for (2998ds/17Mi)
% 2.73/1.72 % (10952)Instruction limit reached!
% 2.73/1.72 % (10952)------------------------------
% 2.73/1.72 % (10952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.72 % (10952)Termination reason: Unknown
% 2.73/1.72 % (10952)Termination phase: shuffling
% 2.73/1.72
% 2.73/1.72 % (10952)Memory used [KB]: 1791
% 2.73/1.72 % (10952)Time elapsed: 0.006 s
% 2.73/1.72 % (10952)Instructions burned: 17 (million)
% 2.73/1.72 % (10952)------------------------------
% 2.73/1.72 % (10952)------------------------------
% 2.73/1.73 % (10959)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_10607 for (2998ds/3Mi)
% 2.73/1.73 % (10959)Instruction limit reached!
% 2.73/1.73 % (10959)------------------------------
% 2.73/1.73 % (10959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.73 % (10959)Termination reason: Unknown
% 2.73/1.73 % (10959)Termination phase: shuffling
% 2.73/1.73
% 2.73/1.73 % (10959)Memory used [KB]: 1535
% 2.73/1.73 % (10959)Time elapsed: 0.002 s
% 2.73/1.73 % (10959)Instructions burned: 3 (million)
% 2.73/1.73 % (10959)------------------------------
% 2.73/1.73 % (10959)------------------------------
% 2.73/1.75 % (10962)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_10607 for (2997ds/30Mi)
% 2.73/1.76 % (10962)Instruction limit reached!
% 2.73/1.76 % (10962)------------------------------
% 2.73/1.76 % (10962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.76 % (10962)Termination reason: Unknown
% 2.73/1.76 % (10962)Termination phase: shuffling
% 2.73/1.76
% 2.73/1.76 % (10962)Memory used [KB]: 2046
% 2.73/1.76 % (10962)Time elapsed: 0.018 s
% 2.73/1.76 % (10962)Instructions burned: 31 (million)
% 2.73/1.76 % (10962)------------------------------
% 2.73/1.76 % (10962)------------------------------
% 2.73/1.78 % (10971)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_10607 for (2997ds/127Mi)
% 2.73/1.79 % (10918)First to succeed.
% 2.73/1.79 % (10930)Instruction limit reached!
% 2.73/1.79 % (10930)------------------------------
% 2.73/1.79 % (10930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.79 % (10930)Termination reason: Unknown
% 2.73/1.79 % (10930)Termination phase: Saturation
% 2.73/1.79
% 2.73/1.79 % (10930)Memory used [KB]: 9338
% 2.73/1.79 % (10930)Time elapsed: 0.126 s
% 2.73/1.79 % (10930)Instructions burned: 379 (million)
% 2.73/1.79 % (10930)------------------------------
% 2.73/1.79 % (10930)------------------------------
% 2.73/1.79 % (10918)Refutation found. Thanks to Tanya!
% 2.73/1.79 % SZS status Theorem for DTF2THF_10607
% 2.73/1.79 % SZS output start Proof for DTF2THF_10607
% 2.73/1.79 thf(func_def_0, type, in: $i > $i > $o).
% 2.73/1.79 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.73/1.79 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.73/1.79 thf(func_def_8, type, powerset: $i > $i).
% 2.73/1.79 thf(func_def_10, type, setunion: $i > $i).
% 2.73/1.79 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.73/1.79 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_26, type, prop2set: $o > $i).
% 2.73/1.79 thf(func_def_36, type, nonempty: $i > $o).
% 2.73/1.79 thf(func_def_69, type, set2prop: $i > $o).
% 2.73/1.79 thf(func_def_88, type, subset: $i > $i > $o).
% 2.73/1.79 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.73/1.79 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.73/1.79 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.73/1.79 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.73/1.79 thf(func_def_135, type, regular: $i > $o).
% 2.73/1.79 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.73/1.79 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.73/1.79 thf(func_def_153, type, iskpair: $i > $o).
% 2.73/1.79 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.73/1.79 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.73/1.79 thf(func_def_177, type, singleton: $i > $o).
% 2.73/1.79 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.73/1.79 thf(func_def_184, type, atmost1p: $i > $o).
% 2.73/1.79 thf(func_def_185, type, atleast2p: $i > $o).
% 2.73/1.79 thf(func_def_186, type, atmost2p: $i > $o).
% 2.73/1.79 thf(func_def_187, type, upairsetp: $i > $o).
% 2.73/1.79 thf(func_def_191, type, kfst: $i > $i).
% 2.73/1.79 thf(func_def_203, type, ksnd: $i > $i).
% 2.73/1.79 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.73/1.79 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.73/1.79 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.73/1.79 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.73/1.79 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.73/1.79 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.73/1.79 thf(func_def_304, type, sP0: $i > $o > $i > $i > $o).
% 2.73/1.79 thf(func_def_308, type, sP4: $i > $i > $i > $i > $o).
% 2.73/1.79 thf(func_def_310, type, sP6: $i > $i > $o).
% 2.73/1.79 thf(func_def_311, type, sP7: $i > $o).
% 2.73/1.79 thf(func_def_312, type, sP8: $i > $i > $o).
% 2.73/1.79 thf(func_def_313, type, sP9: $i > $i > $o).
% 2.73/1.79 thf(func_def_314, type, sP10: $i > $i > $i > $o).
% 2.73/1.79 thf(func_def_317, type, sK13: $i > $i > $o).
% 2.73/1.79 thf(func_def_325, type, sK21: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_326, type, sK22: $i > $o).
% 2.73/1.79 thf(func_def_344, type, sK40: $i > $i > $i).
% 2.73/1.79 thf(func_def_368, type, sK64: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_369, type, sK65: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_376, type, sK72: $i > $o).
% 2.73/1.79 thf(func_def_385, type, sK81: $i > $i > $i).
% 2.73/1.79 thf(func_def_397, type, sK93: $i > $o).
% 2.73/1.79 thf(func_def_398, type, sK94: ($i > $o) > $i).
% 2.73/1.79 thf(func_def_445, type, sK141: $i > $o).
% 2.73/1.79 thf(func_def_448, type, sK144: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_459, type, sK155: $i > $o).
% 2.73/1.79 thf(func_def_462, type, sK158: $i > $o).
% 2.73/1.79 thf(func_def_464, type, sK160: ($i > $o) > $i > $i).
% 2.73/1.79 thf(func_def_465, type, sK161: ($i > $o) > $i > $i).
% 2.73/1.79 thf(func_def_471, type, sK167: $i > $i).
% 2.73/1.79 thf(func_def_474, type, sK170: $i > ($i > $i) > $i > $i).
% 2.73/1.79 thf(func_def_477, type, sK173: $o > $i > $i > $i).
% 2.73/1.79 thf(func_def_489, type, sK185: $i > $i > $i).
% 2.73/1.79 thf(func_def_515, type, sK211: $i > $i > $o).
% 2.73/1.79 thf(func_def_525, type, sK221: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_526, type, sK222: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_529, type, sK225: $i > $i > $i).
% 2.73/1.79 thf(func_def_532, type, sK228: $i > $o).
% 2.73/1.79 thf(func_def_533, type, sK229: $i > $i).
% 2.73/1.79 thf(func_def_534, type, sK230: ($i > $o) > $i).
% 2.73/1.79 thf(func_def_567, type, sK263: $i > $i > $o).
% 2.73/1.79 thf(func_def_578, type, sK274: $i > $i).
% 2.73/1.79 thf(func_def_587, type, sK283: $i > $i > $o).
% 2.73/1.79 thf(func_def_600, type, sK296: $i > $i > $o).
% 2.73/1.79 thf(func_def_620, type, sK316: $i > $i > $i).
% 2.73/1.79 thf(func_def_629, type, sK325: $i > $o).
% 2.73/1.79 thf(func_def_636, type, sK332: $i > $i).
% 2.73/1.79 thf(func_def_638, type, sK334: $i > $i).
% 2.73/1.79 thf(func_def_650, type, sK346: $i > $i).
% 2.73/1.79 thf(func_def_660, type, sK356: $i > $i).
% 2.73/1.79 thf(func_def_662, type, sK358: $i > ($i > $i) > $i > $i).
% 2.73/1.79 thf(func_def_676, type, sK372: $i > $o).
% 2.73/1.79 thf(func_def_677, type, sK373: $i > ($i > $i) > $i > $i).
% 2.73/1.79 thf(func_def_679, type, sK375: $i > $i).
% 2.73/1.79 thf(func_def_684, type, sK380: $i > $o).
% 2.73/1.79 thf(func_def_717, type, sK413: $i > $i > $o).
% 2.73/1.79 thf(func_def_718, type, sK414: $i > $i).
% 2.73/1.79 thf(func_def_719, type, sK415: $i > $i).
% 2.73/1.79 thf(func_def_720, type, sK416: ($i > $i > $o) > $i > $i).
% 2.73/1.79 thf(func_def_721, type, sK417: ($i > $i > $o) > $i > $i).
% 2.73/1.79 thf(func_def_722, type, sK418: $i > ($i > $i > $o) > $i > $i).
% 2.73/1.79 thf(func_def_724, type, sK420: $i > $o).
% 2.73/1.79 thf(func_def_727, type, sK423: $i > $i > $i).
% 2.73/1.79 thf(func_def_728, type, sK424: $i > $i > $i).
% 2.73/1.79 thf(func_def_734, type, sK430: $i > $o).
% 2.73/1.79 thf(func_def_737, type, sK433: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_738, type, sK434: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_748, type, sK444: $i > $o).
% 2.73/1.79 thf(func_def_749, type, sK445: $i > $i).
% 2.73/1.79 thf(func_def_750, type, sK446: ($i > $o) > $i).
% 2.73/1.79 thf(func_def_751, type, sK447: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_752, type, sK448: $i > $o).
% 2.73/1.79 thf(func_def_754, type, sK450: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_762, type, sK458: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_763, type, sK459: $i > $o).
% 2.73/1.79 thf(func_def_765, type, sK461: ($i > $o) > $i).
% 2.73/1.79 thf(func_def_766, type, sK462: ($i > $o) > $i).
% 2.73/1.79 thf(func_def_767, type, sK463: $i > $o).
% 2.73/1.79 thf(func_def_783, type, sK479: ($i > $o) > $i > $i).
% 2.73/1.79 thf(func_def_785, type, sK481: $i > $o).
% 2.73/1.79 thf(func_def_789, type, sK485: $i > $o).
% 2.73/1.79 thf(func_def_792, type, sK488: $i > $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_793, type, sK489: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_798, type, sK494: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_799, type, sK495: $i > $o).
% 2.73/1.79 thf(func_def_806, type, sK502: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_807, type, sK503: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_808, type, sK504: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_809, type, sK505: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_813, type, sK509: $i > $i > $o).
% 2.73/1.79 thf(func_def_817, type, sK513: $i > $i).
% 2.73/1.79 thf(func_def_818, type, sK514: ($i > $i) > $i > $i > $i).
% 2.73/1.79 thf(func_def_831, type, sK527: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_832, type, sK528: $i > $o).
% 2.73/1.79 thf(func_def_842, type, sK538: $i > $i > $i).
% 2.73/1.79 thf(func_def_847, type, sK543: $i > $o).
% 2.73/1.79 thf(func_def_849, type, sK545: $i > $o).
% 2.73/1.79 thf(func_def_850, type, sK546: $i > $o).
% 2.73/1.79 thf(func_def_851, type, sK547: ($i > $o) > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_852, type, sK548: ($i > $o) > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_854, type, sK550: $i > $o).
% 2.73/1.79 thf(func_def_864, type, sK560: $i > $i).
% 2.73/1.79 thf(func_def_866, type, sK562: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.73/1.79 thf(func_def_867, type, sK563: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.73/1.79 thf(func_def_870, type, sK566: $i > $o).
% 2.73/1.79 thf(func_def_871, type, sK567: $i > $o).
% 2.73/1.79 thf(func_def_872, type, sK568: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_873, type, sK569: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_877, type, sK573: $i > $o).
% 2.73/1.79 thf(func_def_887, type, sK583: $i > $i).
% 2.73/1.79 thf(func_def_889, type, sK585: $i > ($i > $i) > $i > $i).
% 2.73/1.79 thf(func_def_898, type, sK594: $i > $i > $i).
% 2.73/1.79 thf(func_def_909, type, sK605: $i > $i).
% 2.73/1.79 thf(func_def_920, type, sK616: $i > $o).
% 2.73/1.79 thf(func_def_921, type, sK617: $i > $o).
% 2.73/1.79 thf(func_def_922, type, sK618: ($i > $o) > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_923, type, sK619: ($i > $o) > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_925, type, sK621: $i > $i > $i).
% 2.73/1.79 thf(func_def_926, type, sK622: $i > $i > $i).
% 2.73/1.79 thf(func_def_927, type, sK623: $i > $i > $i).
% 2.73/1.79 thf(func_def_928, type, sK624: $i > $i > $i).
% 2.73/1.79 thf(func_def_929, type, sK625: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_930, type, sK626: $i > $i).
% 2.73/1.79 thf(func_def_931, type, sK627: $i > $i).
% 2.73/1.79 thf(func_def_932, type, sK628: $i > $i).
% 2.73/1.79 thf(func_def_933, type, sK629: $i > $i).
% 2.73/1.79 thf(func_def_934, type, sK630: $i > $i > $i).
% 2.73/1.79 thf(func_def_935, type, sK631: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_936, type, sK632: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_937, type, sK633: $i > $i > $i).
% 2.73/1.79 thf(func_def_938, type, sK634: $i > $i > $i).
% 2.73/1.79 thf(func_def_939, type, sK635: $i > $i).
% 2.73/1.79 thf(func_def_940, type, sK636: $i > $i > $i).
% 2.73/1.79 thf(func_def_942, type, sK638: $i > $i).
% 2.73/1.79 thf(func_def_943, type, sK639: $i > $i).
% 2.73/1.79 thf(func_def_951, type, sK647: $i > $o).
% 2.73/1.79 thf(func_def_973, type, sK669: $i > $o).
% 2.73/1.79 thf(func_def_989, type, sK685: $i > $i).
% 2.73/1.79 thf(func_def_991, type, sK687: $i > ($i > $i) > $i > $i).
% 2.73/1.79 thf(func_def_995, type, sK691: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_996, type, sK692: $i > $o).
% 2.73/1.79 thf(func_def_1000, type, sK696: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_1004, type, sK700: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_1006, type, sK702: $i > $i > $o).
% 2.73/1.79 thf(func_def_1018, type, sK714: $i > $i).
% 2.73/1.79 thf(func_def_1020, type, sK716: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_1021, type, sK717: $i > $o).
% 2.73/1.79 thf(func_def_1026, type, sK722: $i > $o).
% 2.73/1.79 thf(func_def_1028, type, sK724: $i > ($i > $o) > $i).
% 2.73/1.79 thf(func_def_1035, type, sK731: $i > $o).
% 2.73/1.79 thf(func_def_1052, type, sK748: $i > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_1057, type, sK753: $i > $i > $i).
% 2.73/1.79 thf(func_def_1060, type, sK756: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_1061, type, sK757: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_1065, type, sK761: $i > $o).
% 2.73/1.79 thf(func_def_1072, type, sK768: $i > $i > $i > $i).
% 2.73/1.79 thf(func_def_1098, type, sK794: $i > $o).
% 2.73/1.79 thf(func_def_1104, type, ph800: !>[X0: $tType]:(X0)).
% 2.73/1.79 thf(f5341,plain,(
% 2.73/1.79 $false),
% 2.73/1.79 inference(avatar_sat_refutation,[status(thm)],[f3982,f4021,f4082,f4144,f4198,f5095])).
% 2.73/1.79 thf(f5095,plain,(
% 2.73/1.79 ~spl799_11 | ~spl799_10 | ~spl799_17),
% 2.73/1.79 inference(avatar_split_clause,[status(thm)],[f5094,f4015,f3954,f3958])).
% 2.73/1.79 thf(f3958,definition,(
% 2.73/1.79 spl799_11 <=> ((in @ sK165 @ sK163) = $true)),
% 2.73/1.79 introduced(definition,[new_symbols(naming,[spl799_11])],[avatar_definition])).
% 2.73/1.79 thf(f3954,definition,(
% 2.73/1.79 spl799_10 <=> ($true = (in @ sK165 @ sK162))),
% 2.73/1.79 introduced(definition,[new_symbols(naming,[spl799_10])],[avatar_definition])).
% 2.73/1.79 thf(f4015,definition,(
% 2.73/1.79 spl799_17 <=> ! [X0 : $i] : (((in @ X0 @ sK164) != $true) | ((in @ X0 @ sK163) != $true) | ((in @ X0 @ sK162) != $true))),
% 2.73/1.79 introduced(definition,[new_symbols(naming,[spl799_17])],[avatar_definition])).
% 2.73/1.79 thf(f5094,plain,(
% 2.73/1.79 ((in @ sK165 @ sK163) != $true) | (~spl799_10 | ~spl799_17)),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f5093])).
% 2.73/1.79 thf(f5093,plain,(
% 2.73/1.79 ($true != $true) | ((in @ sK165 @ sK163) != $true) | (~spl799_10 | ~spl799_17)),
% 2.73/1.79 inference(forward_demodulation,[status(thm)],[f5037,f3956])).
% 2.73/1.79 thf(f3956,plain,(
% 2.73/1.79 ($true = (in @ sK165 @ sK162)) | ~spl799_10),
% 2.73/1.79 inference(avatar_component_clause,[status(thm)],[f3954])).
% 2.73/1.79 thf(f5037,plain,(
% 2.73/1.79 ($true != (in @ sK165 @ sK162)) | ((in @ sK165 @ sK163) != $true) | ~spl799_17),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f4958])).
% 2.73/1.79 thf(f4958,plain,(
% 2.73/1.79 ($true != (in @ sK165 @ sK162)) | ((in @ sK165 @ sK163) != $true) | ($true != $true) | ~spl799_17),
% 2.73/1.79 inference(superposition,[status(thm)],[f4016,f2788])).
% 2.73/1.79 thf(f2788,plain,(
% 2.73/1.79 ((in @ sK165 @ sK164) = $true)),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.73/1.79 thf(f1362,plain,(
% 2.73/1.79 (cartprodpairsurjEq = $true) & (lamProp = $true) & (setminusSubset1 = $true) & (doubleComplementI1 = $true) & (iftrueProp1 = $true) & (setunion__Cong = $true) & (notsubsetI = $true) & (in__Cong = $true) & (setadjoinOr = $true) & (cartprodmempair1 = $true) & (funcGraphProp4 = $true) & (setminusERneg = $true) & (((in @ sK162 @ (powerset @ sK163)) = $true) & (((subset @ sK162 @ (setminus @ sK163 @ sK164)) = $true) & ((in @ sK164 @ (powerset @ sK163)) = $true) & (((in @ sK165 @ sK163) = $true) & ((in @ sK165 @ sK164) = $true) & ((in @ sK165 @ (setminus @ sK163 @ sK162)) != $true)))) & (subset2powerset = $true) & (funcext = $true) & (upairset2IR = $true) & (iftrueProp2 = $true) & (singletoninpowerset = $true) & (setOfPairsIsBReln = $true) & (powerset__Cong = $true) & (nonemptyI = $true) & (lam2lamEq = $true) & (dpsetconstrSub = $true) & (cartprodmempaircEq = $true) & (exuE2 = $true) & (setadjoin__Cong = $true) & (setminusSubset2 = $true) & (nonemptyI1 = $true) & (emptysetimpfalse = $true) & (ap2apEq1 = $true) & (setukpairinjL1 = $true) & (binunionE = $true) & (cartprodfstpairEq = $true) & (setminusILneg = $true) & (emptyI = $true) & (app = $true) & (brelnall2 = $true) & (upairsubunion = $true) & (wellorderingAx = $true) & (setukpairinjR1 = $true) & (dsetconstrI = $true) & (setukpairinjR11 = $true) & (binunionT_lem = $true) & (notdexE = $true) & (binintersectT_lem = $true) & (binintersectSubset5 = $true) & (setukpairinjR = $true) & (setukpairinjR2 = $true) & (binunionRsub = $true) & (funcinfuncset = $true) & (omega0Ax = $true) & (sepInPowerset = $true) & (setminusLsub = $true) & (eqimpsubset2 = $true) & (eqbreln = $true) & (brelnall1 = $true) & (binintersectTELcontra = $true) & (kpairiskpair = $true) & (quantDeMorgan1 = $true) & (singletonprop = $true) & (setunionsingleton2 = $true) & (notequalI1 = $true) & (singletonsuniq = $true) & (binintersectSubset4 = $true) & (apProp = $true) & (powersetE1 = $true) & (contrasubsetT1 = $true) & (funcImageSingleton = $true) & (cartprodsndin = $true) & (binunionIL = $true) & (cartprodpairmemEL = $true) & (dsetconstrEL = $true) & (setminusER = $true) & (cartprodmempair = $true) & (noeltsimpempty = $true) & (upairsetE = $true) & (contrasubsetT2 = $true) & (iffalseProp1 = $true) & (theprop = $true) & (cartprodpairmemER = $true) & (subsetRefl = $true) & (singletoninpowunion = $true) & (complementImpComplementIntersect = $true) & (subsetI1 = $true) & (binunionEcases = $true) & (symdiffE = $true) & (emptysetAx = $true) & (powersetTI1 = $true) & (cartprodpairin = $true) & (exuEu = $true) & (prop2set2propI = $true) & (iftrueorfalse = $true) & (notinemptyset = $true) & (lamp = $true) & (binintersectRsub = $true) & (emptyset__Cong = $true) & (binintersectTERcontra = $true) & (exu__Cong = $true) & (foundationAx = $true) & (setukpairinjL2 = $true) & (setunionsingleton1 = $true) & (emptyinPowerset = $true) & (dpsetconstrEL2 = $true) & (theeq = $true) & (binintersectER = $true) & (ubforcartprodlem2 = $true) & (setukpairinjR12 = $true) & (eta2 = $true) & (omegaIndAx = $true) & (setadjoinSub = $true) & (complementT_lem = $true) & (uniqinunit = $true) & (dpsetconstrI = $true) & (subsetTI = $true) & (symdiffIneg1 = $true) & (upairinpowunion = $true) & (ex1I = $true) & (iffalse = $true) & (cartprodsndpairEq = $true) & (beta2 = $true) & (nonemptyE1 = $true) & (prop2setE = $true) & (setbeta = $true) & (setadjoinE = $true) & (omega__Cong = $true) & (setext = $true) & (setadjoinIL = $true) & (notdallE = $true) & (setunionAx = $true) & (dsetconstrER = $true) & (singletonsswitch = $true) & (upairequniteq = $true) & (prop2setI = $true) & (binintersectLsub = $true) & (powersetAx = $true) & (sepSubset = $true) & (complementTE1 = $true) & (vacuousDall = $true) & (subsetE2 = $true) & (ksndpairEq = $true) & (complementTI1 = $true) & (quantDeMorgan4 = $true) & (emptyInPowerset = $true) & (symdiffIneg2 = $true) & (dpsetconstrER = $true) & (complementTnotintersectT = $true) & (binintersectSubset1 = $true) & (upairsetIR = $true) & (symdiffI1 = $true) & (ap2p = $true) & (setadjoinSub2 = $true) & (doubleComplementSub2 = $true) & (eqinunit = $true) & (exuE3u = $true) & (emptyE1 = $true) & (setukpairIR = $true) & (contrasubsetT = $true) & (subsetE = $true) & (beta1 = $true) & (ksndsingleton = $true) & (iftrue = $true) & (setunionsingleton = $true) & (descr__Cong = $true) & (setextsub = $true) & (subsetI2 = $true) & (omegaSAx = $true) & (setadjoinIR = $true) & (doubleComplementSub1 = $true) & (kfstsingleton = $true) & (subsetTrans = $true) & (powersetsubset = $true) & (setunionE = $true) & (ap2apEq2 = $true) & (inPowerset = $true) & (setextT = $true) & (funcGraphProp1 = $true) & (quantDeMorgan3 = $true) & (subsetemptysetimpeq = $true) & (replAx = $true) & (powersetTE1 = $true) & (funcGraphProp3 = $true) & (upairsetIL = $true) & (binunionLsub = $true) & (subbreln = $true) & (setminusELneg = $true) & (emptysetE = $true) & (kfstpairEq = $true) & (iffalseProp2 = $true) & (kpairsurjEq = $true) & (symdiffI2 = $true) & (inCongP = $true) & (nonemptyImpWitness = $true) & (cartprodfstin = $true) & (descrp = $true) & (funcGraphProp2 = $true) & (powersetI1 = $true) & (exuI2 = $true) & (binintersectSubset2 = $true) & (lam2p = $true) & (setunionI = $true) & (doubleComplementEq = $true) & (ubforcartprodlem1 = $true) & (funcextLem = $true) & (emptyinunitempty = $true) & (powersetI = $true) & (exuE3e = $true) & (ex1E1 = $true) & (setminusEL = $true) & (dsetconstr__Cong = $true) & (powersetE = $true) & (notequalI2 = $true) & (setukpairIL = $true) & (ubforcartprodlem3 = $true) & (secondinupair = $true) & (dpsetconstrEL1 = $true) & (ex1I2 = $true) & (binintersectSubset3 = $true) & (ex1E2 = $true) & (setoftrueEq = $true) & (powersetT_lem = $true) & (quantDeMorgan2 = $true) & (subPowSU = $true) & (disjointsetsI1 = $true) & (setminusI = $true) & (complementInPowersetComplementIntersect = $true) & (kpairp = $true) & (binunionIR = $true) & (eta1 = $true) & (doubleComplementE1 = $true) & (exuI1 = $true) & (infuncsetfunc = $true) & (exuI3 = $true) & (notinsingleton = $true) & (complementSubsetComplementIntersect = $true) & (contrasubsetT3 = $true) & (ifp = $true) & (exuE1 = $true) & (binintersectEL = $true) & (setadjoinAx = $true) & (ifSingleton = $true) & (setminusT_lem = $true) & (binintersectI = $true) & (dpsetconstrERa = $true) & (singletonsubset = $true) & (funcext2 = $true) & (emptysetsubset = $true) & (setukpairinjL = $true) & (setunionE2 = $true) & (setminusIRneg = $true) & (eqimpsubset1 = $true) & (bs114d = $true) & (setextAx = $true) & (upairset2E = $true)),
% 2.73/1.79 inference(skolemisation,[status(esa),new_symbols(skolem,[sK162,sK163,sK164,sK165])],[f952,f1361,f1360,f1359])).
% 2.73/1.79 thf(f1359,plain,(
% 2.73/1.79 ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (((subset @ X0 @ (setminus @ X1 @ X2)) = $true) & ((in @ X2 @ (powerset @ X1)) = $true) & ? [X3 : $i] : (((in @ X3 @ X1) = $true) & ((in @ X3 @ X2) = $true) & ((in @ X3 @ (setminus @ X1 @ X0)) != $true)))) => (((in @ sK162 @ (powerset @ sK163)) = $true) & ? [X2 : $i] : (((subset @ sK162 @ (setminus @ sK163 @ X2)) = $true) & ((in @ X2 @ (powerset @ sK163)) = $true) & ? [X3 : $i] : (((in @ X3 @ sK163) = $true) & ((in @ X3 @ X2) = $true) & ((in @ X3 @ (setminus @ sK163 @ sK162)) != $true))))),
% 2.73/1.79 introduced(definition,[],[choice_axiom])).
% 2.73/1.79 thf(f1360,plain,(
% 2.73/1.79 ? [X2 : $i] : (((subset @ sK162 @ (setminus @ sK163 @ X2)) = $true) & ((in @ X2 @ (powerset @ sK163)) = $true) & ? [X3 : $i] : (((in @ X3 @ sK163) = $true) & ((in @ X3 @ X2) = $true) & ((in @ X3 @ (setminus @ sK163 @ sK162)) != $true))) => (((subset @ sK162 @ (setminus @ sK163 @ sK164)) = $true) & ((in @ sK164 @ (powerset @ sK163)) = $true) & ? [X3 : $i] : (((in @ X3 @ sK163) = $true) & ((in @ X3 @ sK164) = $true) & ((in @ X3 @ (setminus @ sK163 @ sK162)) != $true)))),
% 2.73/1.79 introduced(definition,[],[choice_axiom])).
% 2.73/1.79 thf(f1361,plain,(
% 2.73/1.79 ? [X3 : $i] : (((in @ X3 @ sK163) = $true) & ((in @ X3 @ sK164) = $true) & ((in @ X3 @ (setminus @ sK163 @ sK162)) != $true)) => (((in @ sK165 @ sK163) = $true) & ((in @ sK165 @ sK164) = $true) & ((in @ sK165 @ (setminus @ sK163 @ sK162)) != $true))),
% 2.73/1.79 introduced(definition,[],[choice_axiom])).
% 2.73/1.79 thf(f952,plain,(
% 2.73/1.79 (cartprodpairsurjEq = $true) & (lamProp = $true) & (setminusSubset1 = $true) & (doubleComplementI1 = $true) & (iftrueProp1 = $true) & (setunion__Cong = $true) & (notsubsetI = $true) & (in__Cong = $true) & (setadjoinOr = $true) & (cartprodmempair1 = $true) & (funcGraphProp4 = $true) & (setminusERneg = $true) & ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (((subset @ X0 @ (setminus @ X1 @ X2)) = $true) & ((in @ X2 @ (powerset @ X1)) = $true) & ? [X3 : $i] : (((in @ X3 @ X1) = $true) & ((in @ X3 @ X2) = $true) & ((in @ X3 @ (setminus @ X1 @ X0)) != $true)))) & (subset2powerset = $true) & (funcext = $true) & (upairset2IR = $true) & (iftrueProp2 = $true) & (singletoninpowerset = $true) & (setOfPairsIsBReln = $true) & (powerset__Cong = $true) & (nonemptyI = $true) & (lam2lamEq = $true) & (dpsetconstrSub = $true) & (cartprodmempaircEq = $true) & (exuE2 = $true) & (setadjoin__Cong = $true) & (setminusSubset2 = $true) & (nonemptyI1 = $true) & (emptysetimpfalse = $true) & (ap2apEq1 = $true) & (setukpairinjL1 = $true) & (binunionE = $true) & (cartprodfstpairEq = $true) & (setminusILneg = $true) & (emptyI = $true) & (app = $true) & (brelnall2 = $true) & (upairsubunion = $true) & (wellorderingAx = $true) & (setukpairinjR1 = $true) & (dsetconstrI = $true) & (setukpairinjR11 = $true) & (binunionT_lem = $true) & (notdexE = $true) & (binintersectT_lem = $true) & (binintersectSubset5 = $true) & (setukpairinjR = $true) & (setukpairinjR2 = $true) & (binunionRsub = $true) & (funcinfuncset = $true) & (omega0Ax = $true) & (sepInPowerset = $true) & (setminusLsub = $true) & (eqimpsubset2 = $true) & (eqbreln = $true) & (brelnall1 = $true) & (binintersectTELcontra = $true) & (kpairiskpair = $true) & (quantDeMorgan1 = $true) & (singletonprop = $true) & (setunionsingleton2 = $true) & (notequalI1 = $true) & (singletonsuniq = $true) & (binintersectSubset4 = $true) & (apProp = $true) & (powersetE1 = $true) & (contrasubsetT1 = $true) & (funcImageSingleton = $true) & (cartprodsndin = $true) & (binunionIL = $true) & (cartprodpairmemEL = $true) & (dsetconstrEL = $true) & (setminusER = $true) & (cartprodmempair = $true) & (noeltsimpempty = $true) & (upairsetE = $true) & (contrasubsetT2 = $true) & (iffalseProp1 = $true) & (theprop = $true) & (cartprodpairmemER = $true) & (subsetRefl = $true) & (singletoninpowunion = $true) & (complementImpComplementIntersect = $true) & (subsetI1 = $true) & (binunionEcases = $true) & (symdiffE = $true) & (emptysetAx = $true) & (powersetTI1 = $true) & (cartprodpairin = $true) & (exuEu = $true) & (prop2set2propI = $true) & (iftrueorfalse = $true) & (notinemptyset = $true) & (lamp = $true) & (binintersectRsub = $true) & (emptyset__Cong = $true) & (binintersectTERcontra = $true) & (exu__Cong = $true) & (foundationAx = $true) & (setukpairinjL2 = $true) & (setunionsingleton1 = $true) & (emptyinPowerset = $true) & (dpsetconstrEL2 = $true) & (theeq = $true) & (binintersectER = $true) & (ubforcartprodlem2 = $true) & (setukpairinjR12 = $true) & (eta2 = $true) & (omegaIndAx = $true) & (setadjoinSub = $true) & (complementT_lem = $true) & (uniqinunit = $true) & (dpsetconstrI = $true) & (subsetTI = $true) & (symdiffIneg1 = $true) & (upairinpowunion = $true) & (ex1I = $true) & (iffalse = $true) & (cartprodsndpairEq = $true) & (beta2 = $true) & (nonemptyE1 = $true) & (prop2setE = $true) & (setbeta = $true) & (setadjoinE = $true) & (omega__Cong = $true) & (setext = $true) & (setadjoinIL = $true) & (notdallE = $true) & (setunionAx = $true) & (dsetconstrER = $true) & (singletonsswitch = $true) & (upairequniteq = $true) & (prop2setI = $true) & (binintersectLsub = $true) & (powersetAx = $true) & (sepSubset = $true) & (complementTE1 = $true) & (vacuousDall = $true) & (subsetE2 = $true) & (ksndpairEq = $true) & (complementTI1 = $true) & (quantDeMorgan4 = $true) & (emptyInPowerset = $true) & (symdiffIneg2 = $true) & (dpsetconstrER = $true) & (complementTnotintersectT = $true) & (binintersectSubset1 = $true) & (upairsetIR = $true) & (symdiffI1 = $true) & (ap2p = $true) & (setadjoinSub2 = $true) & (doubleComplementSub2 = $true) & (eqinunit = $true) & (exuE3u = $true) & (emptyE1 = $true) & (setukpairIR = $true) & (contrasubsetT = $true) & (subsetE = $true) & (beta1 = $true) & (ksndsingleton = $true) & (iftrue = $true) & (setunionsingleton = $true) & (descr__Cong = $true) & (setextsub = $true) & (subsetI2 = $true) & (omegaSAx = $true) & (setadjoinIR = $true) & (doubleComplementSub1 = $true) & (kfstsingleton = $true) & (subsetTrans = $true) & (powersetsubset = $true) & (setunionE = $true) & (ap2apEq2 = $true) & (inPowerset = $true) & (setextT = $true) & (funcGraphProp1 = $true) & (quantDeMorgan3 = $true) & (subsetemptysetimpeq = $true) & (replAx = $true) & (powersetTE1 = $true) & (funcGraphProp3 = $true) & (upairsetIL = $true) & (binunionLsub = $true) & (subbreln = $true) & (setminusELneg = $true) & (emptysetE = $true) & (kfstpairEq = $true) & (iffalseProp2 = $true) & (kpairsurjEq = $true) & (symdiffI2 = $true) & (inCongP = $true) & (nonemptyImpWitness = $true) & (cartprodfstin = $true) & (descrp = $true) & (funcGraphProp2 = $true) & (powersetI1 = $true) & (exuI2 = $true) & (binintersectSubset2 = $true) & (lam2p = $true) & (setunionI = $true) & (doubleComplementEq = $true) & (ubforcartprodlem1 = $true) & (funcextLem = $true) & (emptyinunitempty = $true) & (powersetI = $true) & (exuE3e = $true) & (ex1E1 = $true) & (setminusEL = $true) & (dsetconstr__Cong = $true) & (powersetE = $true) & (notequalI2 = $true) & (setukpairIL = $true) & (ubforcartprodlem3 = $true) & (secondinupair = $true) & (dpsetconstrEL1 = $true) & (ex1I2 = $true) & (binintersectSubset3 = $true) & (ex1E2 = $true) & (setoftrueEq = $true) & (powersetT_lem = $true) & (quantDeMorgan2 = $true) & (subPowSU = $true) & (disjointsetsI1 = $true) & (setminusI = $true) & (complementInPowersetComplementIntersect = $true) & (kpairp = $true) & (binunionIR = $true) & (eta1 = $true) & (doubleComplementE1 = $true) & (exuI1 = $true) & (infuncsetfunc = $true) & (exuI3 = $true) & (notinsingleton = $true) & (complementSubsetComplementIntersect = $true) & (contrasubsetT3 = $true) & (ifp = $true) & (exuE1 = $true) & (binintersectEL = $true) & (setadjoinAx = $true) & (ifSingleton = $true) & (setminusT_lem = $true) & (binintersectI = $true) & (dpsetconstrERa = $true) & (singletonsubset = $true) & (funcext2 = $true) & (emptysetsubset = $true) & (setukpairinjL = $true) & (setunionE2 = $true) & (setminusIRneg = $true) & (eqimpsubset1 = $true) & (bs114d = $true) & (setextAx = $true) & (upairset2E = $true)),
% 2.73/1.79 inference(flattening,[status(thm)],[f951])).
% 2.73/1.79 thf(f951,plain,(
% 2.73/1.79 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : ((? [X3 : $i] : ((((in @ X3 @ (setminus @ X1 @ X0)) != $true) & ((in @ X3 @ X2) = $true)) & ((in @ X3 @ X1) = $true)) & ((subset @ X0 @ (setminus @ X1 @ X2)) = $true)) & ((in @ X2 @ (powerset @ X1)) = $true)) & ((in @ X0 @ (powerset @ X1)) = $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)),
% 2.73/1.79 inference(ennf_transformation,[status(thm)],[f417])).
% 2.73/1.79 thf(f417,plain,(
% 2.73/1.79 ~((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) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) => (((subset @ X0 @ (setminus @ X1 @ X2)) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => (((in @ X3 @ X2) = $true) => ((in @ X3 @ (setminus @ X1 @ X0)) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.79 inference(fool_elimination,[status(thm)],[f416])).
% 2.73/1.79 thf(f416,plain,(
% 2.73/1.79 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X1)) => ((subset @ X0 @ (setminus @ X1 @ X2)) => ! [X3 : $i] : ((in @ X3 @ X1) => ((in @ X3 @ X2) => (in @ X3 @ (setminus @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.79 inference(rectify,[status(thm)],[f258])).
% 2.73/1.79 thf(f258,negated_conjecture,(
% 2.73/1.79 ~(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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ((subset @ X11 @ (setminus @ X3 @ X16)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ X16) => (in @ X1 @ (setminus @ X3 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.79 inference(negated_conjecture,[status(cth)],[f257])).
% 2.73/1.79 thf(f257,conjecture,(
% 2.73/1.79 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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ((subset @ X11 @ (setminus @ X3 @ X16)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ X16) => (in @ X1 @ (setminus @ X3 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.79 file('/export/starexec/sandbox2/tmp/tmp.1oBuXnfUb8/DTF2THF_10607.p',contraSubsetComplement)).
% 2.73/1.79 thf(f4016,plain,(
% 2.73/1.79 ( ! [X0 : $i] : (((in @ X0 @ sK164) != $true) | ((in @ X0 @ sK162) != $true) | ((in @ X0 @ sK163) != $true)) ) | ~spl799_17),
% 2.73/1.79 inference(avatar_component_clause,[status(thm)],[f4015])).
% 2.73/1.79 thf(f4198,plain,(
% 2.73/1.79 spl799_18),
% 2.73/1.79 inference(avatar_contradiction_clause,[status(thm)],[f4197])).
% 2.73/1.79 thf(f4197,plain,(
% 2.73/1.79 $false | spl799_18),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f4189])).
% 2.73/1.79 thf(f4189,plain,(
% 2.73/1.79 ($true != $true) | spl799_18),
% 2.73/1.79 inference(superposition,[status(thm)],[f4020,f2790])).
% 2.73/1.79 thf(f2790,plain,(
% 2.73/1.79 ((in @ sK164 @ (powerset @ sK163)) = $true)),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.73/1.79 thf(f4020,plain,(
% 2.73/1.79 ((in @ sK164 @ (powerset @ sK163)) != $true) | spl799_18),
% 2.73/1.79 inference(avatar_component_clause,[status(thm)],[f4018])).
% 2.73/1.79 thf(f4018,definition,(
% 2.73/1.79 spl799_18 <=> ((in @ sK164 @ (powerset @ sK163)) = $true)),
% 2.73/1.79 introduced(definition,[new_symbols(naming,[spl799_18])],[avatar_definition])).
% 2.73/1.79 thf(f4144,plain,(
% 2.73/1.79 spl799_12),
% 2.73/1.79 inference(avatar_contradiction_clause,[status(thm)],[f4143])).
% 2.73/1.79 thf(f4143,plain,(
% 2.73/1.79 $false | spl799_12),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f4133])).
% 2.73/1.79 thf(f4133,plain,(
% 2.73/1.79 ($true != $true) | spl799_12),
% 2.73/1.79 inference(superposition,[status(thm)],[f3981,f2792])).
% 2.73/1.79 thf(f2792,plain,(
% 2.73/1.79 ((in @ sK162 @ (powerset @ sK163)) = $true)),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.73/1.79 thf(f3981,plain,(
% 2.73/1.79 ((in @ sK162 @ (powerset @ sK163)) != $true) | spl799_12),
% 2.73/1.79 inference(avatar_component_clause,[status(thm)],[f3979])).
% 2.73/1.79 thf(f3979,definition,(
% 2.73/1.79 spl799_12 <=> ((in @ sK162 @ (powerset @ sK163)) = $true)),
% 2.73/1.79 introduced(definition,[new_symbols(naming,[spl799_12])],[avatar_definition])).
% 2.73/1.79 thf(f4082,plain,(
% 2.73/1.79 spl799_11),
% 2.73/1.79 inference(avatar_contradiction_clause,[status(thm)],[f4081])).
% 2.73/1.79 thf(f4081,plain,(
% 2.73/1.79 $false | spl799_11),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f4072])).
% 2.73/1.79 thf(f4072,plain,(
% 2.73/1.79 ($true != $true) | spl799_11),
% 2.73/1.79 inference(superposition,[status(thm)],[f3960,f2789])).
% 2.73/1.79 thf(f2789,plain,(
% 2.73/1.79 ((in @ sK165 @ sK163) = $true)),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.73/1.79 thf(f3960,plain,(
% 2.73/1.79 ((in @ sK165 @ sK163) != $true) | spl799_11),
% 2.73/1.79 inference(avatar_component_clause,[status(thm)],[f3958])).
% 2.73/1.79 thf(f4021,plain,(
% 2.73/1.79 spl799_17 | ~spl799_18),
% 2.73/1.79 inference(avatar_split_clause,[status(thm)],[f4013,f4018,f4015])).
% 2.73/1.79 thf(f4013,plain,(
% 2.73/1.79 ( ! [X0 : $i] : (((in @ sK164 @ (powerset @ sK163)) != $true) | ((in @ X0 @ sK164) != $true) | ((in @ X0 @ sK162) != $true) | ((in @ X0 @ sK163) != $true)) )),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f4012])).
% 2.73/1.79 thf(f4012,plain,(
% 2.73/1.79 ( ! [X0 : $i] : (((in @ X0 @ sK162) != $true) | ($true != $true) | ((in @ X0 @ sK164) != $true) | ((in @ X0 @ sK163) != $true) | ((in @ sK164 @ (powerset @ sK163)) != $true)) )),
% 2.73/1.79 inference(forward_demodulation,[status(thm)],[f4011,f2792])).
% 2.73/1.79 thf(f4011,plain,(
% 2.73/1.79 ( ! [X0 : $i] : (((in @ sK164 @ (powerset @ sK163)) != $true) | ((in @ X0 @ sK164) != $true) | ((in @ sK162 @ (powerset @ sK163)) != $true) | ((in @ X0 @ sK162) != $true) | ((in @ X0 @ sK163) != $true)) )),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f4010])).
% 2.73/1.79 thf(f4010,plain,(
% 2.73/1.79 ( ! [X0 : $i] : (((in @ X0 @ sK162) != $true) | ((in @ X0 @ sK163) != $true) | ((in @ sK162 @ (powerset @ sK163)) != $true) | ((in @ X0 @ sK164) != $true) | ((in @ sK164 @ (powerset @ sK163)) != $true) | ($true != $true)) )),
% 2.73/1.79 inference(forward_demodulation,[status(thm)],[f3986,f2643])).
% 2.73/1.79 thf(f2643,plain,(
% 2.73/1.79 (contrasubsetT = $true)),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.73/1.79 thf(f3986,plain,(
% 2.73/1.79 ( ! [X0 : $i] : (((in @ sK164 @ (powerset @ sK163)) != $true) | (contrasubsetT != $true) | ((in @ X0 @ sK164) != $true) | ((in @ sK162 @ (powerset @ sK163)) != $true) | ((in @ X0 @ sK162) != $true) | ((in @ X0 @ sK163) != $true)) )),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f3984])).
% 2.73/1.79 thf(f3984,plain,(
% 2.73/1.79 ( ! [X0 : $i] : (((in @ X0 @ sK164) != $true) | ((in @ sK162 @ (powerset @ sK163)) != $true) | ((in @ X0 @ sK163) != $true) | ((in @ X0 @ sK162) != $true) | ($true != $true) | ((in @ sK164 @ (powerset @ sK163)) != $true) | (contrasubsetT != $true)) )),
% 2.73/1.79 inference(superposition,[status(thm)],[f2938,f2791])).
% 2.73/1.79 thf(f2791,plain,(
% 2.73/1.79 ((subset @ sK162 @ (setminus @ sK163 @ sK164)) = $true)),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.73/1.79 thf(f2938,plain,(
% 2.73/1.79 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((subset @ X0 @ (setminus @ X1 @ X2)) != $true) | ((in @ X3 @ X0) != $true) | ((in @ X0 @ (powerset @ X1)) != $true) | ((in @ X3 @ X1) != $true) | ((in @ X2 @ (powerset @ X1)) != $true) | ((in @ X3 @ X2) != $true) | (contrasubsetT != $true)) )),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1522])).
% 2.73/1.79 thf(f1522,plain,(
% 2.73/1.79 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ X0 @ (setminus @ X1 @ X2)) != $true) | ((in @ X3 @ X0) != $true) | ((in @ X3 @ X1) != $true)) | ((in @ X2 @ (powerset @ X1)) != $true)) | ((in @ X0 @ (powerset @ X1)) != $true)) | (contrasubsetT != $true)) & ((contrasubsetT = $true) | (((((in @ sK272 @ sK271) = $true) & ((subset @ sK269 @ (setminus @ sK270 @ sK271)) = $true) & ((in @ sK272 @ sK269) = $true) & ((in @ sK272 @ sK270) = $true)) & ((in @ sK271 @ (powerset @ sK270)) = $true)) & ((in @ sK269 @ (powerset @ sK270)) = $true)))),
% 2.73/1.79 inference(skolemisation,[status(esa),new_symbols(skolem,[sK269,sK270,sK271,sK272])],[f1518,f1521,f1520,f1519])).
% 2.73/1.79 thf(f1519,plain,(
% 2.73/1.79 ? [X4 : $i,X5 : $i] : (? [X6 : $i] : (? [X7 : $i] : (((in @ X7 @ X6) = $true) & ((subset @ X4 @ (setminus @ X5 @ X6)) = $true) & ((in @ X7 @ X4) = $true) & ((in @ X7 @ X5) = $true)) & ($true = (in @ X6 @ (powerset @ X5)))) & ((in @ X4 @ (powerset @ X5)) = $true)) => (? [X6 : $i] : (? [X7 : $i] : (((in @ X7 @ X6) = $true) & ($true = (subset @ sK269 @ (setminus @ sK270 @ X6))) & ((in @ X7 @ sK269) = $true) & ((in @ X7 @ sK270) = $true)) & ((in @ X6 @ (powerset @ sK270)) = $true)) & ((in @ sK269 @ (powerset @ sK270)) = $true))),
% 2.73/1.79 introduced(definition,[],[choice_axiom])).
% 2.73/1.79 thf(f1520,plain,(
% 2.73/1.79 ? [X6 : $i] : (? [X7 : $i] : (((in @ X7 @ X6) = $true) & ($true = (subset @ sK269 @ (setminus @ sK270 @ X6))) & ((in @ X7 @ sK269) = $true) & ((in @ X7 @ sK270) = $true)) & ((in @ X6 @ (powerset @ sK270)) = $true)) => (? [X7 : $i] : (((in @ X7 @ sK271) = $true) & ((subset @ sK269 @ (setminus @ sK270 @ sK271)) = $true) & ((in @ X7 @ sK269) = $true) & ((in @ X7 @ sK270) = $true)) & ((in @ sK271 @ (powerset @ sK270)) = $true))),
% 2.73/1.79 introduced(definition,[],[choice_axiom])).
% 2.73/1.79 thf(f1521,plain,(
% 2.73/1.79 ? [X7 : $i] : (((in @ X7 @ sK271) = $true) & ((subset @ sK269 @ (setminus @ sK270 @ sK271)) = $true) & ((in @ X7 @ sK269) = $true) & ((in @ X7 @ sK270) = $true)) => (((in @ sK272 @ sK271) = $true) & ((subset @ sK269 @ (setminus @ sK270 @ sK271)) = $true) & ((in @ sK272 @ sK269) = $true) & ((in @ sK272 @ sK270) = $true))),
% 2.73/1.79 introduced(definition,[],[choice_axiom])).
% 2.73/1.79 thf(f1518,plain,(
% 2.73/1.79 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ X0 @ (setminus @ X1 @ X2)) != $true) | ((in @ X3 @ X0) != $true) | ((in @ X3 @ X1) != $true)) | ((in @ X2 @ (powerset @ X1)) != $true)) | ((in @ X0 @ (powerset @ X1)) != $true)) | (contrasubsetT != $true)) & ((contrasubsetT = $true) | ? [X4 : $i,X5 : $i] : (? [X6 : $i] : (? [X7 : $i] : (((in @ X7 @ X6) = $true) & ((subset @ X4 @ (setminus @ X5 @ X6)) = $true) & ((in @ X7 @ X4) = $true) & ((in @ X7 @ X5) = $true)) & ($true = (in @ X6 @ (powerset @ X5)))) & ((in @ X4 @ (powerset @ X5)) = $true)))),
% 2.73/1.79 inference(rectify,[status(thm)],[f1517])).
% 2.73/1.79 thf(f1517,plain,(
% 2.73/1.79 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ X0 @ (setminus @ X1 @ X2)) != $true) | ((in @ X3 @ X0) != $true) | ((in @ X3 @ X1) != $true)) | ((in @ X2 @ (powerset @ X1)) != $true)) | ((in @ X0 @ (powerset @ X1)) != $true)) | (contrasubsetT != $true)) & ((contrasubsetT = $true) | ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((subset @ X0 @ (setminus @ X1 @ X2)) = $true) & ((in @ X3 @ X0) = $true) & ((in @ X3 @ X1) = $true)) & ((in @ X2 @ (powerset @ X1)) = $true)) & ((in @ X0 @ (powerset @ X1)) = $true)))),
% 2.73/1.79 inference(nnf_transformation,[status(thm)],[f995])).
% 2.73/1.79 thf(f995,plain,(
% 2.73/1.79 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ X0 @ (setminus @ X1 @ X2)) != $true) | ((in @ X3 @ X0) != $true) | ((in @ X3 @ X1) != $true)) | ((in @ X2 @ (powerset @ X1)) != $true)) | ((in @ X0 @ (powerset @ X1)) != $true)) <=> (contrasubsetT = $true)),
% 2.73/1.79 inference(flattening,[status(thm)],[f994])).
% 2.73/1.79 thf(f994,plain,(
% 2.73/1.79 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (! [X3 : $i] : (((((in @ X3 @ X0) != $true) | ((in @ X3 @ X2) != $true)) | ((subset @ X0 @ (setminus @ X1 @ X2)) != $true)) | ((in @ X3 @ X1) != $true)) | ((in @ X2 @ (powerset @ X1)) != $true)) | ((in @ X0 @ (powerset @ X1)) != $true)) <=> (contrasubsetT = $true)),
% 2.73/1.79 inference(ennf_transformation,[status(thm)],[f806])).
% 2.73/1.79 thf(f806,plain,(
% 2.73/1.79 ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => (((subset @ X0 @ (setminus @ X1 @ X2)) = $true) => (((in @ X3 @ X2) = $true) => ((in @ X3 @ X0) != $true)))))) <=> (contrasubsetT = $true)),
% 2.73/1.79 inference(flattening,[status(thm)],[f696])).
% 2.73/1.79 thf(f696,plain,(
% 2.73/1.79 (contrasubsetT = $true) <=> ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => (((subset @ X0 @ (setminus @ X1 @ X2)) = $true) => (((in @ X3 @ X2) = $true) => ~((in @ X3 @ X0) = $true))))))),
% 2.73/1.79 inference(fool_elimination,[status(thm)],[f695])).
% 2.73/1.79 thf(f695,plain,(
% 2.73/1.79 (contrasubsetT = ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X1)) => ! [X3 : $i] : ((in @ X3 @ X1) => ((subset @ X0 @ (setminus @ X1 @ X2)) => ((in @ X3 @ X2) => ~(in @ X3 @ X0)))))))),
% 2.73/1.79 inference(rectify,[status(thm)],[f244])).
% 2.73/1.79 thf(f244,axiom,(
% 2.73/1.79 (contrasubsetT = ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((subset @ X11 @ (setminus @ X3 @ X16)) => ((in @ X1 @ X16) => ~(in @ X1 @ X11)))))))),
% 2.73/1.79 file('/export/starexec/sandbox2/tmp/tmp.1oBuXnfUb8/DTF2THF_10607.p',contrasubsetT)).
% 2.73/1.79 thf(f3982,plain,(
% 2.73/1.79 spl799_10 | ~spl799_12),
% 2.73/1.79 inference(avatar_split_clause,[status(thm)],[f3977,f3979,f3954])).
% 2.73/1.79 thf(f3977,plain,(
% 2.73/1.79 ($true = (in @ sK165 @ sK162)) | ((in @ sK162 @ (powerset @ sK163)) != $true)),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f3976])).
% 2.73/1.79 thf(f3976,plain,(
% 2.73/1.79 ((in @ sK162 @ (powerset @ sK163)) != $true) | ($true != $true) | ($true = (in @ sK165 @ sK162))),
% 2.73/1.79 inference(forward_demodulation,[status(thm)],[f3975,f2789])).
% 2.73/1.79 thf(f3975,plain,(
% 2.73/1.79 ((in @ sK165 @ sK163) != $true) | ((in @ sK162 @ (powerset @ sK163)) != $true) | ($true = (in @ sK165 @ sK162))),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f3974])).
% 2.73/1.79 thf(f3974,plain,(
% 2.73/1.79 ($true != $true) | ((in @ sK165 @ sK163) != $true) | ($true = (in @ sK165 @ sK162)) | ((in @ sK162 @ (powerset @ sK163)) != $true)),
% 2.73/1.79 inference(forward_demodulation,[status(thm)],[f3932,f2663])).
% 2.73/1.79 thf(f2663,plain,(
% 2.73/1.79 (complementTE1 = $true)),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.73/1.79 thf(f3932,plain,(
% 2.73/1.79 (complementTE1 != $true) | ((in @ sK162 @ (powerset @ sK163)) != $true) | ((in @ sK165 @ sK163) != $true) | ($true = (in @ sK165 @ sK162))),
% 2.73/1.79 inference(trivial_inequality_removal,[status(thm)],[f3925])).
% 2.73/1.79 thf(f3925,plain,(
% 2.73/1.79 ($true != $true) | (complementTE1 != $true) | ((in @ sK165 @ sK163) != $true) | ((in @ sK162 @ (powerset @ sK163)) != $true) | ($true = (in @ sK165 @ sK162))),
% 2.73/1.79 inference(superposition,[status(thm)],[f2787,f2892])).
% 2.73/1.79 thf(f2892,plain,(
% 2.73/1.79 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ (setminus @ X4 @ X3)) = $true) | (complementTE1 != $true) | ((in @ X3 @ (powerset @ X4)) != $true) | ((in @ X5 @ X4) != $true) | ((in @ X5 @ X3) = $true)) )),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1473])).
% 2.73/1.79 thf(f1473,plain,(
% 2.73/1.79 ((complementTE1 = $true) | (((in @ sK237 @ (powerset @ sK238)) = $true) & (((in @ sK239 @ sK238) = $true) & ((in @ sK239 @ sK237) != $true) & ((in @ sK239 @ (setminus @ sK238 @ sK237)) != $true)))) & (! [X3 : $i,X4 : $i] : (((in @ X3 @ (powerset @ X4)) != $true) | ! [X5 : $i] : (((in @ X5 @ X4) != $true) | ((in @ X5 @ X3) = $true) | ((in @ X5 @ (setminus @ X4 @ X3)) = $true))) | (complementTE1 != $true))),
% 2.73/1.79 inference(skolemisation,[status(esa),new_symbols(skolem,[sK237,sK238,sK239])],[f1470,f1472,f1471])).
% 2.73/1.79 thf(f1471,plain,(
% 2.73/1.79 ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (((in @ X2 @ X1) = $true) & ($true != (in @ X2 @ X0)) & ((in @ X2 @ (setminus @ X1 @ X0)) != $true))) => (((in @ sK237 @ (powerset @ sK238)) = $true) & ? [X2 : $i] : (((in @ X2 @ sK238) = $true) & ((in @ X2 @ sK237) != $true) & ((in @ X2 @ (setminus @ sK238 @ sK237)) != $true)))),
% 2.73/1.79 introduced(definition,[],[choice_axiom])).
% 2.73/1.79 thf(f1472,plain,(
% 2.73/1.79 ? [X2 : $i] : (((in @ X2 @ sK238) = $true) & ((in @ X2 @ sK237) != $true) & ((in @ X2 @ (setminus @ sK238 @ sK237)) != $true)) => (((in @ sK239 @ sK238) = $true) & ((in @ sK239 @ sK237) != $true) & ((in @ sK239 @ (setminus @ sK238 @ sK237)) != $true))),
% 2.73/1.79 introduced(definition,[],[choice_axiom])).
% 2.73/1.79 thf(f1470,plain,(
% 2.73/1.79 ((complementTE1 = $true) | ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (((in @ X2 @ X1) = $true) & ($true != (in @ X2 @ X0)) & ((in @ X2 @ (setminus @ X1 @ X0)) != $true)))) & (! [X3 : $i,X4 : $i] : (((in @ X3 @ (powerset @ X4)) != $true) | ! [X5 : $i] : (((in @ X5 @ X4) != $true) | ((in @ X5 @ X3) = $true) | ((in @ X5 @ (setminus @ X4 @ X3)) = $true))) | (complementTE1 != $true))),
% 2.73/1.79 inference(rectify,[status(thm)],[f1469])).
% 2.73/1.79 thf(f1469,plain,(
% 2.73/1.79 ((complementTE1 = $true) | ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (((in @ X2 @ X1) = $true) & ($true != (in @ X2 @ X0)) & ((in @ X2 @ (setminus @ X1 @ X0)) != $true)))) & (! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) != $true) | ! [X2 : $i] : (((in @ X2 @ X1) != $true) | ($true = (in @ X2 @ X0)) | ((in @ X2 @ (setminus @ X1 @ X0)) = $true))) | (complementTE1 != $true))),
% 2.73/1.79 inference(nnf_transformation,[status(thm)],[f1073])).
% 2.73/1.79 thf(f1073,plain,(
% 2.73/1.79 (complementTE1 = $true) <=> ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) != $true) | ! [X2 : $i] : (((in @ X2 @ X1) != $true) | ($true = (in @ X2 @ X0)) | ((in @ X2 @ (setminus @ X1 @ X0)) = $true)))),
% 2.73/1.79 inference(flattening,[status(thm)],[f1072])).
% 2.73/1.79 thf(f1072,plain,(
% 2.73/1.79 (complementTE1 = $true) <=> ! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((($true = (in @ X2 @ X0)) | ((in @ X2 @ (setminus @ X1 @ X0)) = $true)) | ((in @ X2 @ X1) != $true)) | ((in @ X0 @ (powerset @ X1)) != $true))),
% 2.73/1.79 inference(ennf_transformation,[status(thm)],[f771])).
% 2.73/1.79 thf(f771,plain,(
% 2.73/1.79 (complementTE1 = $true) <=> ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (((in @ X2 @ X1) = $true) => (((in @ X2 @ (setminus @ X1 @ X0)) != $true) => ($true = (in @ X2 @ X0)))))),
% 2.73/1.79 inference(flattening,[status(thm)],[f283])).
% 2.73/1.79 thf(f283,plain,(
% 2.73/1.79 ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (((in @ X2 @ X1) = $true) => (~((in @ X2 @ (setminus @ X1 @ X0)) = $true) => ($true = (in @ X2 @ X0))))) <=> (complementTE1 = $true)),
% 2.73/1.79 inference(fool_elimination,[status(thm)],[f282])).
% 2.73/1.79 thf(f282,plain,(
% 2.73/1.79 (! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ X1) => (~(in @ X2 @ (setminus @ X1 @ X0)) => (in @ X2 @ X0)))) = complementTE1)),
% 2.73/1.79 inference(rectify,[status(thm)],[f241])).
% 2.73/1.79 thf(f241,axiom,(
% 2.73/1.79 (! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ (setminus @ X3 @ X11)) => (in @ X1 @ X11)))) = complementTE1)),
% 2.73/1.79 file('/export/starexec/sandbox2/tmp/tmp.1oBuXnfUb8/DTF2THF_10607.p',complementTE1)).
% 2.73/1.79 thf(f2787,plain,(
% 2.73/1.79 ((in @ sK165 @ (setminus @ sK163 @ sK162)) != $true)),
% 2.73/1.79 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.73/1.79 % SZS output end Proof for DTF2THF_10607
% 2.73/1.79 % (10918)------------------------------
% 2.73/1.79 % (10918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.79 % (10918)Termination reason: Refutation
% 2.73/1.79
% 2.73/1.79 % (10918)Memory used [KB]: 10874
% 2.73/1.79 % (10918)Time elapsed: 0.158 s
% 2.73/1.79 % (10918)Instructions burned: 509 (million)
% 2.73/1.79 % (10918)------------------------------
% 2.73/1.79 % (10918)------------------------------
% 2.73/1.79 % (10848)Success in time 0.265 s
%------------------------------------------------------------------------------