%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU738^1 : 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 : n008.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 09:36:47 AM UTC 2026
% Result : Theorem 2.93s 1.81s
% Output : Refutation 2.93s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 6
% Syntax : Number of formulae : 29 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 4933 (1364 equ; 0 cnn)
% Maximal formula atoms : 259 ( 170 avg)
% Number of connectives : 2432 ( 47 ~; 16 |; 800 &; 270 @)
% ( 2 <=>;1297 =>; 0 <=; 0 <~>)
% Maximal formula depth : 268 ( 76 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 452 ( 449 usr; 263 con; 0-4 aty)
% Number of variables : 71 ( 0 ^; 44 !; 27 ?; 71 :)
% 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_306,type,
sP0: $i > $o > $i > $i > $o ).
thf(func_def_307,type,
sP1: $i > $i > $o ).
thf(func_def_308,type,
sP2: $i > $o ).
thf(func_def_309,type,
sP3: $i > $i > $o ).
thf(func_def_310,type,
sP4: $i > $i > $o ).
thf(func_def_311,type,
sP5: $i > $i > $i > $i > $o ).
thf(func_def_317,type,
sP11: $i > $i > $i > $o ).
thf(func_def_360,type,
sK54: $i > $o ).
thf(func_def_362,type,
sK56: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_363,type,
sK57: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_370,type,
sK64: $i > $i > $i > $i ).
thf(func_def_371,type,
sK65: $i > $i > $i > $i ).
thf(func_def_377,type,
sK71: $i > $o ).
thf(func_def_390,type,
sK84: $i > $i > $i ).
thf(func_def_398,type,
sK92: $i > $i > $i > $i ).
thf(func_def_407,type,
sK101: $i > $i > $o ).
thf(func_def_408,type,
sK102: $i > $i > $i ).
thf(func_def_409,type,
sK103: $i > $i > $i ).
thf(func_def_410,type,
sK104: $i > $i > $i ).
thf(func_def_411,type,
sK105: $i > $i > $i ).
thf(func_def_412,type,
sK106: $i > $i > $i > $i ).
thf(func_def_413,type,
sK107: $i > $i ).
thf(func_def_414,type,
sK108: $i > $i ).
thf(func_def_415,type,
sK109: $i > $i ).
thf(func_def_416,type,
sK110: $i > $i ).
thf(func_def_417,type,
sK111: $i > $i > $i ).
thf(func_def_418,type,
sK112: $i > $i > $i > $i ).
thf(func_def_419,type,
sK113: $i > $i > $i > $i ).
thf(func_def_420,type,
sK114: $i > $i > $i ).
thf(func_def_421,type,
sK115: $i > $i > $i ).
thf(func_def_423,type,
sK117: $i > $i ).
thf(func_def_424,type,
sK118: $i > $i ).
thf(func_def_425,type,
sK119: $i > $i ).
thf(func_def_426,type,
sK120: $i > $i > $i ).
thf(func_def_428,type,
sK122: $i > $i ).
thf(func_def_431,type,
sK125: $i > ( $i > $i ) > $i > $i ).
thf(func_def_447,type,
sK141: $i > $i ).
thf(func_def_459,type,
sK153: ( $i > $o ) > $i ).
thf(func_def_460,type,
sK154: $i > $o ).
thf(func_def_461,type,
sK155: $i > $i ).
thf(func_def_472,type,
sK166: $i > $o ).
thf(func_def_474,type,
sK168: ( $i > $o ) > $i > $i ).
thf(func_def_479,type,
sK173: $i > $o ).
thf(func_def_484,type,
sK178: $o > $i > $i > $i ).
thf(func_def_485,type,
sK179: $i > $i ).
thf(func_def_497,type,
sK191: $i > $o ).
thf(func_def_498,type,
sK192: $i > $o ).
thf(func_def_499,type,
sK193: $i > $o ).
thf(func_def_500,type,
sK194: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_501,type,
sK195: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_530,type,
sK224: $i > $i ).
thf(func_def_531,type,
sK225: ( $i > $i ) > $i > $i > $i ).
thf(func_def_541,type,
sK235: $i > $i > $i > $i > $i ).
thf(func_def_542,type,
sK236: $i > $i > $i > $i > $i ).
thf(func_def_543,type,
sK237: $i > $i > $i > $i > $i ).
thf(func_def_544,type,
sK238: $i > $i > $i > $i > $i ).
thf(func_def_548,type,
sK242: $i > $i > $i ).
thf(func_def_553,type,
sK247: $i > $o ).
thf(func_def_555,type,
sK249: $i > ( $i > $o ) > $i ).
thf(func_def_569,type,
sK263: $i > $i > ( $i > $i ) > $i ).
thf(func_def_570,type,
sK264: $i > $i ).
thf(func_def_577,type,
sK271: $i > $i > $o ).
thf(func_def_580,type,
sK274: $i > $o ).
thf(func_def_585,type,
sK279: $i > $i > $o ).
thf(func_def_595,type,
sK289: $i > $i > $i ).
thf(func_def_600,type,
sK294: $i > $o ).
thf(func_def_609,type,
sK303: $i > $i > $i ).
thf(func_def_635,type,
sK329: $i > $i ).
thf(func_def_637,type,
sK331: ( $i > $i ) > $i > $i > $i ).
thf(func_def_643,type,
sK337: $i > $i > $o ).
thf(func_def_647,type,
sK341: $i > $o ).
thf(func_def_650,type,
sK344: $i > ( $i > $o ) > $i ).
thf(func_def_651,type,
sK345: $i > $o ).
thf(func_def_655,type,
sK349: $i > $o ).
thf(func_def_675,type,
sK369: $i > $i > $o ).
thf(func_def_684,type,
sK378: $i > $i ).
thf(func_def_687,type,
sK381: $i > $i > $o ).
thf(func_def_688,type,
sK382: $i > $i ).
thf(func_def_689,type,
sK383: $i > $i ).
thf(func_def_690,type,
sK384: ( $i > $i > $o ) > $i > $i ).
thf(func_def_691,type,
sK385: ( $i > $i > $o ) > $i > $i ).
thf(func_def_692,type,
sK386: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_717,type,
sK411: $i > $i > $i ).
thf(func_def_719,type,
sK413: $i > $o ).
thf(func_def_750,type,
sK444: $i > ( $i > $i ) > $i > $i ).
thf(func_def_752,type,
sK446: $i > $i ).
thf(func_def_754,type,
sK448: $i > $i > $i > $i > $i ).
thf(func_def_759,type,
sK453: ( $i > $o ) > $i ).
thf(func_def_760,type,
sK454: $i > $o ).
thf(func_def_767,type,
sK461: $i > $i > $i ).
thf(func_def_781,type,
sK475: $i > $i > $o ).
thf(func_def_783,type,
sK477: $i > ( $i > $o ) > $i ).
thf(func_def_784,type,
sK478: $i > $o ).
thf(func_def_801,type,
sK495: $i > $o ).
thf(func_def_804,type,
sK498: $i > ( $i > $o ) > $i ).
thf(func_def_805,type,
sK499: $i > $o ).
thf(func_def_806,type,
sK500: $i > $o ).
thf(func_def_807,type,
sK501: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_808,type,
sK502: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_815,type,
sK509: $i > $i ).
thf(func_def_827,type,
sK521: $i > $o ).
thf(func_def_829,type,
sK523: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_830,type,
sK524: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_837,type,
sK531: $i > $i ).
thf(func_def_849,type,
sK543: $i > $o ).
thf(func_def_878,type,
sK572: ( $i > $o ) > $i > $i > $i ).
thf(func_def_881,type,
sK575: $i > $o ).
thf(func_def_887,type,
sK581: $i > $o ).
thf(func_def_888,type,
sK582: $i > $o ).
thf(func_def_901,type,
sK595: ( $i > $i ) > $i > $i > $i ).
thf(func_def_904,type,
sK598: $i > $i ).
thf(func_def_912,type,
sK606: $i > $o ).
thf(func_def_933,type,
sK627: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_934,type,
sK628: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_937,type,
sK631: $i > $o ).
thf(func_def_938,type,
sK632: $i > $o ).
thf(func_def_953,type,
sK647: $i > $i > $i ).
thf(func_def_956,type,
sK650: $i > $i > $i > $i > $i ).
thf(func_def_961,type,
sK655: $i > $o ).
thf(func_def_962,type,
sK656: $i > $i ).
thf(func_def_963,type,
sK657: ( $i > $o ) > $i ).
thf(func_def_965,type,
sK659: $i > $o ).
thf(func_def_967,type,
sK661: ( $i > $o ) > $i > $i ).
thf(func_def_976,type,
sK670: $i > $o ).
thf(func_def_979,type,
sK673: $i > ( $i > $o ) > $i ).
thf(func_def_980,type,
sK674: $i > ( $i > $o ) > $i ).
thf(func_def_984,type,
sK678: $i > $i > $i > $i ).
thf(func_def_989,type,
sK683: $i > $i > $i > $i > $i ).
thf(func_def_990,type,
sK684: $i > $i > $i > $i > $i ).
thf(func_def_999,type,
sK693: $i > $i > $o ).
thf(func_def_1008,type,
sK702: $i > $i > $i ).
thf(func_def_1009,type,
sK703: $i > $i > $i ).
thf(func_def_1010,type,
sK704: $i > $o ).
thf(func_def_1013,type,
sK707: $i > ( $i > $o ) > $i ).
thf(func_def_1014,type,
sK708: $i > ( $i > $o ) > $i ).
thf(func_def_1015,type,
sK709: $i > $o ).
thf(func_def_1020,type,
sK714: $i > $i ).
thf(func_def_1023,type,
sK717: $i > $i ).
thf(func_def_1024,type,
sK718: ( $i > $o ) > $i ).
thf(func_def_1025,type,
sK719: ( $i > $o ) > $i ).
thf(func_def_1026,type,
sK720: $i > $o ).
thf(func_def_1036,type,
sK730: $i > $o ).
thf(func_def_1039,type,
sK733: $i > ( $i > $o ) > $i ).
thf(func_def_1040,type,
sK734: $i > ( $i > $o ) > $i ).
thf(func_def_1044,type,
sK738: $i > $i > $i ).
thf(func_def_1067,type,
sK761: $i > $o ).
thf(func_def_1083,type,
sK777: $i > $i > $i > $i > $i ).
thf(func_def_1088,type,
sK782: $i > $o ).
thf(func_def_1089,type,
sK783: ( $i > $o ) > $i > $i ).
thf(func_def_1097,type,
sK791: $i > $i > $i > $i ).
thf(func_def_1101,type,
sK795: $i > $i > $i > $i ).
thf(func_def_1113,type,
ph807:
!>[X0: $tType] : X0 ).
thf(f4014,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f4006]) ).
thf(f4006,plain,
$true != $true,
inference(superposition,[status(thm)],[f3975,f2567]) ).
thf(f2567,plain,
( ( in @ sK134 @ sK132 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1322]) ).
thf(f1322,plain,
( ( sepInPowerset = $true )
& ( brelnall2 = $true )
& ( binunionIL = $true )
& ( exuE3e = $true )
& ( dpsetconstrER = $true )
& ( subbreln = $true )
& ( setukpairIL = $true )
& ( vacuousDall = $true )
& ( inCongP = $true )
& ( setadjoinIR = $true )
& ( dpsetconstrEL2 = $true )
& ( apProp = $true )
& ( ex1E2 = $true )
& ( ex1I2 = $true )
& ( cartprodpairsurjEq = $true )
& ( quantDeMorgan4 = $true )
& ( setukpairinjL1 = $true )
& ( eqinunit = $true )
& ( dpsetconstrSub = $true )
& ( complementTI1 = $true )
& ( disjointsetsI1 = $true )
& ( ifp = $true )
& ( cartprodsndpairEq = $true )
& ( binintersectTERcontra = $true )
& ( descr__Cong = $true )
& ( exuI1 = $true )
& ( binintersectLsub = $true )
& ( funcGraphProp3 = $true )
& ( setminusLsub = $true )
& ( contrasubsetT = $true )
& ( subsetI1 = $true )
& ( descrp = $true )
& ( iffalse = $true )
& ( uniqinunit = $true )
& ( dsetconstrI = $true )
& ( ex1E1 = $true )
& ( omegaSAx = $true )
& ( cartprodpairmemEL = $true )
& ( brelnall1 = $true )
& ( binintersectEL = $true )
& ( singletonsubset = $true )
& ( setoftrueEq = $true )
& ( complementImpComplementIntersect = $true )
& ( ksndpairEq = $true )
& ( dsetconstr__Cong = $true )
& ( subsetTrans = $true )
& ( foundationAx = $true )
& ( binunionIR = $true )
& ( notsubsetI = $true )
& ( nonemptyI = $true )
& ( symdiffIneg1 = $true )
& ( binintersectSubset3 = $true )
& ( eqimpsubset2 = $true )
& ( doubleComplementSub1 = $true )
& ( exuE3u = $true )
& ( setukpairinjR12 = $true )
& ( symdiffE = $true )
& ( notinemptyset = $true )
& ( kpairp = $true )
& ( powersetAx = $true )
& ( beta1 = $true )
& ( upairsetIL = $true )
& ( noeltsimpempty = $true )
& ( exuE2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setukpairinjR2 = $true )
& ( dsetconstrER = $true )
& ( upairsubunion = $true )
& ( cartprodmempair = $true )
& ( subsetRefl = $true )
& ( powersetsubset = $true )
& ( emptysetE = $true )
& ( iffalseProp2 = $true )
& ( binintersectSubset4 = $true )
& ( iftrue = $true )
& ( setminusELneg = $true )
& ( ap2apEq1 = $true )
& ( setunionsingleton1 = $true )
& ( subset2powerset = $true )
& ( setOfPairsIsBReln = $true )
& ( ap2apEq2 = $true )
& ( binintersectSubset1 = $true )
& ( eta1 = $true )
& ( setminusIRneg = $true )
& ( contraSubsetComplement = $true )
& ( setunionE = $true )
& ( notequalI2 = $true )
& ( funcGraphProp4 = $true )
& ( kpairsurjEq = $true )
& ( funcext = $true )
& ( omega__Cong = $true )
& ( setextT = $true )
& ( subPowSU = $true )
& ( setadjoinE = $true )
& ( emptyI = $true )
& ( kfstsingleton = $true )
& ( kpairiskpair = $true )
& ( ex1I = $true )
& ( setukpairinjL = $true )
& ( complementTcontraSubset = $true )
& ( powersetE = $true )
& ( binintersectSubset2 = $true )
& ( theeq = $true )
& ( setunionsingleton = $true )
& ( in__Cong = $true )
& ( subsetemptysetimpeq = $true )
& ( eqimpsubset1 = $true )
& ( emptysetAx = $true )
& ( bs114d = $true )
& ( notinsingleton = $true )
& ( dpsetconstrI = $true )
& ( upairset2E = $true )
& ( kfstpairEq = $true )
& ( funcGraphProp1 = $true )
& ( upairinpowunion = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( emptysetimpfalse = $true )
& ( setextsub = $true )
& ( emptyE1 = $true )
& ( doubleComplementSub2 = $true )
& ( subsetI2 = $true )
& ( lam2lamEq = $true )
& ( complementT_lem = $true )
& ( infuncsetfunc = $true )
& ( funcext2 = $true )
& ( beta2 = $true )
& ( binintersectSubset5 = $true )
& ( setminusSubset2 = $true )
& ( subsetE2 = $true )
& ( upairset2IR = $true )
& ( prop2set2propI = $true )
& ( notequalI1 = $true )
& ( funcextLem = $true )
& ( ap2p = $true )
& ( binintersectTELcontra = $true )
& ( omegaIndAx = $true )
& ( app = $true )
& ( exuI2 = $true )
& ( binunionLsub = $true )
& ( upairsetIR = $true )
& ( lamp = $true )
& ( setadjoinSub2 = $true )
& ( setukpairinjR1 = $true )
& ( prop2setI = $true )
& ( nonemptyE1 = $true )
& ( setminusT_lem = $true )
& ( wellorderingAx = $true )
& ( setminusI = $true )
& ( setadjoinSub = $true )
& ( iftrueProp1 = $true )
& ( omega0Ax = $true )
& ( setukpairinjR = $true )
& ( emptyinPowerset = $true )
& ( binintersectER = $true )
& ( singletoninpowerset = $true )
& ( emptysetsubset = $true )
& ( sepSubset = $true )
& ( upairequniteq = $true )
& ( funcGraphProp2 = $true )
& ( powersetI1 = $true )
& ( setunionAx = $true )
& ( nonemptyI1 = $true )
& ( cartprodpairmemER = $true )
& ( binintersectRsub = $true )
& ( emptyset__Cong = $true )
& ( powersetTE1 = $true )
& ( powerset__Cong = $true )
& ( setadjoinOr = $true )
& ( subsetE = $true )
& ( contrasubsetT1 = $true )
& ( setadjoinAx = $true )
& ( setunionsingleton2 = $true )
& ( symdiffIneg2 = $true )
& ( singletonsswitch = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( binunionEcases = $true )
& ( doubleComplementE1 = $true )
& ( setminusEL = $true )
& ( cartprodfstpairEq = $true )
& ( ksndsingleton = $true )
& ( setminusSubset1 = $true )
& ( iffalseProp1 = $true )
& ( subsetTI = $true )
& ( dpsetconstrERa = $true )
& ( contrasubsetT2 = $true )
& ( binintersectI = $true )
& ( doubleComplementI1 = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( complementTnotintersectT = $true )
& ( theprop = $true )
& ( singletonprop = $true )
& ( upairsetE = $true )
& ( iftrueorfalse = $true )
& ( powersetTI1 = $true )
& ( funcImageSingleton = $true )
& ( setext = $true )
& ( prop2setE = $true )
& ( notdexE = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( cartprodsndin = $true )
& ( exu__Cong = $true )
& ( singletonsuniq = $true )
& ( setunionI = $true )
& ( inPowerset = $true )
& ( setunion__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( ifSingleton = $true )
& ( binunionRsub = $true )
& ( exuEu = $true )
& ( contrasubsetT3 = $true )
& ( replAx = $true )
& ( setadjoin__Cong = $true )
& ( ubforcartprodlem2 = $true )
& ( setminusERneg = $true )
& ( setukpairinjL2 = $true )
& ( exuI3 = $true )
& ( lamProp = $true )
& ( emptyInPowerset = $true )
& ( setukpairinjR11 = $true )
& ( setukpairIR = $true )
& ( ( in @ sK132 @ ( powerset @ sK131 ) )
= $true )
& ( ( in @ sK134 @ sK132 )
= $true )
& ( ( in @ sK134 @ sK131 )
= $true )
& ( ( in @ sK134 @ ( binunion @ sK132 @ sK133 ) )
!= $true )
& ( $true
= ( in @ sK133 @ ( powerset @ sK131 ) ) )
& ( complementTE1 = $true )
& ( eqbreln = $true )
& ( setunionE2 = $true )
& ( funcinfuncset = $true )
& ( binunionT_lem = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodfstin = $true )
& ( setminusER = $true )
& ( emptyinunitempty = $true )
& ( setminusILneg = $true )
& ( setextAx = $true )
& ( cartprodmempaircEq = $true )
& ( secondinupair = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinIL = $true )
& ( binintersectT_lem = $true )
& ( powersetT_lem = $true )
& ( notdallE = $true )
& ( iftrueProp2 = $true )
& ( setbeta = $true )
& ( powersetE1 = $true )
& ( powersetI = $true )
& ( singletoninpowunion = $true )
& ( binunionE = $true )
& ( dsetconstrEL = $true )
& ( lam2p = $true )
& ( dpsetconstrEL1 = $true )
& ( doubleComplementEq = $true )
& ( eta2 = $true )
& ( exuE1 = $true )
& ( ubforcartprodlem1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK131,sK132,sK133,sK134])],[f851,f1321,f1320,f1319]) ).
thf(f1319,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( $true
= ( in @ X3 @ X0 ) )
& ( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
!= $true ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
=> ( ( ( in @ sK132 @ ( powerset @ sK131 ) )
= $true )
& ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK132 ) )
& ( ( in @ X3 @ sK131 )
= $true )
& ( ( in @ X3 @ ( binunion @ sK132 @ X2 ) )
!= $true ) )
& ( ( in @ X2 @ ( powerset @ sK131 ) )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1320,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK132 ) )
& ( ( in @ X3 @ sK131 )
= $true )
& ( ( in @ X3 @ ( binunion @ sK132 @ X2 ) )
!= $true ) )
& ( ( in @ X2 @ ( powerset @ sK131 ) )
= $true ) )
=> ( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK132 ) )
& ( ( in @ X3 @ sK131 )
= $true )
& ( ( in @ X3 @ ( binunion @ sK132 @ sK133 ) )
!= $true ) )
& ( $true
= ( in @ sK133 @ ( powerset @ sK131 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1321,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK132 ) )
& ( ( in @ X3 @ sK131 )
= $true )
& ( ( in @ X3 @ ( binunion @ sK132 @ sK133 ) )
!= $true ) )
=> ( ( ( in @ sK134 @ sK132 )
= $true )
& ( ( in @ sK134 @ sK131 )
= $true )
& ( ( in @ sK134 @ ( binunion @ sK132 @ sK133 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f851,plain,
( ( sepInPowerset = $true )
& ( brelnall2 = $true )
& ( binunionIL = $true )
& ( exuE3e = $true )
& ( dpsetconstrER = $true )
& ( subbreln = $true )
& ( setukpairIL = $true )
& ( vacuousDall = $true )
& ( inCongP = $true )
& ( setadjoinIR = $true )
& ( dpsetconstrEL2 = $true )
& ( apProp = $true )
& ( ex1E2 = $true )
& ( ex1I2 = $true )
& ( cartprodpairsurjEq = $true )
& ( quantDeMorgan4 = $true )
& ( setukpairinjL1 = $true )
& ( eqinunit = $true )
& ( dpsetconstrSub = $true )
& ( complementTI1 = $true )
& ( disjointsetsI1 = $true )
& ( ifp = $true )
& ( cartprodsndpairEq = $true )
& ( binintersectTERcontra = $true )
& ( descr__Cong = $true )
& ( exuI1 = $true )
& ( binintersectLsub = $true )
& ( funcGraphProp3 = $true )
& ( setminusLsub = $true )
& ( contrasubsetT = $true )
& ( subsetI1 = $true )
& ( descrp = $true )
& ( iffalse = $true )
& ( uniqinunit = $true )
& ( dsetconstrI = $true )
& ( ex1E1 = $true )
& ( omegaSAx = $true )
& ( cartprodpairmemEL = $true )
& ( brelnall1 = $true )
& ( binintersectEL = $true )
& ( singletonsubset = $true )
& ( setoftrueEq = $true )
& ( complementImpComplementIntersect = $true )
& ( ksndpairEq = $true )
& ( dsetconstr__Cong = $true )
& ( subsetTrans = $true )
& ( foundationAx = $true )
& ( binunionIR = $true )
& ( notsubsetI = $true )
& ( nonemptyI = $true )
& ( symdiffIneg1 = $true )
& ( binintersectSubset3 = $true )
& ( eqimpsubset2 = $true )
& ( doubleComplementSub1 = $true )
& ( exuE3u = $true )
& ( setukpairinjR12 = $true )
& ( symdiffE = $true )
& ( notinemptyset = $true )
& ( kpairp = $true )
& ( powersetAx = $true )
& ( beta1 = $true )
& ( upairsetIL = $true )
& ( noeltsimpempty = $true )
& ( exuE2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setukpairinjR2 = $true )
& ( dsetconstrER = $true )
& ( upairsubunion = $true )
& ( cartprodmempair = $true )
& ( subsetRefl = $true )
& ( powersetsubset = $true )
& ( emptysetE = $true )
& ( iffalseProp2 = $true )
& ( binintersectSubset4 = $true )
& ( iftrue = $true )
& ( setminusELneg = $true )
& ( ap2apEq1 = $true )
& ( setunionsingleton1 = $true )
& ( subset2powerset = $true )
& ( setOfPairsIsBReln = $true )
& ( ap2apEq2 = $true )
& ( binintersectSubset1 = $true )
& ( eta1 = $true )
& ( setminusIRneg = $true )
& ( contraSubsetComplement = $true )
& ( setunionE = $true )
& ( notequalI2 = $true )
& ( funcGraphProp4 = $true )
& ( kpairsurjEq = $true )
& ( funcext = $true )
& ( omega__Cong = $true )
& ( setextT = $true )
& ( subPowSU = $true )
& ( setadjoinE = $true )
& ( emptyI = $true )
& ( kfstsingleton = $true )
& ( kpairiskpair = $true )
& ( ex1I = $true )
& ( setukpairinjL = $true )
& ( complementTcontraSubset = $true )
& ( powersetE = $true )
& ( binintersectSubset2 = $true )
& ( theeq = $true )
& ( setunionsingleton = $true )
& ( in__Cong = $true )
& ( subsetemptysetimpeq = $true )
& ( eqimpsubset1 = $true )
& ( emptysetAx = $true )
& ( bs114d = $true )
& ( notinsingleton = $true )
& ( dpsetconstrI = $true )
& ( upairset2E = $true )
& ( kfstpairEq = $true )
& ( funcGraphProp1 = $true )
& ( upairinpowunion = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( emptysetimpfalse = $true )
& ( setextsub = $true )
& ( emptyE1 = $true )
& ( doubleComplementSub2 = $true )
& ( subsetI2 = $true )
& ( lam2lamEq = $true )
& ( complementT_lem = $true )
& ( infuncsetfunc = $true )
& ( funcext2 = $true )
& ( beta2 = $true )
& ( binintersectSubset5 = $true )
& ( setminusSubset2 = $true )
& ( subsetE2 = $true )
& ( upairset2IR = $true )
& ( prop2set2propI = $true )
& ( notequalI1 = $true )
& ( funcextLem = $true )
& ( ap2p = $true )
& ( binintersectTELcontra = $true )
& ( omegaIndAx = $true )
& ( app = $true )
& ( exuI2 = $true )
& ( binunionLsub = $true )
& ( upairsetIR = $true )
& ( lamp = $true )
& ( setadjoinSub2 = $true )
& ( setukpairinjR1 = $true )
& ( prop2setI = $true )
& ( nonemptyE1 = $true )
& ( setminusT_lem = $true )
& ( wellorderingAx = $true )
& ( setminusI = $true )
& ( setadjoinSub = $true )
& ( iftrueProp1 = $true )
& ( omega0Ax = $true )
& ( setukpairinjR = $true )
& ( emptyinPowerset = $true )
& ( binintersectER = $true )
& ( singletoninpowerset = $true )
& ( emptysetsubset = $true )
& ( sepSubset = $true )
& ( upairequniteq = $true )
& ( funcGraphProp2 = $true )
& ( powersetI1 = $true )
& ( setunionAx = $true )
& ( nonemptyI1 = $true )
& ( cartprodpairmemER = $true )
& ( binintersectRsub = $true )
& ( emptyset__Cong = $true )
& ( powersetTE1 = $true )
& ( powerset__Cong = $true )
& ( setadjoinOr = $true )
& ( subsetE = $true )
& ( contrasubsetT1 = $true )
& ( setadjoinAx = $true )
& ( setunionsingleton2 = $true )
& ( symdiffIneg2 = $true )
& ( singletonsswitch = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( binunionEcases = $true )
& ( doubleComplementE1 = $true )
& ( setminusEL = $true )
& ( cartprodfstpairEq = $true )
& ( ksndsingleton = $true )
& ( setminusSubset1 = $true )
& ( iffalseProp1 = $true )
& ( subsetTI = $true )
& ( dpsetconstrERa = $true )
& ( contrasubsetT2 = $true )
& ( binintersectI = $true )
& ( doubleComplementI1 = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( complementTnotintersectT = $true )
& ( theprop = $true )
& ( singletonprop = $true )
& ( upairsetE = $true )
& ( iftrueorfalse = $true )
& ( powersetTI1 = $true )
& ( funcImageSingleton = $true )
& ( setext = $true )
& ( prop2setE = $true )
& ( notdexE = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( cartprodsndin = $true )
& ( exu__Cong = $true )
& ( singletonsuniq = $true )
& ( setunionI = $true )
& ( inPowerset = $true )
& ( setunion__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( ifSingleton = $true )
& ( binunionRsub = $true )
& ( exuEu = $true )
& ( contrasubsetT3 = $true )
& ( replAx = $true )
& ( setadjoin__Cong = $true )
& ( ubforcartprodlem2 = $true )
& ( setminusERneg = $true )
& ( setukpairinjL2 = $true )
& ( exuI3 = $true )
& ( lamProp = $true )
& ( emptyInPowerset = $true )
& ( setukpairinjR11 = $true )
& ( setukpairIR = $true )
& ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( $true
= ( in @ X3 @ X0 ) )
& ( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
!= $true ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
& ( complementTE1 = $true )
& ( eqbreln = $true )
& ( setunionE2 = $true )
& ( funcinfuncset = $true )
& ( binunionT_lem = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodfstin = $true )
& ( setminusER = $true )
& ( emptyinunitempty = $true )
& ( setminusILneg = $true )
& ( setextAx = $true )
& ( cartprodmempaircEq = $true )
& ( secondinupair = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinIL = $true )
& ( binintersectT_lem = $true )
& ( powersetT_lem = $true )
& ( notdallE = $true )
& ( iftrueProp2 = $true )
& ( setbeta = $true )
& ( powersetE1 = $true )
& ( powersetI = $true )
& ( singletoninpowunion = $true )
& ( binunionE = $true )
& ( dsetconstrEL = $true )
& ( lam2p = $true )
& ( dpsetconstrEL1 = $true )
& ( doubleComplementEq = $true )
& ( eta2 = $true )
& ( exuE1 = $true )
& ( ubforcartprodlem1 = $true ) ),
inference(flattening,[status(thm)],[f850]) ).
thf(f850,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
!= $true )
& ( $true
= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f794]) ).
thf(f794,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ( ( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
!= $true )
=> ( ( in @ X3 @ X1 )
!= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f435]) ).
thf(f435,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ( ( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
!= $true )
=> ( ( in @ X3 @ X1 )
!= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f434]) ).
thf(f434,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ~ ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f260]) ).
thf(f260,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ ( binunion @ X11 @ X16 ) )
=> ~ ( in @ X1 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f259]) ).
thf(f259,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ ( binunion @ X11 @ X16 ) )
=> ~ ( in @ X1 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.ebZAxzcvYS/DTF2THF_11747.p',binunionTILcontra) ).
thf(f3975,plain,
( ( in @ sK134 @ sK132 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3974]) ).
thf(f3974,plain,
( ( $true != $true )
| ( ( in @ sK134 @ sK132 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3966,f2789]) ).
thf(f2789,plain,
binunionIL = $true,
inference(cnf_transformation,[status(thm)],[f1322]) ).
thf(f3966,plain,
( ( binunionIL != $true )
| ( ( in @ sK134 @ sK132 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3957]) ).
thf(f3957,plain,
( ( ( in @ sK134 @ sK132 )
!= $true )
| ( binunionIL != $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2565,f2973]) ).
thf(f2973,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( binunion @ X5 @ X4 ) )
= $true )
| ( ( in @ X3 @ X5 )
!= $true )
| ( binunionIL != $true ) ),
inference(cnf_transformation,[status(thm)],[f1543]) ).
thf(f1543,plain,
( ( ( binunionIL = $true )
| ( ( ( in @ sK267 @ ( binunion @ sK269 @ sK268 ) )
!= $true )
& ( ( in @ sK267 @ sK269 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( binunion @ X5 @ X4 ) )
= $true )
| ( ( in @ X3 @ X5 )
!= $true ) )
| ( binunionIL != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK267,sK268,sK269])],[f1541,f1542]) ).
thf(f1542,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
!= $true )
& ( $true
= ( in @ X0 @ X2 ) ) )
=> ( ( ( in @ sK267 @ ( binunion @ sK269 @ sK268 ) )
!= $true )
& ( ( in @ sK267 @ sK269 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1541,plain,
( ( ( binunionIL = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
!= $true )
& ( $true
= ( in @ X0 @ X2 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( binunion @ X5 @ X4 ) )
= $true )
| ( ( in @ X3 @ X5 )
!= $true ) )
| ( binunionIL != $true ) ) ),
inference(rectify,[status(thm)],[f1540]) ).
thf(f1540,plain,
( ( ( binunionIL = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
!= $true )
& ( $true
= ( in @ X0 @ X2 ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
| ( $true
!= ( in @ X0 @ X2 ) ) )
| ( binunionIL != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f824]) ).
thf(f824,plain,
( ( binunionIL = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
| ( $true
!= ( in @ X0 @ X2 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f533]) ).
thf(f533,plain,
( ( binunionIL = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X2 ) )
=> ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f532]) ).
thf(f532,plain,
( binunionIL
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ( in @ X0 @ ( binunion @ X2 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f101]) ).
thf(f101,axiom,
( binunionIL
= ( ! [X1: $i,X4: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ ( binunion @ X3 @ X4 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.ebZAxzcvYS/DTF2THF_11747.p',binunionIL) ).
thf(f2565,plain,
( ( in @ sK134 @ ( binunion @ sK132 @ sK133 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1322]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU738^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n008.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:32:39 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.47 ---- Original DTF file ---
% 0.31/0.47 thf(spec,logic,$$dhol).
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % File : SEU738^1 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.47 % Domain : Set Theory
% 0.31/0.47 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.31/0.47 % Version : Especial.
% 0.31/0.47 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.31/0.47 % (! x:i.in x A -> ~(in x (binunion X Y)) -> ~(in x X))))
% 0.31/0.47
% 0.31/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.47 % Source : [Bro08]
% 0.31/0.47 % Names : ZFC240g [Bro08]
% 0.31/0.47
% 0.31/0.47 % Status : Theorem
% 0.31/0.47 % Rating : 0.11 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.31/0.47 % Syntax : Number of formulae : 551 ( 258 unt; 292 typ; 258 def)
% 0.31/0.47 % Number of atoms : 1491 ( 384 equ; 0 cnn)
% 0.31/0.47 % Maximal formula atoms : 259 ( 5 avg)
% 0.31/0.47 % Number of connectives : 2993 ( 67 ~; 12 |; 44 &;2130 @)
% 0.31/0.47 % ( 14 <=>; 726 =>; 0 <=; 0 <~>)
% 0.31/0.47 % Maximal formula depth : 267 ( 2 avg)
% 0.31/0.47 % Number of types : 2 ( 0 usr)
% 0.31/0.47 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.31/0.47 % Number of symbols : 295 ( 292 usr; 258 con; 0-4 aty)
% 0.31/0.47 % Number of variables : 867 ( 74 ^; 758 !; 35 ?; 867 :)
% 0.31/0.47 % SPC : TH0_THM_EQU_NAR
% 0.31/0.47
% 0.31/0.47 % Comments : http://mathgate.info/detsetitem.php?id=300
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 thf(in_type,type,
% 0.31/0.47 in: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(exu_type,type,
% 0.31/0.47 exu: ( $i > $o ) > $o ).
% 0.31/0.47
% 0.31/0.47 thf(exu,definition,
% 0.31/0.47 ( exu
% 0.31/0.47 = ( ^ [Xphi: $i > $o] :
% 0.31/0.47 ? [Xx: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 & ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setextAx_type,type,
% 0.31/0.47 setextAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setextAx,definition,
% 0.31/0.47 ( setextAx
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 <=> ( in @ Xx @ B ) )
% 0.31/0.47 => ( A = B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyset_type,type,
% 0.31/0.47 emptyset: $i ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetAx_type,type,
% 0.31/0.47 emptysetAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetAx,definition,
% 0.31/0.47 ( emptysetAx
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoin_type,type,
% 0.31/0.47 setadjoin: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinAx_type,type,
% 0.31/0.47 setadjoinAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinAx,definition,
% 0.31/0.47 ( setadjoinAx
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.47 <=> ( ( Xy = Xx )
% 0.31/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powerset_type,type,
% 0.31/0.47 powerset: $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(powersetAx_type,type,
% 0.31/0.47 powersetAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetAx,definition,
% 0.31/0.47 ( powersetAx
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.47 <=> ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunion_type,type,
% 0.31/0.47 setunion: $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(setunionAx_type,type,
% 0.31/0.47 setunionAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionAx,definition,
% 0.31/0.47 ( setunionAx
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.47 <=> ? [B: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(omega_type,type,
% 0.31/0.47 omega: $i ).
% 0.31/0.47
% 0.31/0.47 thf(omega0Ax_type,type,
% 0.31/0.47 omega0Ax: $o ).
% 0.31/0.47
% 0.31/0.47 thf(omega0Ax,definition,
% 0.31/0.47 ( omega0Ax
% 0.31/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(omegaSAx_type,type,
% 0.31/0.47 omegaSAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(omegaSAx,definition,
% 0.31/0.47 ( omegaSAx
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ omega )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(omegaIndAx_type,type,
% 0.31/0.47 omegaIndAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(omegaIndAx,definition,
% 0.31/0.47 ( omegaIndAx
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( ( in @ emptyset @ A )
% 0.31/0.47 & ! [Xx: $i] :
% 0.31/0.47 ( ( ( in @ Xx @ omega )
% 0.31/0.47 & ( in @ Xx @ A ) )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ omega )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(replAx_type,type,
% 0.31/0.47 replAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(replAx,definition,
% 0.31/0.47 ( replAx
% 0.31/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( exu
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.47 => ? [B: $i] :
% 0.31/0.47 ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 <=> ? [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(foundationAx_type,type,
% 0.31/0.47 foundationAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(foundationAx,definition,
% 0.31/0.47 ( foundationAx
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.47 => ? [B: $i] :
% 0.31/0.47 ( ( in @ B @ A )
% 0.31/0.47 & ~ ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(wellorderingAx_type,type,
% 0.31/0.47 wellorderingAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(wellorderingAx,definition,
% 0.31/0.47 ( wellorderingAx
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ? [B: $i] :
% 0.31/0.47 ( ! [C: $i] :
% 0.31/0.47 ( ( in @ C @ B )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ C )
% 0.31/0.47 => ( in @ Xx @ A ) ) )
% 0.31/0.47 & ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( ( in @ Xx @ A )
% 0.31/0.47 & ( in @ Xy @ A ) )
% 0.31/0.47 => ( ! [C: $i] :
% 0.31/0.47 ( ( in @ C @ B )
% 0.31/0.47 => ( ( in @ Xx @ C )
% 0.31/0.47 <=> ( in @ Xy @ C ) ) )
% 0.31/0.47 => ( Xx = Xy ) ) )
% 0.31/0.47 & ! [C: $i,D: $i] :
% 0.31/0.47 ( ( ( in @ C @ B )
% 0.31/0.47 & ( in @ D @ B ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ C )
% 0.31/0.47 => ( in @ Xx @ D ) )
% 0.31/0.47 | ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ D )
% 0.31/0.47 => ( in @ Xx @ C ) ) ) )
% 0.31/0.47 & ! [C: $i] :
% 0.31/0.47 ( ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ C )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/0.47 => ? [D: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ D @ B )
% 0.31/0.47 & ( in @ Xx @ C )
% 0.31/0.47 & ~ ? [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ D )
% 0.31/0.47 & ( in @ Xy @ C ) )
% 0.31/0.47 & ! [E: $i] :
% 0.31/0.47 ( ( in @ E @ B )
% 0.31/0.47 => ( ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ E )
% 0.31/0.47 => ( in @ Xy @ D ) )
% 0.31/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(descr_type,type,
% 0.31/0.47 descr: ( $i > $o ) > $i ).
% 0.31/0.47
% 0.31/0.47 thf(descrp_type,type,
% 0.31/0.47 descrp: $o ).
% 0.31/0.47
% 0.31/0.47 thf(descrp,definition,
% 0.31/0.47 ( descrp
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ( Xphi
% 0.31/0.47 @ ( descr
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstr_type,type,
% 0.31/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrI_type,type,
% 0.31/0.47 dsetconstrI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrI,definition,
% 0.31/0.47 ( dsetconstrI
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrEL_type,type,
% 0.31/0.47 dsetconstrEL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrEL,definition,
% 0.31/0.47 ( dsetconstrEL
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrER_type,type,
% 0.31/0.47 dsetconstrER: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrER,definition,
% 0.31/0.47 ( dsetconstrER
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE1_type,type,
% 0.31/0.47 exuE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE1,definition,
% 0.31/0.47 ( exuE1
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 & ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(prop2set_type,type,
% 0.31/0.47 prop2set: $o > $i ).
% 0.31/0.47
% 0.31/0.47 thf(prop2setE_type,type,
% 0.31/0.47 prop2setE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(prop2setE,definition,
% 0.31/0.47 ( prop2setE
% 0.31/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/0.47 => Xphi ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetE_type,type,
% 0.31/0.47 emptysetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetE,definition,
% 0.31/0.47 ( emptysetE
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetimpfalse_type,type,
% 0.31/0.47 emptysetimpfalse: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetimpfalse,definition,
% 0.31/0.47 ( emptysetimpfalse
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => $false ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notinemptyset_type,type,
% 0.31/0.47 notinemptyset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notinemptyset,definition,
% 0.31/0.47 ( notinemptyset
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE3e_type,type,
% 0.31/0.47 exuE3e: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE3e,definition,
% 0.31/0.47 ( exuE3e
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setext_type,type,
% 0.31/0.47 setext: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setext,definition,
% 0.31/0.47 ( setext
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 => ( A = B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyI_type,type,
% 0.31/0.47 emptyI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyI,definition,
% 0.31/0.47 ( emptyI
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ A )
% 0.31/0.47 => ( A = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(noeltsimpempty_type,type,
% 0.31/0.47 noeltsimpempty: $o ).
% 0.31/0.47
% 0.31/0.47 thf(noeltsimpempty,definition,
% 0.31/0.47 ( noeltsimpempty
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ A )
% 0.31/0.47 => ( A = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setbeta_type,type,
% 0.31/0.47 setbeta: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setbeta,definition,
% 0.31/0.47 ( setbeta
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonempty_type,type,
% 0.31/0.47 nonempty: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonempty,definition,
% 0.31/0.47 ( nonempty
% 0.31/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyE1_type,type,
% 0.31/0.47 nonemptyE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyE1,definition,
% 0.31/0.47 ( nonemptyE1
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( nonempty @ A )
% 0.31/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI_type,type,
% 0.31/0.47 nonemptyI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI,definition,
% 0.31/0.47 ( nonemptyI
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( nonempty
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI1_type,type,
% 0.31/0.47 nonemptyI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI1,definition,
% 0.31/0.47 ( nonemptyI1
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIL_type,type,
% 0.31/0.47 setadjoinIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIL,definition,
% 0.31/0.47 ( setadjoinIL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinunitempty_type,type,
% 0.31/0.47 emptyinunitempty: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinunitempty,definition,
% 0.31/0.47 ( emptyinunitempty
% 0.31/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIR_type,type,
% 0.31/0.47 setadjoinIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIR,definition,
% 0.31/0.47 ( setadjoinIR
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinE_type,type,
% 0.31/0.47 setadjoinE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinE,definition,
% 0.31/0.47 ( setadjoinE
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.47 => ! [Xphi: $o] :
% 0.31/0.47 ( ( ( Xy = Xx )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => ( ( ( in @ Xy @ A )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinOr_type,type,
% 0.31/0.47 setadjoinOr: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinOr,definition,
% 0.31/0.47 ( setadjoinOr
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.47 => ( ( Xy = Xx )
% 0.31/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setoftrueEq_type,type,
% 0.31/0.47 setoftrueEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setoftrueEq,definition,
% 0.31/0.47 ( setoftrueEq
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : $true )
% 0.31/0.47 = A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI_type,type,
% 0.31/0.47 powersetI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI,definition,
% 0.31/0.47 ( powersetI
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinPowerset_type,type,
% 0.31/0.47 emptyinPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinPowerset,definition,
% 0.31/0.47 ( emptyinPowerset
% 0.31/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyInPowerset_type,type,
% 0.31/0.47 emptyInPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyInPowerset,definition,
% 0.31/0.47 ( emptyInPowerset
% 0.31/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE_type,type,
% 0.31/0.47 powersetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE,definition,
% 0.31/0.47 ( powersetE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionI_type,type,
% 0.31/0.47 setunionI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionI,definition,
% 0.31/0.47 ( setunionI
% 0.31/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( ( in @ B @ A )
% 0.31/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE_type,type,
% 0.31/0.47 setunionE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE,definition,
% 0.31/0.47 ( setunionE
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.47 => ! [Xphi: $o] :
% 0.31/0.47 ( ! [B: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( ( in @ B @ A )
% 0.31/0.47 => Xphi ) )
% 0.31/0.47 => Xphi ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subPowSU_type,type,
% 0.31/0.47 subPowSU: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subPowSU,definition,
% 0.31/0.47 ( subPowSU
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE2_type,type,
% 0.31/0.47 exuE2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE2,definition,
% 0.31/0.47 ( exuE2
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyImpWitness_type,type,
% 0.31/0.47 nonemptyImpWitness: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyImpWitness,definition,
% 0.31/0.47 ( nonemptyImpWitness
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( nonempty @ A )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & $true ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(uniqinunit_type,type,
% 0.31/0.47 uniqinunit: $o ).
% 0.31/0.47
% 0.31/0.47 thf(uniqinunit,definition,
% 0.31/0.47 ( uniqinunit
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notinsingleton_type,type,
% 0.31/0.47 notinsingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notinsingleton,definition,
% 0.31/0.47 ( notinsingleton
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx != Xy )
% 0.31/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eqinunit_type,type,
% 0.31/0.47 eqinunit: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eqinunit,definition,
% 0.31/0.47 ( eqinunit
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsswitch_type,type,
% 0.31/0.47 singletonsswitch: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsswitch,definition,
% 0.31/0.47 ( singletonsswitch
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetE_type,type,
% 0.31/0.47 upairsetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetE,definition,
% 0.31/0.47 ( upairsetE
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.47 => ( ( Xz = Xx )
% 0.31/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIL_type,type,
% 0.31/0.47 upairsetIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIL,definition,
% 0.31/0.47 ( upairsetIL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIR_type,type,
% 0.31/0.47 upairsetIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIR,definition,
% 0.31/0.47 ( upairsetIR
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyE1_type,type,
% 0.31/0.47 emptyE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyE1,definition,
% 0.31/0.47 ( emptyE1
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ( ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 = emptyset )
% 0.31/0.47 => $false ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(vacuousDall_type,type,
% 0.31/0.47 vacuousDall: $o ).
% 0.31/0.47
% 0.31/0.47 thf(vacuousDall,definition,
% 0.31/0.47 ( vacuousDall
% 0.31/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan1_type,type,
% 0.31/0.47 quantDeMorgan1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan1,definition,
% 0.31/0.47 ( quantDeMorgan1
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ~ ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan2_type,type,
% 0.31/0.47 quantDeMorgan2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan2,definition,
% 0.31/0.47 ( quantDeMorgan2
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ~ ( Xphi @ Xx ) )
% 0.31/0.47 => ~ ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan3_type,type,
% 0.31/0.47 quantDeMorgan3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan3,definition,
% 0.31/0.47 ( quantDeMorgan3
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan4_type,type,
% 0.31/0.48 quantDeMorgan4: $o ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan4,definition,
% 0.31/0.48 ( quantDeMorgan4
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ~ ( Xphi @ Xx ) )
% 0.31/0.48 => ~ ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(prop2setI_type,type,
% 0.31/0.48 prop2setI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(prop2setI,definition,
% 0.31/0.48 ( prop2setI
% 0.31/0.48 = ( ! [Xphi: $o] :
% 0.31/0.48 ( Xphi
% 0.31/0.48 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(set2prop_type,type,
% 0.31/0.48 set2prop: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(prop2set2propI_type,type,
% 0.31/0.48 prop2set2propI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(prop2set2propI,definition,
% 0.31/0.48 ( prop2set2propI
% 0.31/0.48 = ( ! [Xphi: $o] :
% 0.31/0.48 ( Xphi
% 0.31/0.48 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notdexE_type,type,
% 0.31/0.48 notdexE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notdexE,definition,
% 0.31/0.48 ( notdexE
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notdallE_type,type,
% 0.31/0.48 notdallE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notdallE,definition,
% 0.31/0.48 ( notdallE
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ~ ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuI1_type,type,
% 0.31/0.48 exuI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuI1,definition,
% 0.31/0.48 ( exuI1
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 & ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) )
% 0.31/0.48 => ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuI3_type,type,
% 0.31/0.48 exuI3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuI3,definition,
% 0.31/0.48 ( exuI3
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.31/0.48 => ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) )
% 0.31/0.48 => ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuI2_type,type,
% 0.31/0.48 exuI2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuI2,definition,
% 0.31/0.48 ( exuI2
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] :
% 0.31/0.48 ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 <=> ( Xy = Xx ) )
% 0.31/0.48 => ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(inCongP_type,type,
% 0.31/0.48 inCongP: $o ).
% 0.31/0.48
% 0.31/0.48 thf(inCongP,definition,
% 0.31/0.48 ( inCongP
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(in__Cong_type,type,
% 0.31/0.48 in__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(in__Cong,definition,
% 0.31/0.48 ( in__Cong
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuE3u_type,type,
% 0.31/0.48 exuE3u: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuE3u,definition,
% 0.31/0.48 ( exuE3u
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exu__Cong_type,type,
% 0.31/0.48 exu__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exu__Cong,definition,
% 0.31/0.48 ( exu__Cong
% 0.31/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.48 => ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 <=> ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyset__Cong_type,type,
% 0.31/0.48 emptyset__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptyset__Cong,definition,
% 0.31/0.48 ( emptyset__Cong
% 0.31/0.48 = ( emptyset = emptyset ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoin__Cong_type,type,
% 0.31/0.48 setadjoin__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoin__Cong,definition,
% 0.31/0.48 ( setadjoin__Cong
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ! [Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( Xz = Xu )
% 0.31/0.48 => ( ( setadjoin @ Xx @ Xz )
% 0.31/0.48 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powerset__Cong_type,type,
% 0.31/0.48 powerset__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powerset__Cong,definition,
% 0.31/0.48 ( powerset__Cong
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ( ( powerset @ A )
% 0.31/0.48 = ( powerset @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunion__Cong_type,type,
% 0.31/0.48 setunion__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunion__Cong,definition,
% 0.31/0.48 ( setunion__Cong
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ( ( setunion @ A )
% 0.31/0.48 = ( setunion @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(omega__Cong_type,type,
% 0.31/0.48 omega__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(omega__Cong,definition,
% 0.31/0.48 ( omega__Cong
% 0.31/0.48 = ( omega = omega ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuEu_type,type,
% 0.31/0.48 exuEu: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuEu,definition,
% 0.31/0.48 ( exuEu
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(descr__Cong_type,type,
% 0.31/0.48 descr__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(descr__Cong,definition,
% 0.31/0.48 ( descr__Cong
% 0.31/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.48 => ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/0.48 => ( ( descr
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 = ( descr
% 0.31/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstr__Cong_type,type,
% 0.31/0.48 dsetconstr__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstr__Cong,definition,
% 0.31/0.48 ( dsetconstr__Cong
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( Xx = Xy )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/0.48 => ( ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 = ( dsetconstr @ B
% 0.31/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subset_type,type,
% 0.31/0.48 subset: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(disjoint_type,type,
% 0.31/0.48 disjoint: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(setsmeet_type,type,
% 0.31/0.48 setsmeet: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetI1_type,type,
% 0.31/0.48 subsetI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetI1,definition,
% 0.31/0.48 ( subsetI1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(eqimpsubset2_type,type,
% 0.31/0.48 eqimpsubset2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(eqimpsubset2,definition,
% 0.31/0.48 ( eqimpsubset2
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(eqimpsubset1_type,type,
% 0.31/0.48 eqimpsubset1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(eqimpsubset1,definition,
% 0.31/0.48 ( eqimpsubset1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetI2_type,type,
% 0.31/0.48 subsetI2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetI2,definition,
% 0.31/0.48 ( subsetI2
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetsubset_type,type,
% 0.31/0.48 emptysetsubset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetsubset,definition,
% 0.31/0.48 ( emptysetsubset
% 0.31/0.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetE_type,type,
% 0.31/0.48 subsetE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetE,definition,
% 0.31/0.48 ( subsetE
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetE2_type,type,
% 0.31/0.48 subsetE2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetE2,definition,
% 0.31/0.48 ( subsetE2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notsubsetI_type,type,
% 0.31/0.48 notsubsetI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notsubsetI,definition,
% 0.31/0.48 ( notsubsetI
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notequalI1_type,type,
% 0.31/0.48 notequalI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notequalI1,definition,
% 0.31/0.48 ( notequalI1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ~ ( subset @ A @ B )
% 0.31/0.48 => ( A != B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notequalI2_type,type,
% 0.31/0.48 notequalI2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notequalI2,definition,
% 0.31/0.48 ( notequalI2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ( A != B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetRefl_type,type,
% 0.31/0.48 subsetRefl: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetRefl,definition,
% 0.31/0.48 ( subsetRefl
% 0.31/0.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetTrans_type,type,
% 0.31/0.48 subsetTrans: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetTrans,definition,
% 0.31/0.48 ( subsetTrans
% 0.31/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( subset @ B @ C )
% 0.31/0.48 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinSub_type,type,
% 0.31/0.48 setadjoinSub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinSub,definition,
% 0.31/0.48 ( setadjoinSub
% 0.31/0.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinSub2_type,type,
% 0.31/0.48 setadjoinSub2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinSub2,definition,
% 0.31/0.48 ( setadjoinSub2
% 0.31/0.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subset2powerset_type,type,
% 0.31/0.48 subset2powerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subset2powerset,definition,
% 0.31/0.48 ( subset2powerset
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setextsub_type,type,
% 0.31/0.48 setextsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setextsub,definition,
% 0.31/0.48 ( setextsub
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( subset @ B @ A )
% 0.31/0.48 => ( A = B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetemptysetimpeq_type,type,
% 0.31/0.48 subsetemptysetimpeq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetemptysetimpeq,definition,
% 0.31/0.48 ( subsetemptysetimpeq
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ( subset @ A @ emptyset )
% 0.31/0.48 => ( A = emptyset ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetI1_type,type,
% 0.31/0.48 powersetI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetI1,definition,
% 0.31/0.48 ( powersetI1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ B @ A )
% 0.31/0.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetE1_type,type,
% 0.31/0.48 powersetE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetE1,definition,
% 0.31/0.48 ( powersetE1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(inPowerset_type,type,
% 0.31/0.48 inPowerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(inPowerset,definition,
% 0.31/0.48 ( inPowerset
% 0.31/0.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetsubset_type,type,
% 0.31/0.48 powersetsubset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetsubset,definition,
% 0.31/0.48 ( powersetsubset
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(sepInPowerset_type,type,
% 0.31/0.48 sepInPowerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(sepInPowerset,definition,
% 0.31/0.48 ( sepInPowerset
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( in
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 @ ( powerset @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(sepSubset_type,type,
% 0.31/0.48 sepSubset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(sepSubset,definition,
% 0.31/0.48 ( sepSubset
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( subset
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunion_type,type,
% 0.31/0.48 binunion: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(binunionIL_type,type,
% 0.31/0.48 binunionIL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionIL,definition,
% 0.31/0.48 ( binunionIL
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairset2IR_type,type,
% 0.31/0.48 upairset2IR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairset2IR,definition,
% 0.31/0.48 ( upairset2IR
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionIR_type,type,
% 0.31/0.48 binunionIR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionIR,definition,
% 0.31/0.48 ( binunionIR
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionEcases_type,type,
% 0.31/0.48 binunionEcases: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionEcases,definition,
% 0.31/0.48 ( binunionEcases
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.48 => ( ( ( in @ Xx @ A )
% 0.31/0.48 => Xphi )
% 0.31/0.48 => ( ( ( in @ Xx @ B )
% 0.31/0.48 => Xphi )
% 0.31/0.48 => Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionE_type,type,
% 0.31/0.48 binunionE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionE,definition,
% 0.31/0.48 ( binunionE
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionLsub_type,type,
% 0.31/0.48 binunionLsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionLsub,definition,
% 0.31/0.48 ( binunionLsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionRsub_type,type,
% 0.31/0.48 binunionRsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionRsub,definition,
% 0.31/0.48 ( binunionRsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersect_type,type,
% 0.31/0.48 binintersect: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectI_type,type,
% 0.31/0.48 binintersectI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectI,definition,
% 0.31/0.48 ( binintersectI
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset5_type,type,
% 0.31/0.48 binintersectSubset5: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset5,definition,
% 0.31/0.48 ( binintersectSubset5
% 0.31/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.48 ( ( subset @ C @ A )
% 0.31/0.48 => ( ( subset @ C @ B )
% 0.31/0.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectEL_type,type,
% 0.31/0.48 binintersectEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectEL,definition,
% 0.31/0.48 ( binintersectEL
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectLsub_type,type,
% 0.31/0.48 binintersectLsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectLsub,definition,
% 0.31/0.48 ( binintersectLsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset2_type,type,
% 0.31/0.48 binintersectSubset2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset2,definition,
% 0.31/0.48 ( binintersectSubset2
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( binintersect @ A @ B )
% 0.31/0.48 = A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset3_type,type,
% 0.31/0.48 binintersectSubset3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset3,definition,
% 0.31/0.48 ( binintersectSubset3
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( ( binintersect @ A @ B )
% 0.31/0.48 = B )
% 0.31/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectER_type,type,
% 0.31/0.48 binintersectER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectER,definition,
% 0.31/0.48 ( binintersectER
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.48 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(disjointsetsI1_type,type,
% 0.31/0.48 disjointsetsI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(disjointsetsI1,definition,
% 0.31/0.48 ( disjointsetsI1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( in @ Xx @ B ) )
% 0.31/0.48 => ( ( binintersect @ A @ B )
% 0.31/0.48 = emptyset ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectRsub_type,type,
% 0.31/0.48 binintersectRsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectRsub,definition,
% 0.31/0.48 ( binintersectRsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset4_type,type,
% 0.31/0.48 binintersectSubset4: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset4,definition,
% 0.31/0.48 ( binintersectSubset4
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ B @ A )
% 0.31/0.48 => ( ( binintersect @ A @ B )
% 0.31/0.48 = B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset1_type,type,
% 0.31/0.48 binintersectSubset1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset1,definition,
% 0.31/0.48 ( binintersectSubset1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( ( binintersect @ A @ B )
% 0.31/0.48 = A )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(bs114d_type,type,
% 0.31/0.48 bs114d: $o ).
% 0.31/0.48
% 0.31/0.48 thf(bs114d,definition,
% 0.31/0.48 ( bs114d
% 0.31/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/0.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(regular_type,type,
% 0.31/0.48 regular: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminus_type,type,
% 0.31/0.48 setminus: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(setminusI_type,type,
% 0.31/0.48 setminusI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusI,definition,
% 0.31/0.48 ( setminusI
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusEL_type,type,
% 0.31/0.48 setminusEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusEL,definition,
% 0.31/0.48 ( setminusEL
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusER_type,type,
% 0.31/0.48 setminusER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusER,definition,
% 0.31/0.48 ( setminusER
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusSubset2_type,type,
% 0.31/0.48 setminusSubset2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusSubset2,definition,
% 0.31/0.48 ( setminusSubset2
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( setminus @ A @ B )
% 0.31/0.48 = emptyset ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusERneg_type,type,
% 0.31/0.48 setminusERneg: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusERneg,definition,
% 0.31/0.48 ( setminusERneg
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusELneg_type,type,
% 0.31/0.48 setminusELneg: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusELneg,definition,
% 0.31/0.48 ( setminusELneg
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusILneg_type,type,
% 0.31/0.48 setminusILneg: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusILneg,definition,
% 0.31/0.48 ( setminusILneg
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ A )
% 0.31/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusIRneg_type,type,
% 0.31/0.48 setminusIRneg: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusIRneg,definition,
% 0.31/0.48 ( setminusIRneg
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusLsub_type,type,
% 0.31/0.48 setminusLsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusLsub,definition,
% 0.31/0.48 ( setminusLsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusSubset1_type,type,
% 0.31/0.48 setminusSubset1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusSubset1,definition,
% 0.31/0.48 ( setminusSubset1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( ( setminus @ A @ B )
% 0.31/0.48 = emptyset )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiff_type,type,
% 0.31/0.48 symdiff: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffE_type,type,
% 0.31/0.48 symdiffE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffE,definition,
% 0.31/0.48 ( symdiffE
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/0.48 => ! [Xphi: $o] :
% 0.31/0.48 ( ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => Xphi ) )
% 0.31/0.48 => ( ( ~ ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => Xphi ) )
% 0.31/0.48 => Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffI1_type,type,
% 0.31/0.48 symdiffI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffI1,definition,
% 0.31/0.48 ( symdiffI1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffI2_type,type,
% 0.31/0.48 symdiffI2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffI2,definition,
% 0.31/0.48 ( symdiffI2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffIneg1_type,type,
% 0.31/0.48 symdiffIneg1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffIneg1,definition,
% 0.31/0.48 ( symdiffIneg1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffIneg2_type,type,
% 0.31/0.48 symdiffIneg2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffIneg2,definition,
% 0.31/0.48 ( symdiffIneg2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iskpair_type,type,
% 0.31/0.48 iskpair: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(secondinupair_type,type,
% 0.31/0.48 secondinupair: $o ).
% 0.31/0.48
% 0.31/0.48 thf(secondinupair,definition,
% 0.31/0.48 ( secondinupair
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairIL_type,type,
% 0.31/0.48 setukpairIL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairIL,definition,
% 0.31/0.48 ( setukpairIL
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairIR_type,type,
% 0.31/0.48 setukpairIR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairIR,definition,
% 0.31/0.48 ( setukpairIR
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kpairiskpair_type,type,
% 0.31/0.48 kpairiskpair: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kpairiskpair,definition,
% 0.31/0.48 ( kpairiskpair
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kpair_type,type,
% 0.31/0.48 kpair: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(kpairp_type,type,
% 0.31/0.48 kpairp: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kpairp,definition,
% 0.31/0.48 ( kpairp
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprod_type,type,
% 0.31/0.48 cartprod: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsubset_type,type,
% 0.31/0.48 singletonsubset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsubset,definition,
% 0.31/0.48 ( singletonsubset
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singletoninpowerset_type,type,
% 0.31/0.48 singletoninpowerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletoninpowerset,definition,
% 0.31/0.48 ( singletoninpowerset
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singletoninpowunion_type,type,
% 0.31/0.48 singletoninpowunion: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletoninpowunion,definition,
% 0.31/0.48 ( singletoninpowunion
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairset2E_type,type,
% 0.31/0.48 upairset2E: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairset2E,definition,
% 0.31/0.48 ( upairset2E
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.48 => ( ( Xz = Xx )
% 0.31/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairsubunion_type,type,
% 0.31/0.48 upairsubunion: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairsubunion,definition,
% 0.31/0.48 ( upairsubunion
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairinpowunion_type,type,
% 0.31/0.48 upairinpowunion: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairinpowunion,definition,
% 0.31/0.48 ( upairinpowunion
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem1_type,type,
% 0.31/0.48 ubforcartprodlem1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem1,definition,
% 0.31/0.48 ( ubforcartprodlem1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem2_type,type,
% 0.31/0.48 ubforcartprodlem2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem2,definition,
% 0.31/0.48 ( ubforcartprodlem2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem3_type,type,
% 0.31/0.48 ubforcartprodlem3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem3,definition,
% 0.31/0.48 ( ubforcartprodlem3
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairin_type,type,
% 0.31/0.48 cartprodpairin: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairin,definition,
% 0.31/0.48 ( cartprodpairin
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempair1_type,type,
% 0.31/0.48 cartprodmempair1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempair1,definition,
% 0.31/0.48 ( cartprodmempair1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ? [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 & ( Xu
% 0.31/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempair_type,type,
% 0.31/0.48 cartprodmempair: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempair,definition,
% 0.31/0.48 ( cartprodmempair
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionE2_type,type,
% 0.31/0.48 setunionE2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionE2,definition,
% 0.31/0.48 ( setunionE2
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.48 => ? [X: $i] :
% 0.31/0.48 ( ( in @ X @ A )
% 0.31/0.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton1_type,type,
% 0.31/0.48 setunionsingleton1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton1,definition,
% 0.31/0.48 ( setunionsingleton1
% 0.31/0.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton2_type,type,
% 0.31/0.48 setunionsingleton2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton2,definition,
% 0.31/0.48 ( setunionsingleton2
% 0.31/0.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton_type,type,
% 0.31/0.48 setunionsingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton,definition,
% 0.31/0.48 ( setunionsingleton
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/0.48 = Xx ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singleton_type,type,
% 0.31/0.48 singleton: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletonprop_type,type,
% 0.31/0.48 singletonprop: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletonprop,definition,
% 0.31/0.48 ( singletonprop
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) )
% 0.31/0.48 => ( ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ( singleton
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1_type,type,
% 0.31/0.48 ex1: $i > ( $i > $o ) > $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1,definition,
% 0.31/0.48 ( ex1
% 0.31/0.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( singleton
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1E1_type,type,
% 0.31/0.48 ex1E1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1E1,definition,
% 0.31/0.48 ( ex1E1
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ( ex1 @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1I_type,type,
% 0.31/0.48 ex1I: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1I,definition,
% 0.31/0.48 ( ex1I
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xy = Xx ) ) )
% 0.31/0.48 => ( ex1 @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1I2_type,type,
% 0.31/0.48 ex1I2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1I2,definition,
% 0.31/0.48 ( ex1I2
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) )
% 0.31/0.48 => ( ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ( ex1 @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsuniq_type,type,
% 0.31/0.48 singletonsuniq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsuniq,definition,
% 0.31/0.48 ( singletonsuniq
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/0.48 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(atmost1p_type,type,
% 0.31/0.48 atmost1p: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(atleast2p_type,type,
% 0.31/0.48 atleast2p: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(atmost2p_type,type,
% 0.31/0.48 atmost2p: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairsetp_type,type,
% 0.31/0.48 upairsetp: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL1_type,type,
% 0.31/0.48 setukpairinjL1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL1,definition,
% 0.31/0.48 ( setukpairinjL1
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.48 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kfstsingleton_type,type,
% 0.31/0.48 kfstsingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kfstsingleton,definition,
% 0.31/0.48 ( kfstsingleton
% 0.31/0.48 = ( ! [Xu: $i] :
% 0.31/0.48 ( ( iskpair @ Xu )
% 0.31/0.48 => ( singleton
% 0.31/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.48 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(theprop_type,type,
% 0.31/0.48 theprop: $o ).
% 0.31/0.48
% 0.31/0.48 thf(theprop,definition,
% 0.31/0.48 ( theprop
% 0.31/0.48 = ( ! [X: $i] :
% 0.31/0.48 ( ( singleton @ X )
% 0.31/0.48 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kfst_type,type,
% 0.31/0.48 kfst: $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(kfstpairEq_type,type,
% 0.31/0.48 kfstpairEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kfstpairEq,definition,
% 0.31/0.48 ( kfstpairEq
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.48 = Xx ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodfstin_type,type,
% 0.31/0.48 cartprodfstin: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodfstin,definition,
% 0.31/0.48 ( cartprodfstin
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL2_type,type,
% 0.31/0.48 setukpairinjL2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL2,definition,
% 0.31/0.48 ( setukpairinjL2
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL_type,type,
% 0.31/0.48 setukpairinjL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL,definition,
% 0.31/0.48 ( setukpairinjL
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.48 = ( kpair @ Xz @ Xu ) )
% 0.31/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR11_type,type,
% 0.31/0.48 setukpairinjR11: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR11,definition,
% 0.31/0.48 ( setukpairinjR11
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR12_type,type,
% 0.31/0.48 setukpairinjR12: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR12,definition,
% 0.31/0.48 ( setukpairinjR12
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR1_type,type,
% 0.31/0.48 setukpairinjR1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR1,definition,
% 0.31/0.48 ( setukpairinjR1
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.48 => ( ( Xz = Xu )
% 0.31/0.48 => ( Xy = Xu ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairequniteq_type,type,
% 0.31/0.48 upairequniteq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairequniteq,definition,
% 0.31/0.48 ( upairequniteq
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ Xz @ emptyset ) )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR2_type,type,
% 0.31/0.48 setukpairinjR2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR2,definition,
% 0.31/0.48 ( setukpairinjR2
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR_type,type,
% 0.31/0.48 setukpairinjR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR,definition,
% 0.31/0.48 ( setukpairinjR
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.48 = ( kpair @ Xz @ Xu ) )
% 0.31/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ksndsingleton_type,type,
% 0.31/0.48 ksndsingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ksndsingleton,definition,
% 0.31/0.48 ( ksndsingleton
% 0.31/0.48 = ( ! [Xu: $i] :
% 0.31/0.48 ( ( iskpair @ Xu )
% 0.31/0.48 => ( singleton
% 0.31/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.48 @ ^ [Xx: $i] :
% 0.31/0.48 ( Xu
% 0.31/0.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ksnd_type,type,
% 0.31/0.48 ksnd: $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(ksndpairEq_type,type,
% 0.31/0.48 ksndpairEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ksndpairEq,definition,
% 0.31/0.48 ( ksndpairEq
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.48 = Xy ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kpairsurjEq_type,type,
% 0.31/0.48 kpairsurjEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kpairsurjEq,definition,
% 0.31/0.48 ( kpairsurjEq
% 0.31/0.48 = ( ! [Xu: $i] :
% 0.31/0.48 ( ( iskpair @ Xu )
% 0.31/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.48 = Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodsndin_type,type,
% 0.31/0.48 cartprodsndin: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodsndin,definition,
% 0.31/0.48 ( cartprodsndin
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemEL_type,type,
% 0.31/0.48 cartprodpairmemEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemEL,definition,
% 0.31/0.48 ( cartprodpairmemEL
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemER_type,type,
% 0.31/0.48 cartprodpairmemER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemER,definition,
% 0.31/0.48 ( cartprodpairmemER
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempaircEq_type,type,
% 0.31/0.48 cartprodmempaircEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempaircEq,definition,
% 0.31/0.48 ( cartprodmempaircEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( kpair @ Xx @ Xy )
% 0.31/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodfstpairEq_type,type,
% 0.31/0.48 cartprodfstpairEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodfstpairEq,definition,
% 0.31/0.48 ( cartprodfstpairEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.48 = Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodsndpairEq_type,type,
% 0.31/0.48 cartprodsndpairEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodsndpairEq,definition,
% 0.31/0.48 ( cartprodsndpairEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.48 = Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairsurjEq_type,type,
% 0.31/0.48 cartprodpairsurjEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairsurjEq,definition,
% 0.31/0.48 ( cartprodpairsurjEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.48 = Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(breln_type,type,
% 0.31/0.48 breln: $i > $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(breln,definition,
% 0.31/0.48 ( breln
% 0.31/0.48 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstr_type,type,
% 0.31/0.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrI_type,type,
% 0.31/0.48 dpsetconstrI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrI,definition,
% 0.31/0.48 ( dpsetconstrI
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( Xphi @ Xx @ Xy )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrSub_type,type,
% 0.31/0.48 dpsetconstrSub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrSub,definition,
% 0.31/0.48 ( dpsetconstrSub
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.48 ( subset
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.31/0.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setOfPairsIsBReln_type,type,
% 0.31/0.48 setOfPairsIsBReln: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setOfPairsIsBReln,definition,
% 0.31/0.48 ( setOfPairsIsBReln
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.48 ( breln @ A @ B
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrERa_type,type,
% 0.31/0.48 dpsetconstrERa: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrERa,definition,
% 0.31/0.48 ( dpsetconstrERa
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL1_type,type,
% 0.31/0.48 dpsetconstrEL1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL1,definition,
% 0.31/0.48 ( dpsetconstrEL1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL2_type,type,
% 0.31/0.48 dpsetconstrEL2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL2,definition,
% 0.31/0.48 ( dpsetconstrEL2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrER_type,type,
% 0.31/0.48 dpsetconstrER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrER,definition,
% 0.31/0.48 ( dpsetconstrER
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(func_type,type,
% 0.31/0.48 func: $i > $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcSet_type,type,
% 0.31/0.48 funcSet: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(funcImageSingleton_type,type,
% 0.31/0.48 funcImageSingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcImageSingleton,definition,
% 0.31/0.48 ( funcImageSingleton
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( singleton
% 0.31/0.48 @ ( dsetconstr @ B
% 0.31/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(apProp_type,type,
% 0.31/0.48 apProp: $o ).
% 0.31/0.48
% 0.31/0.48 thf(apProp,definition,
% 0.31/0.48 ( apProp
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in
% 0.31/0.48 @ ( setunion
% 0.31/0.48 @ ( dsetconstr @ B
% 0.31/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.31/0.48 @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ap_type,type,
% 0.31/0.48 ap: $i > $i > $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(app_type,type,
% 0.31/0.48 app: $o ).
% 0.31/0.48
% 0.31/0.48 thf(app,definition,
% 0.31/0.48 ( app
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(infuncsetfunc_type,type,
% 0.31/0.48 infuncsetfunc: $o ).
% 0.31/0.48
% 0.31/0.48 thf(infuncsetfunc,definition,
% 0.31/0.48 ( infuncsetfunc
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.48 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ap2p_type,type,
% 0.31/0.48 ap2p: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ap2p,definition,
% 0.31/0.48 ( ap2p
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(funcinfuncset_type,type,
% 0.31/0.48 funcinfuncset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcinfuncset,definition,
% 0.31/0.48 ( funcinfuncset
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(lamProp_type,type,
% 0.31/0.48 lamProp: $o ).
% 0.31/0.48
% 0.31/0.48 thf(lamProp,definition,
% 0.31/0.48 ( lamProp
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.48 => ( func @ A @ B
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xf @ Xx )
% 0.31/0.48 = Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(lam_type,type,
% 0.31/0.48 lam: $i > $i > ( $i > $i ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(lamp_type,type,
% 0.31/0.48 lamp: $o ).
% 0.31/0.48
% 0.31/0.48 thf(lamp,definition,
% 0.31/0.48 ( lamp
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.48 => ( func @ A @ B
% 0.31/0.48 @ ( lam @ A @ B
% 0.31/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(lam2p_type,type,
% 0.31/0.48 lam2p: $o ).
% 0.31/0.48
% 0.31/0.48 thf(lam2p,definition,
% 0.31/0.48 ( lam2p
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.48 => ( in
% 0.31/0.48 @ ( lam @ A @ B
% 0.31/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/0.48 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(brelnall1_type,type,
% 0.31/0.48 brelnall1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(brelnall1,definition,
% 0.31/0.48 ( brelnall1
% 0.31/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.48 ( ( breln @ A @ B @ R )
% 0.31/0.48 => ! [Xphi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ R )
% 0.31/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(brelnall2_type,type,
% 0.31/0.48 brelnall2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(brelnall2,definition,
% 0.31/0.48 ( brelnall2
% 0.31/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.48 ( ( breln @ A @ B @ R )
% 0.31/0.48 => ! [Xphi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ R )
% 0.31/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1E2_type,type,
% 0.31/0.48 ex1E2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1E2,definition,
% 0.31/0.48 ( ex1E2
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ( ex1 @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(funcGraphProp1_type,type,
% 0.31/0.48 funcGraphProp1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcGraphProp1,definition,
% 0.31/0.48 ( funcGraphProp1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(funcGraphProp3_type,type,
% 0.31/0.48 funcGraphProp3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcGraphProp3,definition,
% 0.31/0.48 ( funcGraphProp3
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(funcGraphProp2_type,type,
% 0.31/0.48 funcGraphProp2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcGraphProp2,definition,
% 0.31/0.48 ( funcGraphProp2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(funcextLem_type,type,
% 0.31/0.48 funcextLem: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcextLem,definition,
% 0.31/0.48 ( funcextLem
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ! [Xg: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xg )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(funcGraphProp4_type,type,
% 0.31/0.48 funcGraphProp4: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcGraphProp4,definition,
% 0.31/0.48 ( funcGraphProp4
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subbreln_type,type,
% 0.31/0.48 subbreln: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subbreln,definition,
% 0.31/0.48 ( subbreln
% 0.31/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.48 ( ( breln @ A @ B @ R )
% 0.31/0.48 => ! [S: $i] :
% 0.31/0.48 ( ( breln @ A @ B @ S )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(eqbreln_type,type,
% 0.31/0.48 eqbreln: $o ).
% 0.31/0.48
% 0.31/0.48 thf(eqbreln,definition,
% 0.31/0.48 ( eqbreln
% 0.31/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.48 ( ( breln @ A @ B @ R )
% 0.31/0.48 => ! [S: $i] :
% 0.31/0.48 ( ( breln @ A @ B @ S )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.31/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(funcext_type,type,
% 0.31/0.48 funcext: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcext,definition,
% 0.31/0.48 ( funcext
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ! [Xg: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xg )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(funcext2_type,type,
% 0.31/0.48 funcext2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(funcext2,definition,
% 0.31/0.48 ( funcext2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.48 => ! [Xg: $i] :
% 0.31/0.48 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ap2apEq1_type,type,
% 0.31/0.48 ap2apEq1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ap2apEq1,definition,
% 0.31/0.48 ( ap2apEq1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ap2apEq2_type,type,
% 0.31/0.48 ap2apEq2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ap2apEq2,definition,
% 0.31/0.48 ( ap2apEq2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(beta1_type,type,
% 0.31/0.48 beta1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(beta1,definition,
% 0.31/0.48 ( beta1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( ap @ A @ B
% 0.31/0.48 @ ( lam @ A @ B
% 0.31/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/0.48 @ Xx )
% 0.31/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(eta1_type,type,
% 0.31/0.48 eta1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(eta1,definition,
% 0.31/0.48 ( eta1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( func @ A @ B @ Xf )
% 0.31/0.48 => ( ( lam @ A @ B
% 0.31/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/0.48 = Xf ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(lam2lamEq_type,type,
% 0.31/0.48 lam2lamEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(lam2lamEq,definition,
% 0.31/0.48 ( lam2lamEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.48 => ( ( lam @ A @ B
% 0.31/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/0.48 = ( lam @ A @ B
% 0.31/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(beta2_type,type,
% 0.31/0.48 beta2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(beta2,definition,
% 0.31/0.48 ( beta2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( ap @ A @ B
% 0.31/0.48 @ ( lam @ A @ B
% 0.31/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/0.48 @ Xx )
% 0.31/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(eta2_type,type,
% 0.31/0.48 eta2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(eta2,definition,
% 0.31/0.48 ( eta2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.48 => ( ( lam @ A @ B
% 0.31/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/0.48 = Xf ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iffalseProp1_type,type,
% 0.31/0.48 iffalseProp1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(iffalseProp1,definition,
% 0.31/0.48 ( iffalseProp1
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ~ Xphi
% 0.31/0.48 => ( in @ Xy
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xz: $i] :
% 0.31/0.48 ( ( Xphi
% 0.31/0.48 & ( Xz = Xx ) )
% 0.31/0.48 | ( ~ Xphi
% 0.31/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iffalseProp2_type,type,
% 0.31/0.48 iffalseProp2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(iffalseProp2,definition,
% 0.31/0.48 ( iffalseProp2
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ~ Xphi
% 0.31/0.48 => ( ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xz: $i] :
% 0.31/0.48 ( ( Xphi
% 0.31/0.48 & ( Xz = Xx ) )
% 0.31/0.48 | ( ~ Xphi
% 0.31/0.48 & ( Xz = Xy ) ) ) )
% 0.31/0.48 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iftrueProp1_type,type,
% 0.31/0.48 iftrueProp1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(iftrueProp1,definition,
% 0.31/0.48 ( iftrueProp1
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( Xphi
% 0.31/0.48 => ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xz: $i] :
% 0.31/0.48 ( ( Xphi
% 0.31/0.48 & ( Xz = Xx ) )
% 0.31/0.48 | ( ~ Xphi
% 0.31/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iftrueProp2_type,type,
% 0.31/0.48 iftrueProp2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(iftrueProp2,definition,
% 0.31/0.48 ( iftrueProp2
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( Xphi
% 0.31/0.48 => ( ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xz: $i] :
% 0.31/0.48 ( ( Xphi
% 0.31/0.48 & ( Xz = Xx ) )
% 0.31/0.48 | ( ~ Xphi
% 0.31/0.48 & ( Xz = Xy ) ) ) )
% 0.31/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ifSingleton_type,type,
% 0.31/0.48 ifSingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ifSingleton,definition,
% 0.31/0.48 ( ifSingleton
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( singleton
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xz: $i] :
% 0.31/0.48 ( ( Xphi
% 0.31/0.48 & ( Xz = Xx ) )
% 0.31/0.48 | ( ~ Xphi
% 0.31/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(if_type,type,
% 0.31/0.48 if: $i > $o > $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(ifp_type,type,
% 0.31/0.48 ifp: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ifp,definition,
% 0.31/0.48 ( ifp
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(theeq_type,type,
% 0.31/0.48 theeq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(theeq,definition,
% 0.31/0.48 ( theeq
% 0.31/0.48 = ( ! [X: $i] :
% 0.31/0.48 ( ( singleton @ X )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ X )
% 0.31/0.48 => ( ( setunion @ X )
% 0.31/0.48 = Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iftrue_type,type,
% 0.31/0.48 iftrue: $o ).
% 0.31/0.48
% 0.31/0.48 thf(iftrue,definition,
% 0.31/0.48 ( iftrue
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( Xphi
% 0.31/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.31/0.48 = Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iffalse_type,type,
% 0.31/0.48 iffalse: $o ).
% 0.31/0.48
% 0.31/0.48 thf(iffalse,definition,
% 0.31/0.48 ( iffalse
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ~ Xphi
% 0.31/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.31/0.48 = Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iftrueorfalse_type,type,
% 0.31/0.48 iftrueorfalse: $o ).
% 0.31/0.48
% 0.31/0.48 thf(iftrueorfalse,definition,
% 0.31/0.48 ( iftrueorfalse
% 0.31/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectT_lem_type,type,
% 0.31/0.48 binintersectT_lem: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectT_lem,definition,
% 0.31/0.48 ( binintersectT_lem
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionT_lem_type,type,
% 0.31/0.48 binunionT_lem: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionT_lem,definition,
% 0.31/0.48 ( binunionT_lem
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetT_lem_type,type,
% 0.31/0.48 powersetT_lem: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetT_lem,definition,
% 0.31/0.48 ( powersetT_lem
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusT_lem_type,type,
% 0.31/0.48 setminusT_lem: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusT_lem,definition,
% 0.31/0.48 ( setminusT_lem
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(complementT_lem_type,type,
% 0.31/0.48 complementT_lem: $o ).
% 0.31/0.48
% 0.31/0.48 thf(complementT_lem,definition,
% 0.31/0.48 ( complementT_lem
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setextT_type,type,
% 0.31/0.48 setextT: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setextT,definition,
% 0.31/0.48 ( setextT
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ X )
% 0.31/0.48 => ( in @ Xx @ Y ) ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ Y )
% 0.31/0.48 => ( in @ Xx @ X ) ) )
% 0.31/0.48 => ( X = Y ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetTI_type,type,
% 0.31/0.48 subsetTI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetTI,definition,
% 0.31/0.48 ( subsetTI
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ X )
% 0.31/0.48 => ( in @ Xx @ Y ) ) )
% 0.31/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetTI1_type,type,
% 0.31/0.48 powersetTI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetTI1,definition,
% 0.31/0.48 ( powersetTI1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ X )
% 0.31/0.48 => ( in @ Xx @ Y ) ) )
% 0.31/0.48 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetTE1_type,type,
% 0.31/0.48 powersetTE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetTE1,definition,
% 0.31/0.48 ( powersetTE1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ X @ ( powerset @ Y ) )
% 0.31/0.48 => ( ( in @ Xx @ X )
% 0.31/0.48 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(complementTI1_type,type,
% 0.31/0.48 complementTI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(complementTI1,definition,
% 0.31/0.48 ( complementTI1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ X )
% 0.31/0.48 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(complementTE1_type,type,
% 0.31/0.48 complementTE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(complementTE1,definition,
% 0.31/0.48 ( complementTE1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectTELcontra_type,type,
% 0.31/0.48 binintersectTELcontra: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectTELcontra,definition,
% 0.31/0.48 ( binintersectTELcontra
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ X )
% 0.31/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectTERcontra_type,type,
% 0.31/0.48 binintersectTERcontra: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectTERcontra,definition,
% 0.31/0.48 ( binintersectTERcontra
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ Y )
% 0.31/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(contrasubsetT_type,type,
% 0.31/0.48 contrasubsetT: $o ).
% 0.31/0.48
% 0.31/0.48 thf(contrasubsetT,definition,
% 0.31/0.48 ( contrasubsetT
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.48 => ( ( in @ Xx @ Y )
% 0.31/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(contrasubsetT1_type,type,
% 0.31/0.48 contrasubsetT1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(contrasubsetT1,definition,
% 0.31/0.48 ( contrasubsetT1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( subset @ X @ Y )
% 0.31/0.48 => ( ~ ( in @ Xx @ Y )
% 0.31/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(contrasubsetT2_type,type,
% 0.31/0.48 contrasubsetT2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(contrasubsetT2,definition,
% 0.31/0.48 ( contrasubsetT2
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( ( subset @ X @ Y )
% 0.31/0.48 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(contrasubsetT3_type,type,
% 0.31/0.48 contrasubsetT3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(contrasubsetT3,definition,
% 0.31/0.48 ( contrasubsetT3
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.31/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementI1_type,type,
% 0.31/0.48 doubleComplementI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementI1,definition,
% 0.31/0.48 ( doubleComplementI1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ X )
% 0.31/0.48 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementE1_type,type,
% 0.31/0.48 doubleComplementE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementE1,definition,
% 0.31/0.48 ( doubleComplementE1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.31/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementSub1_type,type,
% 0.31/0.48 doubleComplementSub1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementSub1,definition,
% 0.31/0.48 ( doubleComplementSub1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementSub2_type,type,
% 0.31/0.48 doubleComplementSub2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementSub2,definition,
% 0.31/0.48 ( doubleComplementSub2
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementEq_type,type,
% 0.31/0.48 doubleComplementEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(doubleComplementEq,definition,
% 0.31/0.48 ( doubleComplementEq
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ( X
% 0.31/0.48 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(complementTnotintersectT_type,type,
% 0.31/0.48 complementTnotintersectT: $o ).
% 0.31/0.48
% 0.31/0.48 thf(complementTnotintersectT,definition,
% 0.31/0.48 ( complementTnotintersectT
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.49 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementImpComplementIntersect_type,type,
% 0.31/0.49 complementImpComplementIntersect: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementImpComplementIntersect,definition,
% 0.31/0.49 ( complementImpComplementIntersect
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementSubsetComplementIntersect_type,type,
% 0.31/0.49 complementSubsetComplementIntersect: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementSubsetComplementIntersect,definition,
% 0.31/0.49 ( complementSubsetComplementIntersect
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementInPowersetComplementIntersect_type,type,
% 0.31/0.49 complementInPowersetComplementIntersect: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementInPowersetComplementIntersect,definition,
% 0.31/0.49 ( complementInPowersetComplementIntersect
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(contraSubsetComplement_type,type,
% 0.31/0.49 contraSubsetComplement: $o ).
% 0.31/0.49
% 0.31/0.49 thf(contraSubsetComplement,definition,
% 0.31/0.49 ( contraSubsetComplement
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ( in @ Xx @ Y )
% 0.31/0.49 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(complementTcontraSubset_type,type,
% 0.31/0.49 complementTcontraSubset: $o ).
% 0.31/0.49
% 0.31/0.49 thf(complementTcontraSubset,definition,
% 0.31/0.49 ( complementTcontraSubset
% 0.31/0.49 = ( ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.49 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(binunionTILcontra,conjecture,
% 0.31/0.49 ( setextAx
% 0.31/0.49 => ( emptysetAx
% 0.31/0.49 => ( setadjoinAx
% 0.31/0.49 => ( powersetAx
% 0.31/0.49 => ( setunionAx
% 0.31/0.49 => ( omega0Ax
% 0.31/0.49 => ( omegaSAx
% 0.31/0.49 => ( omegaIndAx
% 0.31/0.49 => ( replAx
% 0.31/0.49 => ( foundationAx
% 0.31/0.49 => ( wellorderingAx
% 0.31/0.49 => ( descrp
% 0.31/0.49 => ( dsetconstrI
% 0.31/0.49 => ( dsetconstrEL
% 0.31/0.49 => ( dsetconstrER
% 0.31/0.49 => ( exuE1
% 0.31/0.49 => ( prop2setE
% 0.31/0.49 => ( emptysetE
% 0.31/0.49 => ( emptysetimpfalse
% 0.31/0.49 => ( notinemptyset
% 0.31/0.49 => ( exuE3e
% 0.31/0.49 => ( setext
% 0.31/0.49 => ( emptyI
% 0.31/0.49 => ( noeltsimpempty
% 0.31/0.49 => ( setbeta
% 0.31/0.49 => ( nonemptyE1
% 0.31/0.49 => ( nonemptyI
% 0.31/0.49 => ( nonemptyI1
% 0.31/0.49 => ( setadjoinIL
% 0.31/0.49 => ( emptyinunitempty
% 0.31/0.49 => ( setadjoinIR
% 0.31/0.49 => ( setadjoinE
% 0.31/0.49 => ( setadjoinOr
% 0.31/0.49 => ( setoftrueEq
% 0.31/0.49 => ( powersetI
% 0.31/0.49 => ( emptyinPowerset
% 0.31/0.49 => ( emptyInPowerset
% 0.31/0.49 => ( powersetE
% 0.31/0.49 => ( setunionI
% 0.31/0.49 => ( setunionE
% 0.31/0.49 => ( subPowSU
% 0.31/0.49 => ( exuE2
% 0.31/0.49 => ( nonemptyImpWitness
% 0.31/0.49 => ( uniqinunit
% 0.31/0.49 => ( notinsingleton
% 0.31/0.49 => ( eqinunit
% 0.31/0.49 => ( singletonsswitch
% 0.31/0.49 => ( upairsetE
% 0.31/0.49 => ( upairsetIL
% 0.31/0.49 => ( upairsetIR
% 0.31/0.49 => ( emptyE1
% 0.31/0.49 => ( vacuousDall
% 0.31/0.49 => ( quantDeMorgan1
% 0.31/0.49 => ( quantDeMorgan2
% 0.31/0.49 => ( quantDeMorgan3
% 0.31/0.49 => ( quantDeMorgan4
% 0.31/0.49 => ( prop2setI
% 0.31/0.49 => ( prop2set2propI
% 0.31/0.49 => ( notdexE
% 0.31/0.49 => ( notdallE
% 0.31/0.49 => ( exuI1
% 0.31/0.49 => ( exuI3
% 0.31/0.49 => ( exuI2
% 0.31/0.49 => ( inCongP
% 0.31/0.49 => ( in__Cong
% 0.31/0.49 => ( exuE3u
% 0.31/0.49 => ( exu__Cong
% 0.31/0.49 => ( emptyset__Cong
% 0.31/0.49 => ( setadjoin__Cong
% 0.31/0.49 => ( powerset__Cong
% 0.31/0.49 => ( setunion__Cong
% 0.31/0.49 => ( omega__Cong
% 0.31/0.49 => ( exuEu
% 0.31/0.49 => ( descr__Cong
% 0.31/0.49 => ( dsetconstr__Cong
% 0.31/0.49 => ( subsetI1
% 0.31/0.49 => ( eqimpsubset2
% 0.31/0.49 => ( eqimpsubset1
% 0.31/0.49 => ( subsetI2
% 0.31/0.49 => ( emptysetsubset
% 0.31/0.49 => ( subsetE
% 0.31/0.49 => ( subsetE2
% 0.31/0.49 => ( notsubsetI
% 0.31/0.49 => ( notequalI1
% 0.31/0.49 => ( notequalI2
% 0.31/0.49 => ( subsetRefl
% 0.31/0.49 => ( subsetTrans
% 0.31/0.49 => ( setadjoinSub
% 0.31/0.49 => ( setadjoinSub2
% 0.31/0.49 => ( subset2powerset
% 0.31/0.49 => ( setextsub
% 0.31/0.49 => ( subsetemptysetimpeq
% 0.31/0.49 => ( powersetI1
% 0.31/0.49 => ( powersetE1
% 0.31/0.49 => ( inPowerset
% 0.31/0.49 => ( powersetsubset
% 0.31/0.49 => ( sepInPowerset
% 0.31/0.49 => ( sepSubset
% 0.31/0.49 => ( binunionIL
% 0.31/0.49 => ( upairset2IR
% 0.31/0.49 => ( binunionIR
% 0.31/0.49 => ( binunionEcases
% 0.31/0.49 => ( binunionE
% 0.31/0.49 => ( binunionLsub
% 0.31/0.49 => ( binunionRsub
% 0.31/0.49 => ( binintersectI
% 0.31/0.49 => ( binintersectSubset5
% 0.31/0.49 => ( binintersectEL
% 0.31/0.49 => ( binintersectLsub
% 0.31/0.49 => ( binintersectSubset2
% 0.31/0.49 => ( binintersectSubset3
% 0.31/0.49 => ( binintersectER
% 0.31/0.49 => ( disjointsetsI1
% 0.31/0.49 => ( binintersectRsub
% 0.31/0.49 => ( binintersectSubset4
% 0.31/0.49 => ( binintersectSubset1
% 0.31/0.49 => ( bs114d
% 0.31/0.49 => ( setminusI
% 0.31/0.49 => ( setminusEL
% 0.31/0.49 => ( setminusER
% 0.31/0.49 => ( setminusSubset2
% 0.31/0.49 => ( setminusERneg
% 0.31/0.49 => ( setminusELneg
% 0.31/0.49 => ( setminusILneg
% 0.31/0.49 => ( setminusIRneg
% 0.31/0.49 => ( setminusLsub
% 0.31/0.49 => ( setminusSubset1
% 0.31/0.49 => ( symdiffE
% 0.31/0.49 => ( symdiffI1
% 0.31/0.49 => ( symdiffI2
% 0.31/0.49 => ( symdiffIneg1
% 0.31/0.49 => ( symdiffIneg2
% 0.31/0.49 => ( secondinupair
% 0.31/0.49 => ( setukpairIL
% 0.31/0.49 => ( setukpairIR
% 0.31/0.49 => ( kpairiskpair
% 0.31/0.49 => ( kpairp
% 0.31/0.49 => ( singletonsubset
% 0.31/0.49 => ( singletoninpowerset
% 0.31/0.49 => ( singletoninpowunion
% 0.31/0.49 => ( upairset2E
% 0.31/0.49 => ( upairsubunion
% 0.31/0.49 => ( upairinpowunion
% 0.31/0.49 => ( ubforcartprodlem1
% 0.31/0.49 => ( ubforcartprodlem2
% 0.31/0.49 => ( ubforcartprodlem3
% 0.31/0.49 => ( cartprodpairin
% 0.31/0.49 => ( cartprodmempair1
% 0.31/0.49 => ( cartprodmempair
% 0.31/0.49 => ( setunionE2
% 0.31/0.49 => ( setunionsingleton1
% 0.31/0.49 => ( setunionsingleton2
% 0.31/0.49 => ( setunionsingleton
% 0.31/0.49 => ( singletonprop
% 0.31/0.49 => ( ex1E1
% 0.31/0.49 => ( ex1I
% 0.31/0.49 => ( ex1I2
% 0.31/0.49 => ( singletonsuniq
% 0.31/0.49 => ( setukpairinjL1
% 0.31/0.49 => ( kfstsingleton
% 0.31/0.49 => ( theprop
% 0.31/0.49 => ( kfstpairEq
% 0.31/0.49 => ( cartprodfstin
% 0.31/0.49 => ( setukpairinjL2
% 0.31/0.49 => ( setukpairinjL
% 0.31/0.49 => ( setukpairinjR11
% 0.31/0.49 => ( setukpairinjR12
% 0.31/0.49 => ( setukpairinjR1
% 0.31/0.49 => ( upairequniteq
% 0.31/0.49 => ( setukpairinjR2
% 0.31/0.49 => ( setukpairinjR
% 0.31/0.49 => ( ksndsingleton
% 0.31/0.49 => ( ksndpairEq
% 0.31/0.49 => ( kpairsurjEq
% 0.31/0.49 => ( cartprodsndin
% 0.31/0.49 => ( cartprodpairmemEL
% 0.31/0.49 => ( cartprodpairmemER
% 0.31/0.49 => ( cartprodmempaircEq
% 0.31/0.49 => ( cartprodfstpairEq
% 0.31/0.49 => ( cartprodsndpairEq
% 0.31/0.49 => ( cartprodpairsurjEq
% 0.31/0.49 => ( dpsetconstrI
% 0.31/0.49 => ( dpsetconstrSub
% 0.31/0.49 => ( setOfPairsIsBReln
% 0.31/0.49 => ( dpsetconstrERa
% 0.31/0.49 => ( dpsetconstrEL1
% 0.31/0.49 => ( dpsetconstrEL2
% 0.31/0.49 => ( dpsetconstrER
% 0.31/0.49 => ( funcImageSingleton
% 0.31/0.49 => ( apProp
% 0.31/0.49 => ( app
% 0.31/0.49 => ( infuncsetfunc
% 0.31/0.49 => ( ap2p
% 0.31/0.49 => ( funcinfuncset
% 0.31/0.49 => ( lamProp
% 0.31/0.49 => ( lamp
% 0.31/0.49 => ( lam2p
% 0.31/0.49 => ( brelnall1
% 0.31/0.49 => ( brelnall2
% 0.31/0.49 => ( ex1E2
% 0.31/0.49 => ( funcGraphProp1
% 0.31/0.49 => ( funcGraphProp3
% 0.31/0.49 => ( funcGraphProp2
% 0.31/0.49 => ( funcextLem
% 0.31/0.49 => ( funcGraphProp4
% 0.31/0.49 => ( subbreln
% 0.31/0.49 => ( eqbreln
% 0.31/0.49 => ( funcext
% 0.31/0.49 => ( funcext2
% 0.31/0.49 => ( ap2apEq1
% 0.31/0.49 => ( ap2apEq2
% 0.31/0.49 => ( beta1
% 0.31/0.49 => ( eta1
% 0.31/0.49 => ( lam2lamEq
% 0.31/0.49 => ( beta2
% 0.31/0.49 => ( eta2
% 0.31/0.49 => ( iffalseProp1
% 0.31/0.49 => ( iffalseProp2
% 0.31/0.49 => ( iftrueProp1
% 0.31/0.49 => ( iftrueProp2
% 0.31/0.49 => ( ifSingleton
% 0.31/0.49 => ( ifp
% 0.31/0.49 => ( theeq
% 0.31/0.49 => ( iftrue
% 0.31/0.49 => ( iffalse
% 0.31/0.49 => ( iftrueorfalse
% 0.31/0.49 => ( binintersectT_lem
% 0.31/0.49 => ( binunionT_lem
% 0.31/0.49 => ( powersetT_lem
% 0.31/0.49 => ( setminusT_lem
% 0.31/0.49 => ( complementT_lem
% 0.31/0.49 => ( setextT
% 0.31/0.49 => ( subsetTI
% 0.31/0.49 => ( powersetTI1
% 0.31/0.49 => ( powersetTE1
% 0.31/0.49 => ( complementTI1
% 0.31/0.49 => ( complementTE1
% 0.31/0.49 => ( binintersectTELcontra
% 0.31/0.49 => ( binintersectTERcontra
% 0.31/0.49 => ( contrasubsetT
% 0.31/0.49 => ( contrasubsetT1
% 0.31/0.49 => ( contrasubsetT2
% 0.31/0.49 => ( contrasubsetT3
% 0.31/0.49 => ( doubleComplementI1
% 0.31/0.49 => ( doubleComplementE1
% 0.31/0.49 => ( doubleComplementSub1
% 0.31/0.49 => ( doubleComplementSub2
% 0.31/0.49 => ( doubleComplementEq
% 0.31/0.49 => ( complementTnotintersectT
% 0.31/0.49 => ( complementImpComplementIntersect
% 0.31/0.49 => ( complementSubsetComplementIntersect
% 0.31/0.49 => ( complementInPowersetComplementIntersect
% 0.31/0.49 => ( contraSubsetComplement
% 0.31/0.49 => ( complementTcontraSubset
% 0.31/0.49 => ! [A: $i,X: $i] :
% 0.31/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.49 => ! [Y: $i] :
% 0.31/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.49 => ! [Xx: $i] :
% 0.31/0.49 ( ( in @ Xx @ A )
% 0.31/0.49 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 %------------------------------------------------------------------------------
% 0.31/0.49 ------------------------
% 2.78/1.56 ---- Embedded in THF ---
% 2.78/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.78/1.56 %%% Generated on Tue Feb 24 12:32:40 EST 2026
% 2.78/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.78/1.56 %%% Logic specification used:
% 2.78/1.56 %%% thf(spec, logic, $$dhol).
% 2.78/1.56
% 2.78/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ebZAxzcvYS/DTF2DTF11747.p
% See solution above
% 2.78/1.56 ------------------------
% 2.78/1.59 ---- Cleaned THF ---
% 2.78/1.59 thf(in_type,type,
% 2.78/1.59 in: $i > $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(exu_type,type,
% 2.78/1.59 exu: ( $i > $o ) > $o ).
% 2.78/1.59
% 2.78/1.59 thf(exu,definition,
% 2.78/1.59 ( exu
% 2.78/1.59 = ( ^ [Xphi: $i > $o] :
% 2.78/1.59 ? [Xx: $i] :
% 2.78/1.59 ( ( Xphi @ Xx )
% 2.78/1.59 & ! [Xy: $i] :
% 2.78/1.59 ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setextAx_type,type,
% 2.78/1.59 setextAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setextAx,definition,
% 2.78/1.59 ( setextAx
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 <=> ( in @ Xx @ B ) )
% 2.78/1.59 => ( A = B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptyset_type,type,
% 2.78/1.59 emptyset: $i ).
% 2.78/1.59
% 2.78/1.59 thf(emptysetAx_type,type,
% 2.78/1.59 emptysetAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptysetAx,definition,
% 2.78/1.59 ( emptysetAx
% 2.78/1.59 = ( ! [Xx: $i] :
% 2.78/1.59 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoin_type,type,
% 2.78/1.59 setadjoin: $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinAx_type,type,
% 2.78/1.59 setadjoinAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinAx,definition,
% 2.78/1.59 ( setadjoinAx
% 2.78/1.59 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.78/1.59 <=> ( ( Xy = Xx )
% 2.78/1.59 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(powerset_type,type,
% 2.78/1.59 powerset: $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(powersetAx_type,type,
% 2.78/1.59 powersetAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(powersetAx,definition,
% 2.78/1.59 ( powersetAx
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( in @ B @ ( powerset @ A ) )
% 2.78/1.59 <=> ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setunion_type,type,
% 2.78/1.59 setunion: $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(setunionAx_type,type,
% 2.78/1.59 setunionAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setunionAx,definition,
% 2.78/1.59 ( setunionAx
% 2.78/1.59 = ( ! [A: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( setunion @ A ) )
% 2.78/1.59 <=> ? [B: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 & ( in @ B @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(omega_type,type,
% 2.78/1.59 omega: $i ).
% 2.78/1.59
% 2.78/1.59 thf(omega0Ax_type,type,
% 2.78/1.59 omega0Ax: $o ).
% 2.78/1.59
% 2.78/1.59 thf(omega0Ax,definition,
% 2.78/1.59 ( omega0Ax
% 2.78/1.59 = ( in @ emptyset @ omega ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(omegaSAx_type,type,
% 2.78/1.59 omegaSAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(omegaSAx,definition,
% 2.78/1.59 ( omegaSAx
% 2.78/1.59 = ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ omega )
% 2.78/1.59 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(omegaIndAx_type,type,
% 2.78/1.59 omegaIndAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(omegaIndAx,definition,
% 2.78/1.59 ( omegaIndAx
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ( ( in @ emptyset @ A )
% 2.78/1.59 & ! [Xx: $i] :
% 2.78/1.59 ( ( ( in @ Xx @ omega )
% 2.78/1.59 & ( in @ Xx @ A ) )
% 2.78/1.59 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ omega )
% 2.78/1.59 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(replAx_type,type,
% 2.78/1.59 replAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(replAx,definition,
% 2.78/1.59 ( replAx
% 2.78/1.59 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( exu
% 2.78/1.59 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.78/1.59 => ? [B: $i] :
% 2.78/1.59 ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 <=> ? [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ A )
% 2.78/1.59 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(foundationAx_type,type,
% 2.78/1.59 foundationAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(foundationAx,definition,
% 2.78/1.59 ( foundationAx
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.78/1.59 => ? [B: $i] :
% 2.78/1.59 ( ( in @ B @ A )
% 2.78/1.59 & ~ ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(wellorderingAx_type,type,
% 2.78/1.59 wellorderingAx: $o ).
% 2.78/1.59
% 2.78/1.59 thf(wellorderingAx,definition,
% 2.78/1.59 ( wellorderingAx
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ? [B: $i] :
% 2.78/1.59 ( ! [C: $i] :
% 2.78/1.59 ( ( in @ C @ B )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ C )
% 2.78/1.59 => ( in @ Xx @ A ) ) )
% 2.78/1.59 & ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( ( in @ Xx @ A )
% 2.78/1.59 & ( in @ Xy @ A ) )
% 2.78/1.59 => ( ! [C: $i] :
% 2.78/1.59 ( ( in @ C @ B )
% 2.78/1.59 => ( ( in @ Xx @ C )
% 2.78/1.59 <=> ( in @ Xy @ C ) ) )
% 2.78/1.59 => ( Xx = Xy ) ) )
% 2.78/1.59 & ! [C: $i,D: $i] :
% 2.78/1.59 ( ( ( in @ C @ B )
% 2.78/1.59 & ( in @ D @ B ) )
% 2.78/1.59 => ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ C )
% 2.78/1.59 => ( in @ Xx @ D ) )
% 2.78/1.59 | ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ D )
% 2.78/1.59 => ( in @ Xx @ C ) ) ) )
% 2.78/1.59 & ! [C: $i] :
% 2.78/1.59 ( ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ C )
% 2.78/1.59 => ( in @ Xx @ A ) )
% 2.78/1.59 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.78/1.59 => ? [D: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ D @ B )
% 2.78/1.59 & ( in @ Xx @ C )
% 2.78/1.59 & ~ ? [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ D )
% 2.78/1.59 & ( in @ Xy @ C ) )
% 2.78/1.59 & ! [E: $i] :
% 2.78/1.59 ( ( in @ E @ B )
% 2.78/1.59 => ( ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ E )
% 2.78/1.59 => ( in @ Xy @ D ) )
% 2.78/1.59 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(descr_type,type,
% 2.78/1.59 descr: ( $i > $o ) > $i ).
% 2.78/1.59
% 2.78/1.59 thf(descrp_type,type,
% 2.78/1.59 descrp: $o ).
% 2.78/1.59
% 2.78/1.59 thf(descrp,definition,
% 2.78/1.59 ( descrp
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ( Xphi
% 2.78/1.59 @ ( descr
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstr_type,type,
% 2.78/1.59 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstrI_type,type,
% 2.78/1.59 dsetconstrI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstrI,definition,
% 2.78/1.59 ( dsetconstrI
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 => ( in @ Xx
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstrEL_type,type,
% 2.78/1.59 dsetconstrEL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstrEL,definition,
% 2.78/1.59 ( dsetconstrEL
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.78/1.59 => ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstrER_type,type,
% 2.78/1.59 dsetconstrER: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstrER,definition,
% 2.78/1.59 ( dsetconstrER
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.78/1.59 => ( Xphi @ Xx ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exuE1_type,type,
% 2.78/1.59 exuE1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exuE1,definition,
% 2.78/1.59 ( exuE1
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ? [Xx: $i] :
% 2.78/1.59 ( ( Xphi @ Xx )
% 2.78/1.59 & ! [Xy: $i] :
% 2.78/1.59 ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(prop2set_type,type,
% 2.78/1.59 prop2set: $o > $i ).
% 2.78/1.59
% 2.78/1.59 thf(prop2setE_type,type,
% 2.78/1.59 prop2setE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(prop2setE,definition,
% 2.78/1.59 ( prop2setE
% 2.78/1.59 = ( ! [Xphi: $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.78/1.59 => Xphi ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptysetE_type,type,
% 2.78/1.59 emptysetE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptysetE,definition,
% 2.78/1.59 ( emptysetE
% 2.78/1.59 = ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ emptyset )
% 2.78/1.59 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptysetimpfalse_type,type,
% 2.78/1.59 emptysetimpfalse: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptysetimpfalse,definition,
% 2.78/1.59 ( emptysetimpfalse
% 2.78/1.59 = ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ emptyset )
% 2.78/1.59 => $false ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(notinemptyset_type,type,
% 2.78/1.59 notinemptyset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(notinemptyset,definition,
% 2.78/1.59 ( notinemptyset
% 2.78/1.59 = ( ! [Xx: $i] :
% 2.78/1.59 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exuE3e_type,type,
% 2.78/1.59 exuE3e: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exuE3e,definition,
% 2.78/1.59 ( exuE3e
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setext_type,type,
% 2.78/1.59 setext: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setext,definition,
% 2.78/1.59 ( setext
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ Xx @ B ) )
% 2.78/1.59 => ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ A ) )
% 2.78/1.59 => ( A = B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptyI_type,type,
% 2.78/1.59 emptyI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptyI,definition,
% 2.78/1.59 ( emptyI
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ~ ( in @ Xx @ A )
% 2.78/1.59 => ( A = emptyset ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(noeltsimpempty_type,type,
% 2.78/1.59 noeltsimpempty: $o ).
% 2.78/1.59
% 2.78/1.59 thf(noeltsimpempty,definition,
% 2.78/1.59 ( noeltsimpempty
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ~ ( in @ Xx @ A )
% 2.78/1.59 => ( A = emptyset ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setbeta_type,type,
% 2.78/1.59 setbeta: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setbeta,definition,
% 2.78/1.59 ( setbeta
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( in @ Xx
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.78/1.59 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(nonempty_type,type,
% 2.78/1.59 nonempty: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(nonempty,definition,
% 2.78/1.59 ( nonempty
% 2.78/1.59 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(nonemptyE1_type,type,
% 2.78/1.59 nonemptyE1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(nonemptyE1,definition,
% 2.78/1.59 ( nonemptyE1
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ( nonempty @ A )
% 2.78/1.59 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(nonemptyI_type,type,
% 2.78/1.59 nonemptyI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(nonemptyI,definition,
% 2.78/1.59 ( nonemptyI
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 => ( nonempty
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(nonemptyI1_type,type,
% 2.78/1.59 nonemptyI1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(nonemptyI1,definition,
% 2.78/1.59 ( nonemptyI1
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.78/1.59 => ( nonempty @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinIL_type,type,
% 2.78/1.59 setadjoinIL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinIL,definition,
% 2.78/1.59 ( setadjoinIL
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptyinunitempty_type,type,
% 2.78/1.59 emptyinunitempty: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptyinunitempty,definition,
% 2.78/1.59 ( emptyinunitempty
% 2.78/1.59 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinIR_type,type,
% 2.78/1.59 setadjoinIR: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinIR,definition,
% 2.78/1.59 ( setadjoinIR
% 2.78/1.59 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ A )
% 2.78/1.59 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinE_type,type,
% 2.78/1.59 setadjoinE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinE,definition,
% 2.78/1.59 ( setadjoinE
% 2.78/1.59 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.78/1.59 => ! [Xphi: $o] :
% 2.78/1.59 ( ( ( Xy = Xx )
% 2.78/1.59 => Xphi )
% 2.78/1.59 => ( ( ( in @ Xy @ A )
% 2.78/1.59 => Xphi )
% 2.78/1.59 => Xphi ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinOr_type,type,
% 2.78/1.59 setadjoinOr: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinOr,definition,
% 2.78/1.59 ( setadjoinOr
% 2.78/1.59 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.78/1.59 => ( ( Xy = Xx )
% 2.78/1.59 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setoftrueEq_type,type,
% 2.78/1.59 setoftrueEq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setoftrueEq,definition,
% 2.78/1.59 ( setoftrueEq
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xx: $i] : $true )
% 2.78/1.59 = A ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(powersetI_type,type,
% 2.78/1.59 powersetI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(powersetI,definition,
% 2.78/1.59 ( powersetI
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ A ) )
% 2.78/1.59 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptyinPowerset_type,type,
% 2.78/1.59 emptyinPowerset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptyinPowerset,definition,
% 2.78/1.59 ( emptyinPowerset
% 2.78/1.59 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptyInPowerset_type,type,
% 2.78/1.59 emptyInPowerset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptyInPowerset,definition,
% 2.78/1.59 ( emptyInPowerset
% 2.78/1.59 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(powersetE_type,type,
% 2.78/1.59 powersetE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(powersetE,definition,
% 2.78/1.59 ( powersetE
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ B @ ( powerset @ A ) )
% 2.78/1.59 => ( ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setunionI_type,type,
% 2.78/1.59 setunionI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setunionI,definition,
% 2.78/1.59 ( setunionI
% 2.78/1.59 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 => ( ( in @ B @ A )
% 2.78/1.59 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setunionE_type,type,
% 2.78/1.59 setunionE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setunionE,definition,
% 2.78/1.59 ( setunionE
% 2.78/1.59 = ( ! [A: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( setunion @ A ) )
% 2.78/1.59 => ! [Xphi: $o] :
% 2.78/1.59 ( ! [B: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 => ( ( in @ B @ A )
% 2.78/1.59 => Xphi ) )
% 2.78/1.59 => Xphi ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subPowSU_type,type,
% 2.78/1.59 subPowSU: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subPowSU,definition,
% 2.78/1.59 ( subPowSU
% 2.78/1.59 = ( ! [A: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exuE2_type,type,
% 2.78/1.59 exuE2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exuE2,definition,
% 2.78/1.59 ( exuE2
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ? [Xx: $i] :
% 2.78/1.59 ! [Xy: $i] :
% 2.78/1.59 ( ( Xphi @ Xy )
% 2.78/1.59 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(nonemptyImpWitness_type,type,
% 2.78/1.59 nonemptyImpWitness: $o ).
% 2.78/1.59
% 2.78/1.59 thf(nonemptyImpWitness,definition,
% 2.78/1.59 ( nonemptyImpWitness
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ( nonempty @ A )
% 2.78/1.59 => ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & $true ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(uniqinunit_type,type,
% 2.78/1.59 uniqinunit: $o ).
% 2.78/1.59
% 2.78/1.59 thf(uniqinunit,definition,
% 2.78/1.59 ( uniqinunit
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(notinsingleton_type,type,
% 2.78/1.59 notinsingleton: $o ).
% 2.78/1.59
% 2.78/1.59 thf(notinsingleton,definition,
% 2.78/1.59 ( notinsingleton
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx != Xy )
% 2.78/1.59 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(eqinunit_type,type,
% 2.78/1.59 eqinunit: $o ).
% 2.78/1.59
% 2.78/1.59 thf(eqinunit,definition,
% 2.78/1.59 ( eqinunit
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx = Xy )
% 2.78/1.59 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(singletonsswitch_type,type,
% 2.78/1.59 singletonsswitch: $o ).
% 2.78/1.59
% 2.78/1.59 thf(singletonsswitch,definition,
% 2.78/1.59 ( singletonsswitch
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.78/1.59 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(upairsetE_type,type,
% 2.78/1.59 upairsetE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairsetE,definition,
% 2.78/1.59 ( upairsetE
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.78/1.59 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.78/1.59 => ( ( Xz = Xx )
% 2.78/1.59 | ( Xz = Xy ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(upairsetIL_type,type,
% 2.78/1.59 upairsetIL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairsetIL,definition,
% 2.78/1.59 ( upairsetIL
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(upairsetIR_type,type,
% 2.78/1.59 upairsetIR: $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairsetIR,definition,
% 2.78/1.59 ( upairsetIR
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptyE1_type,type,
% 2.78/1.59 emptyE1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptyE1,definition,
% 2.78/1.59 ( emptyE1
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ( Xphi @ Xx ) )
% 2.78/1.59 => ( ( ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 = emptyset )
% 2.78/1.59 => $false ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(vacuousDall_type,type,
% 2.78/1.59 vacuousDall: $o ).
% 2.78/1.59
% 2.78/1.59 thf(vacuousDall,definition,
% 2.78/1.59 ( vacuousDall
% 2.78/1.59 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ emptyset )
% 2.78/1.59 => ( Xphi @ Xx ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(quantDeMorgan1_type,type,
% 2.78/1.59 quantDeMorgan1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(quantDeMorgan1,definition,
% 2.78/1.59 ( quantDeMorgan1
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ~ ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( Xphi @ Xx ) )
% 2.78/1.59 => ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(quantDeMorgan2_type,type,
% 2.78/1.59 quantDeMorgan2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(quantDeMorgan2,definition,
% 2.78/1.59 ( quantDeMorgan2
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ~ ( Xphi @ Xx ) )
% 2.78/1.59 => ~ ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(quantDeMorgan3_type,type,
% 2.78/1.59 quantDeMorgan3: $o ).
% 2.78/1.59
% 2.78/1.59 thf(quantDeMorgan3,definition,
% 2.78/1.59 ( quantDeMorgan3
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ~ ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ( Xphi @ Xx ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(quantDeMorgan4_type,type,
% 2.78/1.59 quantDeMorgan4: $o ).
% 2.78/1.59
% 2.78/1.59 thf(quantDeMorgan4,definition,
% 2.78/1.59 ( quantDeMorgan4
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ~ ( Xphi @ Xx ) )
% 2.78/1.59 => ~ ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(prop2setI_type,type,
% 2.78/1.59 prop2setI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(prop2setI,definition,
% 2.78/1.59 ( prop2setI
% 2.78/1.59 = ( ! [Xphi: $o] :
% 2.78/1.59 ( Xphi
% 2.78/1.59 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(set2prop_type,type,
% 2.78/1.59 set2prop: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(prop2set2propI_type,type,
% 2.78/1.59 prop2set2propI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(prop2set2propI,definition,
% 2.78/1.59 ( prop2set2propI
% 2.78/1.59 = ( ! [Xphi: $o] :
% 2.78/1.59 ( Xphi
% 2.78/1.59 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(notdexE_type,type,
% 2.78/1.59 notdexE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(notdexE,definition,
% 2.78/1.59 ( notdexE
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ~ ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ( Xphi @ Xx ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(notdallE_type,type,
% 2.78/1.59 notdallE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(notdallE,definition,
% 2.78/1.59 ( notdallE
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ~ ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( Xphi @ Xx ) )
% 2.78/1.59 => ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exuI1_type,type,
% 2.78/1.59 exuI1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exuI1,definition,
% 2.78/1.59 ( exuI1
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ? [Xx: $i] :
% 2.78/1.59 ( ( Xphi @ Xx )
% 2.78/1.59 & ! [Xy: $i] :
% 2.78/1.59 ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) )
% 2.78/1.59 => ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exuI3_type,type,
% 2.78/1.59 exuI3: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exuI3,definition,
% 2.78/1.59 ( exuI3
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.78/1.59 => ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xphi @ Xx )
% 2.78/1.59 => ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) )
% 2.78/1.59 => ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exuI2_type,type,
% 2.78/1.59 exuI2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exuI2,definition,
% 2.78/1.59 ( exuI2
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ? [Xx: $i] :
% 2.78/1.59 ! [Xy: $i] :
% 2.78/1.59 ( ( Xphi @ Xy )
% 2.78/1.59 <=> ( Xy = Xx ) )
% 2.78/1.59 => ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(inCongP_type,type,
% 2.78/1.59 inCongP: $o ).
% 2.78/1.59
% 2.78/1.59 thf(inCongP,definition,
% 2.78/1.59 ( inCongP
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( A = B )
% 2.78/1.59 => ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx = Xy )
% 2.78/1.59 => ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(in__Cong_type,type,
% 2.78/1.59 in__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(in__Cong,definition,
% 2.78/1.59 ( in__Cong
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( A = B )
% 2.78/1.59 => ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx = Xy )
% 2.78/1.59 => ( ( in @ Xx @ A )
% 2.78/1.59 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exuE3u_type,type,
% 2.78/1.59 exuE3u: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exuE3u,definition,
% 2.78/1.59 ( exuE3u
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xphi @ Xx )
% 2.78/1.59 => ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exu__Cong_type,type,
% 2.78/1.59 exu__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exu__Cong,definition,
% 2.78/1.59 ( exu__Cong
% 2.78/1.59 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.78/1.59 ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx = Xy )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 <=> ( Xpsi @ Xy ) ) )
% 2.78/1.59 => ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 <=> ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptyset__Cong_type,type,
% 2.78/1.59 emptyset__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptyset__Cong,definition,
% 2.78/1.59 ( emptyset__Cong
% 2.78/1.59 = ( emptyset = emptyset ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoin__Cong_type,type,
% 2.78/1.59 setadjoin__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoin__Cong,definition,
% 2.78/1.59 ( setadjoin__Cong
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx = Xy )
% 2.78/1.59 => ! [Xz: $i,Xu: $i] :
% 2.78/1.59 ( ( Xz = Xu )
% 2.78/1.59 => ( ( setadjoin @ Xx @ Xz )
% 2.78/1.59 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(powerset__Cong_type,type,
% 2.78/1.59 powerset__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(powerset__Cong,definition,
% 2.78/1.59 ( powerset__Cong
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( A = B )
% 2.78/1.59 => ( ( powerset @ A )
% 2.78/1.59 = ( powerset @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setunion__Cong_type,type,
% 2.78/1.59 setunion__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setunion__Cong,definition,
% 2.78/1.59 ( setunion__Cong
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( A = B )
% 2.78/1.59 => ( ( setunion @ A )
% 2.78/1.59 = ( setunion @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(omega__Cong_type,type,
% 2.78/1.59 omega__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(omega__Cong,definition,
% 2.78/1.59 ( omega__Cong
% 2.78/1.59 = ( omega = omega ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(exuEu_type,type,
% 2.78/1.59 exuEu: $o ).
% 2.78/1.59
% 2.78/1.59 thf(exuEu,definition,
% 2.78/1.59 ( exuEu
% 2.78/1.59 = ( ! [Xphi: $i > $o] :
% 2.78/1.59 ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xphi @ Xx )
% 2.78/1.59 => ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(descr__Cong_type,type,
% 2.78/1.59 descr__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(descr__Cong,definition,
% 2.78/1.59 ( descr__Cong
% 2.78/1.59 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.78/1.59 ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx = Xy )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 <=> ( Xpsi @ Xy ) ) )
% 2.78/1.59 => ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ( ( exu
% 2.78/1.59 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.78/1.59 => ( ( descr
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 = ( descr
% 2.78/1.59 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstr__Cong_type,type,
% 2.78/1.59 dsetconstr__Cong: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dsetconstr__Cong,definition,
% 2.78/1.59 ( dsetconstr__Cong
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( A = B )
% 2.78/1.59 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( Xx = Xy )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.78/1.59 => ( ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 = ( dsetconstr @ B
% 2.78/1.59 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subset_type,type,
% 2.78/1.59 subset: $i > $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(disjoint_type,type,
% 2.78/1.59 disjoint: $i > $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(setsmeet_type,type,
% 2.78/1.59 setsmeet: $i > $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(subsetI1_type,type,
% 2.78/1.59 subsetI1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subsetI1,definition,
% 2.78/1.59 ( subsetI1
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ Xx @ B ) )
% 2.78/1.59 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(eqimpsubset2_type,type,
% 2.78/1.59 eqimpsubset2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(eqimpsubset2,definition,
% 2.78/1.59 ( eqimpsubset2
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( A = B )
% 2.78/1.59 => ( subset @ B @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(eqimpsubset1_type,type,
% 2.78/1.59 eqimpsubset1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(eqimpsubset1,definition,
% 2.78/1.59 ( eqimpsubset1
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( A = B )
% 2.78/1.59 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subsetI2_type,type,
% 2.78/1.59 subsetI2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subsetI2,definition,
% 2.78/1.59 ( subsetI2
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ Xx @ B ) )
% 2.78/1.59 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(emptysetsubset_type,type,
% 2.78/1.59 emptysetsubset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(emptysetsubset,definition,
% 2.78/1.59 ( emptysetsubset
% 2.78/1.59 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subsetE_type,type,
% 2.78/1.59 subsetE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subsetE,definition,
% 2.78/1.59 ( subsetE
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subsetE2_type,type,
% 2.78/1.59 subsetE2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subsetE2,definition,
% 2.78/1.59 ( subsetE2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( ~ ( in @ Xx @ B )
% 2.78/1.59 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(notsubsetI_type,type,
% 2.78/1.59 notsubsetI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(notsubsetI,definition,
% 2.78/1.59 ( notsubsetI
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ~ ( in @ Xx @ B )
% 2.78/1.59 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(notequalI1_type,type,
% 2.78/1.59 notequalI1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(notequalI1,definition,
% 2.78/1.59 ( notequalI1
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ~ ( subset @ A @ B )
% 2.78/1.59 => ( A != B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(notequalI2_type,type,
% 2.78/1.59 notequalI2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(notequalI2,definition,
% 2.78/1.59 ( notequalI2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ~ ( in @ Xx @ B )
% 2.78/1.59 => ( A != B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subsetRefl_type,type,
% 2.78/1.59 subsetRefl: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subsetRefl,definition,
% 2.78/1.59 ( subsetRefl
% 2.78/1.59 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subsetTrans_type,type,
% 2.78/1.59 subsetTrans: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subsetTrans,definition,
% 2.78/1.59 ( subsetTrans
% 2.78/1.59 = ( ! [A: $i,B: $i,C: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( ( subset @ B @ C )
% 2.78/1.59 => ( subset @ A @ C ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinSub_type,type,
% 2.78/1.59 setadjoinSub: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinSub,definition,
% 2.78/1.59 ( setadjoinSub
% 2.78/1.59 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinSub2_type,type,
% 2.78/1.59 setadjoinSub2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setadjoinSub2,definition,
% 2.78/1.59 ( setadjoinSub2
% 2.78/1.59 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subset2powerset_type,type,
% 2.78/1.59 subset2powerset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subset2powerset,definition,
% 2.78/1.59 ( subset2powerset
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setextsub_type,type,
% 2.78/1.59 setextsub: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setextsub,definition,
% 2.78/1.59 ( setextsub
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( ( subset @ B @ A )
% 2.78/1.59 => ( A = B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subsetemptysetimpeq_type,type,
% 2.78/1.59 subsetemptysetimpeq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subsetemptysetimpeq,definition,
% 2.78/1.59 ( subsetemptysetimpeq
% 2.78/1.59 = ( ! [A: $i] :
% 2.78/1.59 ( ( subset @ A @ emptyset )
% 2.78/1.59 => ( A = emptyset ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(powersetI1_type,type,
% 2.78/1.59 powersetI1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(powersetI1,definition,
% 2.78/1.59 ( powersetI1
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( subset @ B @ A )
% 2.78/1.59 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(powersetE1_type,type,
% 2.78/1.59 powersetE1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(powersetE1,definition,
% 2.78/1.59 ( powersetE1
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( in @ B @ ( powerset @ A ) )
% 2.78/1.59 => ( subset @ B @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(inPowerset_type,type,
% 2.78/1.59 inPowerset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(inPowerset,definition,
% 2.78/1.59 ( inPowerset
% 2.78/1.59 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(powersetsubset_type,type,
% 2.78/1.59 powersetsubset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(powersetsubset,definition,
% 2.78/1.59 ( powersetsubset
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(sepInPowerset_type,type,
% 2.78/1.59 sepInPowerset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(sepInPowerset,definition,
% 2.78/1.59 ( sepInPowerset
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( in
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 @ ( powerset @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(sepSubset_type,type,
% 2.78/1.59 sepSubset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(sepSubset,definition,
% 2.78/1.59 ( sepSubset
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( subset
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 @ A ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binunion_type,type,
% 2.78/1.59 binunion: $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(binunionIL_type,type,
% 2.78/1.59 binunionIL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binunionIL,definition,
% 2.78/1.59 ( binunionIL
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(upairset2IR_type,type,
% 2.78/1.59 upairset2IR: $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairset2IR,definition,
% 2.78/1.59 ( upairset2IR
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binunionIR_type,type,
% 2.78/1.59 binunionIR: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binunionIR,definition,
% 2.78/1.59 ( binunionIR
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binunionEcases_type,type,
% 2.78/1.59 binunionEcases: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binunionEcases,definition,
% 2.78/1.59 ( binunionEcases
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.78/1.59 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.78/1.59 => ( ( ( in @ Xx @ A )
% 2.78/1.59 => Xphi )
% 2.78/1.59 => ( ( ( in @ Xx @ B )
% 2.78/1.59 => Xphi )
% 2.78/1.59 => Xphi ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binunionE_type,type,
% 2.78/1.59 binunionE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binunionE,definition,
% 2.78/1.59 ( binunionE
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.78/1.59 => ( ( in @ Xx @ A )
% 2.78/1.59 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binunionLsub_type,type,
% 2.78/1.59 binunionLsub: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binunionLsub,definition,
% 2.78/1.59 ( binunionLsub
% 2.78/1.59 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binunionRsub_type,type,
% 2.78/1.59 binunionRsub: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binunionRsub,definition,
% 2.78/1.59 ( binunionRsub
% 2.78/1.59 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersect_type,type,
% 2.78/1.59 binintersect: $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectI_type,type,
% 2.78/1.59 binintersectI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectI,definition,
% 2.78/1.59 ( binintersectI
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset5_type,type,
% 2.78/1.59 binintersectSubset5: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset5,definition,
% 2.78/1.59 ( binintersectSubset5
% 2.78/1.59 = ( ! [A: $i,B: $i,C: $i] :
% 2.78/1.59 ( ( subset @ C @ A )
% 2.78/1.59 => ( ( subset @ C @ B )
% 2.78/1.59 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectEL_type,type,
% 2.78/1.59 binintersectEL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectEL,definition,
% 2.78/1.59 ( binintersectEL
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.78/1.59 => ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectLsub_type,type,
% 2.78/1.59 binintersectLsub: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectLsub,definition,
% 2.78/1.59 ( binintersectLsub
% 2.78/1.59 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset2_type,type,
% 2.78/1.59 binintersectSubset2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset2,definition,
% 2.78/1.59 ( binintersectSubset2
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( ( binintersect @ A @ B )
% 2.78/1.59 = A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset3_type,type,
% 2.78/1.59 binintersectSubset3: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset3,definition,
% 2.78/1.59 ( binintersectSubset3
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( ( binintersect @ A @ B )
% 2.78/1.59 = B )
% 2.78/1.59 => ( subset @ B @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectER_type,type,
% 2.78/1.59 binintersectER: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectER,definition,
% 2.78/1.59 ( binintersectER
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.78/1.59 => ( in @ Xx @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(disjointsetsI1_type,type,
% 2.78/1.59 disjointsetsI1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(disjointsetsI1,definition,
% 2.78/1.59 ( disjointsetsI1
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ~ ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ( in @ Xx @ B ) )
% 2.78/1.59 => ( ( binintersect @ A @ B )
% 2.78/1.59 = emptyset ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectRsub_type,type,
% 2.78/1.59 binintersectRsub: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectRsub,definition,
% 2.78/1.59 ( binintersectRsub
% 2.78/1.59 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset4_type,type,
% 2.78/1.59 binintersectSubset4: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset4,definition,
% 2.78/1.59 ( binintersectSubset4
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( subset @ B @ A )
% 2.78/1.59 => ( ( binintersect @ A @ B )
% 2.78/1.59 = B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset1_type,type,
% 2.78/1.59 binintersectSubset1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(binintersectSubset1,definition,
% 2.78/1.59 ( binintersectSubset1
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( ( binintersect @ A @ B )
% 2.78/1.59 = A )
% 2.78/1.59 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(bs114d_type,type,
% 2.78/1.59 bs114d: $o ).
% 2.78/1.59
% 2.78/1.59 thf(bs114d,definition,
% 2.78/1.59 ( bs114d
% 2.78/1.59 = ( ! [A: $i,B: $i,C: $i] :
% 2.78/1.59 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.78/1.59 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(regular_type,type,
% 2.78/1.59 regular: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminus_type,type,
% 2.78/1.59 setminus: $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(setminusI_type,type,
% 2.78/1.59 setminusI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusI,definition,
% 2.78/1.59 ( setminusI
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ~ ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusEL_type,type,
% 2.78/1.59 setminusEL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusEL,definition,
% 2.78/1.59 ( setminusEL
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.78/1.59 => ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusER_type,type,
% 2.78/1.59 setminusER: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusER,definition,
% 2.78/1.59 ( setminusER
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.78/1.59 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusSubset2_type,type,
% 2.78/1.59 setminusSubset2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusSubset2,definition,
% 2.78/1.59 ( setminusSubset2
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( subset @ A @ B )
% 2.78/1.59 => ( ( setminus @ A @ B )
% 2.78/1.59 = emptyset ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusERneg_type,type,
% 2.78/1.59 setminusERneg: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusERneg,definition,
% 2.78/1.59 ( setminusERneg
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.78/1.59 => ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusELneg_type,type,
% 2.78/1.59 setminusELneg: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusELneg,definition,
% 2.78/1.59 ( setminusELneg
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.78/1.59 => ( ~ ( in @ Xx @ B )
% 2.78/1.59 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusILneg_type,type,
% 2.78/1.59 setminusILneg: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusILneg,definition,
% 2.78/1.59 ( setminusILneg
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ~ ( in @ Xx @ A )
% 2.78/1.59 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusIRneg_type,type,
% 2.78/1.59 setminusIRneg: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusIRneg,definition,
% 2.78/1.59 ( setminusIRneg
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ B )
% 2.78/1.59 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusLsub_type,type,
% 2.78/1.59 setminusLsub: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusLsub,definition,
% 2.78/1.59 ( setminusLsub
% 2.78/1.59 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setminusSubset1_type,type,
% 2.78/1.59 setminusSubset1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setminusSubset1,definition,
% 2.78/1.59 ( setminusSubset1
% 2.78/1.59 = ( ! [A: $i,B: $i] :
% 2.78/1.59 ( ( ( setminus @ A @ B )
% 2.78/1.59 = emptyset )
% 2.78/1.59 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(symdiff_type,type,
% 2.78/1.59 symdiff: $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffE_type,type,
% 2.78/1.59 symdiffE: $o ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffE,definition,
% 2.78/1.59 ( symdiffE
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.78/1.59 => ! [Xphi: $o] :
% 2.78/1.59 ( ( ( in @ Xx @ A )
% 2.78/1.59 => ( ~ ( in @ Xx @ B )
% 2.78/1.59 => Xphi ) )
% 2.78/1.59 => ( ( ~ ( in @ Xx @ A )
% 2.78/1.59 => ( ( in @ Xx @ B )
% 2.78/1.59 => Xphi ) )
% 2.78/1.59 => Xphi ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffI1_type,type,
% 2.78/1.59 symdiffI1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffI1,definition,
% 2.78/1.59 ( symdiffI1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ~ ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffI2_type,type,
% 2.78/1.59 symdiffI2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffI2,definition,
% 2.78/1.59 ( symdiffI2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ~ ( in @ Xx @ A )
% 2.78/1.59 => ( ( in @ Xx @ B )
% 2.78/1.59 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffIneg1_type,type,
% 2.78/1.59 symdiffIneg1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffIneg1,definition,
% 2.78/1.59 ( symdiffIneg1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( in @ Xx @ B )
% 2.78/1.59 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffIneg2_type,type,
% 2.78/1.59 symdiffIneg2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(symdiffIneg2,definition,
% 2.78/1.59 ( symdiffIneg2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ~ ( in @ Xx @ A )
% 2.78/1.59 => ( ~ ( in @ Xx @ B )
% 2.78/1.59 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(iskpair_type,type,
% 2.78/1.59 iskpair: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(secondinupair_type,type,
% 2.78/1.59 secondinupair: $o ).
% 2.78/1.59
% 2.78/1.59 thf(secondinupair,definition,
% 2.78/1.59 ( secondinupair
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairIL_type,type,
% 2.78/1.59 setukpairIL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairIL,definition,
% 2.78/1.59 ( setukpairIL
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairIR_type,type,
% 2.78/1.59 setukpairIR: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairIR,definition,
% 2.78/1.59 ( setukpairIR
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(kpairiskpair_type,type,
% 2.78/1.59 kpairiskpair: $o ).
% 2.78/1.59
% 2.78/1.59 thf(kpairiskpair,definition,
% 2.78/1.59 ( kpairiskpair
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(kpair_type,type,
% 2.78/1.59 kpair: $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(kpairp_type,type,
% 2.78/1.59 kpairp: $o ).
% 2.78/1.59
% 2.78/1.59 thf(kpairp,definition,
% 2.78/1.59 ( kpairp
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprod_type,type,
% 2.78/1.59 cartprod: $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(singletonsubset_type,type,
% 2.78/1.59 singletonsubset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(singletonsubset,definition,
% 2.78/1.59 ( singletonsubset
% 2.78/1.59 = ( ! [A: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(singletoninpowerset_type,type,
% 2.78/1.59 singletoninpowerset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(singletoninpowerset,definition,
% 2.78/1.59 ( singletoninpowerset
% 2.78/1.59 = ( ! [A: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(singletoninpowunion_type,type,
% 2.78/1.59 singletoninpowunion: $o ).
% 2.78/1.59
% 2.78/1.59 thf(singletoninpowunion,definition,
% 2.78/1.59 ( singletoninpowunion
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(upairset2E_type,type,
% 2.78/1.59 upairset2E: $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairset2E,definition,
% 2.78/1.59 ( upairset2E
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.78/1.59 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.78/1.59 => ( ( Xz = Xx )
% 2.78/1.59 | ( Xz = Xy ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(upairsubunion_type,type,
% 2.78/1.59 upairsubunion: $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairsubunion,definition,
% 2.78/1.59 ( upairsubunion
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(upairinpowunion_type,type,
% 2.78/1.59 upairinpowunion: $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairinpowunion,definition,
% 2.78/1.59 ( upairinpowunion
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ubforcartprodlem1_type,type,
% 2.78/1.59 ubforcartprodlem1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ubforcartprodlem1,definition,
% 2.78/1.59 ( ubforcartprodlem1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ubforcartprodlem2_type,type,
% 2.78/1.59 ubforcartprodlem2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ubforcartprodlem2,definition,
% 2.78/1.59 ( ubforcartprodlem2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ubforcartprodlem3_type,type,
% 2.78/1.59 ubforcartprodlem3: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ubforcartprodlem3,definition,
% 2.78/1.59 ( ubforcartprodlem3
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodpairin_type,type,
% 2.78/1.59 cartprodpairin: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodpairin,definition,
% 2.78/1.59 ( cartprodpairin
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodmempair1_type,type,
% 2.78/1.59 cartprodmempair1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodmempair1,definition,
% 2.78/1.59 ( cartprodmempair1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.78/1.59 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.78/1.59 => ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ? [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 & ( Xu
% 2.78/1.59 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodmempair_type,type,
% 2.78/1.59 cartprodmempair: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodmempair,definition,
% 2.78/1.59 ( cartprodmempair
% 2.78/1.59 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.78/1.59 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.78/1.59 => ( iskpair @ Xu ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setunionE2_type,type,
% 2.78/1.59 setunionE2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setunionE2,definition,
% 2.78/1.59 ( setunionE2
% 2.78/1.59 = ( ! [A: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ ( setunion @ A ) )
% 2.78/1.59 => ? [X: $i] :
% 2.78/1.59 ( ( in @ X @ A )
% 2.78/1.59 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setunionsingleton1_type,type,
% 2.78/1.59 setunionsingleton1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setunionsingleton1,definition,
% 2.78/1.59 ( setunionsingleton1
% 2.78/1.59 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setunionsingleton2_type,type,
% 2.78/1.59 setunionsingleton2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setunionsingleton2,definition,
% 2.78/1.59 ( setunionsingleton2
% 2.78/1.59 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setunionsingleton_type,type,
% 2.78/1.59 setunionsingleton: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setunionsingleton,definition,
% 2.78/1.59 ( setunionsingleton
% 2.78/1.59 = ( ! [Xx: $i] :
% 2.78/1.59 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.78/1.59 = Xx ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(singleton_type,type,
% 2.78/1.59 singleton: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(singletonprop_type,type,
% 2.78/1.59 singletonprop: $o ).
% 2.78/1.59
% 2.78/1.59 thf(singletonprop,definition,
% 2.78/1.59 ( singletonprop
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ A )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 => ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) )
% 2.78/1.59 => ( ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ( Xphi @ Xx ) )
% 2.78/1.59 => ( singleton
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ex1_type,type,
% 2.78/1.59 ex1: $i > ( $i > $o ) > $o ).
% 2.78/1.59
% 2.78/1.59 thf(ex1,definition,
% 2.78/1.59 ( ex1
% 2.78/1.59 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( singleton
% 2.78/1.59 @ ( dsetconstr @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ex1E1_type,type,
% 2.78/1.59 ex1E1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ex1E1,definition,
% 2.78/1.59 ( ex1E1
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ( ex1 @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ex1I_type,type,
% 2.78/1.59 ex1I: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ex1I,definition,
% 2.78/1.59 ( ex1I
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 => ( ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ A )
% 2.78/1.59 => ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xy = Xx ) ) )
% 2.78/1.59 => ( ex1 @ A
% 2.78/1.59 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ex1I2_type,type,
% 2.78/1.59 ex1I2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ex1I2,definition,
% 2.78/1.59 ( ex1I2
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ A )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 => ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) )
% 2.78/1.59 => ( ? [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 & ( Xphi @ Xx ) )
% 2.78/1.59 => ( ex1 @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(singletonsuniq_type,type,
% 2.78/1.59 singletonsuniq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(singletonsuniq,definition,
% 2.78/1.59 ( singletonsuniq
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( ( setadjoin @ Xx @ emptyset )
% 2.78/1.59 = ( setadjoin @ Xy @ emptyset ) )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(atmost1p_type,type,
% 2.78/1.59 atmost1p: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(atleast2p_type,type,
% 2.78/1.59 atleast2p: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(atmost2p_type,type,
% 2.78/1.59 atmost2p: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairsetp_type,type,
% 2.78/1.59 upairsetp: $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjL1_type,type,
% 2.78/1.59 setukpairinjL1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjL1,definition,
% 2.78/1.59 ( setukpairinjL1
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.78/1.59 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.78/1.59 => ( Xx = Xz ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(kfstsingleton_type,type,
% 2.78/1.59 kfstsingleton: $o ).
% 2.78/1.59
% 2.78/1.59 thf(kfstsingleton,definition,
% 2.78/1.59 ( kfstsingleton
% 2.78/1.59 = ( ! [Xu: $i] :
% 2.78/1.59 ( ( iskpair @ Xu )
% 2.78/1.59 => ( singleton
% 2.78/1.59 @ ( dsetconstr @ ( setunion @ Xu )
% 2.78/1.59 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(theprop_type,type,
% 2.78/1.59 theprop: $o ).
% 2.78/1.59
% 2.78/1.59 thf(theprop,definition,
% 2.78/1.59 ( theprop
% 2.78/1.59 = ( ! [X: $i] :
% 2.78/1.59 ( ( singleton @ X )
% 2.78/1.59 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(kfst_type,type,
% 2.78/1.59 kfst: $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(kfstpairEq_type,type,
% 2.78/1.59 kfstpairEq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(kfstpairEq,definition,
% 2.78/1.59 ( kfstpairEq
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.78/1.59 = Xx ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodfstin_type,type,
% 2.78/1.59 cartprodfstin: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodfstin,definition,
% 2.78/1.59 ( cartprodfstin
% 2.78/1.59 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.78/1.59 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.78/1.59 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjL2_type,type,
% 2.78/1.59 setukpairinjL2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjL2,definition,
% 2.78/1.59 ( setukpairinjL2
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.78/1.59 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.78/1.59 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.78/1.59 => ( Xx = Xz ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjL_type,type,
% 2.78/1.59 setukpairinjL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjL,definition,
% 2.78/1.59 ( setukpairinjL
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.78/1.59 ( ( ( kpair @ Xx @ Xy )
% 2.78/1.59 = ( kpair @ Xz @ Xu ) )
% 2.78/1.59 => ( Xx = Xz ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR11_type,type,
% 2.78/1.59 setukpairinjR11: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR11,definition,
% 2.78/1.59 ( setukpairinjR11
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx = Xy )
% 2.78/1.59 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.78/1.59 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR12_type,type,
% 2.78/1.59 setukpairinjR12: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR12,definition,
% 2.78/1.59 ( setukpairinjR12
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xx = Xy )
% 2.78/1.59 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.78/1.59 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR1_type,type,
% 2.78/1.59 setukpairinjR1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR1,definition,
% 2.78/1.59 ( setukpairinjR1
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.78/1.59 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.78/1.59 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.78/1.59 => ( ( Xz = Xu )
% 2.78/1.59 => ( Xy = Xu ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(upairequniteq_type,type,
% 2.78/1.59 upairequniteq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(upairequniteq,definition,
% 2.78/1.59 ( upairequniteq
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.78/1.59 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.78/1.59 = ( setadjoin @ Xz @ emptyset ) )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR2_type,type,
% 2.78/1.59 setukpairinjR2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR2,definition,
% 2.78/1.59 ( setukpairinjR2
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.78/1.59 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.78/1.59 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.78/1.59 => ( Xy = Xu ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR_type,type,
% 2.78/1.59 setukpairinjR: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setukpairinjR,definition,
% 2.78/1.59 ( setukpairinjR
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.78/1.59 ( ( ( kpair @ Xx @ Xy )
% 2.78/1.59 = ( kpair @ Xz @ Xu ) )
% 2.78/1.59 => ( Xy = Xu ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ksndsingleton_type,type,
% 2.78/1.59 ksndsingleton: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ksndsingleton,definition,
% 2.78/1.59 ( ksndsingleton
% 2.78/1.59 = ( ! [Xu: $i] :
% 2.78/1.59 ( ( iskpair @ Xu )
% 2.78/1.59 => ( singleton
% 2.78/1.59 @ ( dsetconstr @ ( setunion @ Xu )
% 2.78/1.59 @ ^ [Xx: $i] :
% 2.78/1.59 ( Xu
% 2.78/1.59 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ksnd_type,type,
% 2.78/1.59 ksnd: $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(ksndpairEq_type,type,
% 2.78/1.59 ksndpairEq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ksndpairEq,definition,
% 2.78/1.59 ( ksndpairEq
% 2.78/1.59 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.78/1.59 = Xy ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(kpairsurjEq_type,type,
% 2.78/1.59 kpairsurjEq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(kpairsurjEq,definition,
% 2.78/1.59 ( kpairsurjEq
% 2.78/1.59 = ( ! [Xu: $i] :
% 2.78/1.59 ( ( iskpair @ Xu )
% 2.78/1.59 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.78/1.59 = Xu ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodsndin_type,type,
% 2.78/1.59 cartprodsndin: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodsndin,definition,
% 2.78/1.59 ( cartprodsndin
% 2.78/1.59 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.78/1.59 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.78/1.59 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodpairmemEL_type,type,
% 2.78/1.59 cartprodpairmemEL: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodpairmemEL,definition,
% 2.78/1.59 ( cartprodpairmemEL
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.78/1.59 => ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodpairmemER_type,type,
% 2.78/1.59 cartprodpairmemER: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodpairmemER,definition,
% 2.78/1.59 ( cartprodpairmemER
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.78/1.59 => ( in @ Xy @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodmempaircEq_type,type,
% 2.78/1.59 cartprodmempaircEq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodmempaircEq,definition,
% 2.78/1.59 ( cartprodmempaircEq
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( kpair @ Xx @ Xy )
% 2.78/1.59 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodfstpairEq_type,type,
% 2.78/1.59 cartprodfstpairEq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodfstpairEq,definition,
% 2.78/1.59 ( cartprodfstpairEq
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.78/1.59 = Xx ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodsndpairEq_type,type,
% 2.78/1.59 cartprodsndpairEq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodsndpairEq,definition,
% 2.78/1.59 ( cartprodsndpairEq
% 2.78/1.59 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.78/1.59 = Xy ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodpairsurjEq_type,type,
% 2.78/1.59 cartprodpairsurjEq: $o ).
% 2.78/1.59
% 2.78/1.59 thf(cartprodpairsurjEq,definition,
% 2.78/1.59 ( cartprodpairsurjEq
% 2.78/1.59 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.78/1.59 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.78/1.59 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.78/1.59 = Xu ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(breln_type,type,
% 2.78/1.59 breln: $i > $i > $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(breln,definition,
% 2.78/1.59 ( breln
% 2.78/1.59 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstr_type,type,
% 2.78/1.59 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrI_type,type,
% 2.78/1.59 dpsetconstrI: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrI,definition,
% 2.78/1.59 ( dpsetconstrI
% 2.78/1.59 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( Xphi @ Xx @ Xy )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ Xy )
% 2.78/1.59 @ ( dpsetconstr @ A @ B
% 2.78/1.59 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrSub_type,type,
% 2.78/1.59 dpsetconstrSub: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrSub,definition,
% 2.78/1.59 ( dpsetconstrSub
% 2.78/1.59 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.78/1.59 ( subset
% 2.78/1.59 @ ( dpsetconstr @ A @ B
% 2.78/1.59 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.78/1.59 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(setOfPairsIsBReln_type,type,
% 2.78/1.59 setOfPairsIsBReln: $o ).
% 2.78/1.59
% 2.78/1.59 thf(setOfPairsIsBReln,definition,
% 2.78/1.59 ( setOfPairsIsBReln
% 2.78/1.59 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.78/1.59 ( breln @ A @ B
% 2.78/1.59 @ ( dpsetconstr @ A @ B
% 2.78/1.59 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrERa_type,type,
% 2.78/1.59 dpsetconstrERa: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrERa,definition,
% 2.78/1.59 ( dpsetconstrERa
% 2.78/1.59 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.78/1.59 @ ( dpsetconstr @ A @ B
% 2.78/1.59 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.78/1.59 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrEL1_type,type,
% 2.78/1.59 dpsetconstrEL1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrEL1,definition,
% 2.78/1.59 ( dpsetconstrEL1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ ( kpair @ Xx @ Xy )
% 2.78/1.59 @ ( dpsetconstr @ A @ B
% 2.78/1.59 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.78/1.59 => ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrEL2_type,type,
% 2.78/1.59 dpsetconstrEL2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrEL2,definition,
% 2.78/1.59 ( dpsetconstrEL2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ ( kpair @ Xx @ Xy )
% 2.78/1.59 @ ( dpsetconstr @ A @ B
% 2.78/1.59 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.78/1.59 => ( in @ Xy @ B ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrER_type,type,
% 2.78/1.59 dpsetconstrER: $o ).
% 2.78/1.59
% 2.78/1.59 thf(dpsetconstrER,definition,
% 2.78/1.59 ( dpsetconstrER
% 2.78/1.59 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( in @ ( kpair @ Xx @ Xy )
% 2.78/1.59 @ ( dpsetconstr @ A @ B
% 2.78/1.59 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.78/1.59 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(func_type,type,
% 2.78/1.59 func: $i > $i > $i > $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcSet_type,type,
% 2.78/1.59 funcSet: $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(funcImageSingleton_type,type,
% 2.78/1.59 funcImageSingleton: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcImageSingleton,definition,
% 2.78/1.59 ( funcImageSingleton
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( singleton
% 2.78/1.59 @ ( dsetconstr @ B
% 2.78/1.59 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(apProp_type,type,
% 2.78/1.59 apProp: $o ).
% 2.78/1.59
% 2.78/1.59 thf(apProp,definition,
% 2.78/1.59 ( apProp
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in
% 2.78/1.59 @ ( setunion
% 2.78/1.59 @ ( dsetconstr @ B
% 2.78/1.59 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.78/1.59 @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ap_type,type,
% 2.78/1.59 ap: $i > $i > $i > $i > $i ).
% 2.78/1.59
% 2.78/1.59 thf(app_type,type,
% 2.78/1.59 app: $o ).
% 2.78/1.59
% 2.78/1.59 thf(app,definition,
% 2.78/1.59 ( app
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(infuncsetfunc_type,type,
% 2.78/1.59 infuncsetfunc: $o ).
% 2.78/1.59
% 2.78/1.59 thf(infuncsetfunc,definition,
% 2.78/1.59 ( infuncsetfunc
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.78/1.59 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ap2p_type,type,
% 2.78/1.59 ap2p: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ap2p,definition,
% 2.78/1.59 ( ap2p
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(funcinfuncset_type,type,
% 2.78/1.59 funcinfuncset: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcinfuncset,definition,
% 2.78/1.59 ( funcinfuncset
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(lamProp_type,type,
% 2.78/1.59 lamProp: $o ).
% 2.78/1.59
% 2.78/1.59 thf(lamProp,definition,
% 2.78/1.59 ( lamProp
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.78/1.59 => ( func @ A @ B
% 2.78/1.59 @ ( dpsetconstr @ A @ B
% 2.78/1.59 @ ^ [Xx: $i,Xy: $i] :
% 2.78/1.59 ( ( Xf @ Xx )
% 2.78/1.59 = Xy ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(lam_type,type,
% 2.78/1.59 lam: $i > $i > ( $i > $i ) > $i ).
% 2.78/1.59
% 2.78/1.59 thf(lamp_type,type,
% 2.78/1.59 lamp: $o ).
% 2.78/1.59
% 2.78/1.59 thf(lamp,definition,
% 2.78/1.59 ( lamp
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.78/1.59 => ( func @ A @ B
% 2.78/1.59 @ ( lam @ A @ B
% 2.78/1.59 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(lam2p_type,type,
% 2.78/1.59 lam2p: $o ).
% 2.78/1.59
% 2.78/1.59 thf(lam2p,definition,
% 2.78/1.59 ( lam2p
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.78/1.59 => ( in
% 2.78/1.59 @ ( lam @ A @ B
% 2.78/1.59 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.78/1.59 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(brelnall1_type,type,
% 2.78/1.59 brelnall1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(brelnall1,definition,
% 2.78/1.59 ( brelnall1
% 2.78/1.59 = ( ! [A: $i,B: $i,R: $i] :
% 2.78/1.59 ( ( breln @ A @ B @ R )
% 2.78/1.59 => ! [Xphi: $i > $o] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.78/1.59 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ R )
% 2.78/1.59 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(brelnall2_type,type,
% 2.78/1.59 brelnall2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(brelnall2,definition,
% 2.78/1.59 ( brelnall2
% 2.78/1.59 = ( ! [A: $i,B: $i,R: $i] :
% 2.78/1.59 ( ( breln @ A @ B @ R )
% 2.78/1.59 => ! [Xphi: $i > $o] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.78/1.59 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ R )
% 2.78/1.59 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ex1E2_type,type,
% 2.78/1.59 ex1E2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ex1E2,definition,
% 2.78/1.59 ( ex1E2
% 2.78/1.59 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.59 ( ( ex1 @ A
% 2.78/1.59 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ A )
% 2.78/1.59 => ( ( Xphi @ Xx )
% 2.78/1.59 => ( ( Xphi @ Xy )
% 2.78/1.59 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(funcGraphProp1_type,type,
% 2.78/1.59 funcGraphProp1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcGraphProp1,definition,
% 2.78/1.59 ( funcGraphProp1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(funcGraphProp3_type,type,
% 2.78/1.59 funcGraphProp3: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcGraphProp3,definition,
% 2.78/1.59 ( funcGraphProp3
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(funcGraphProp2_type,type,
% 2.78/1.59 funcGraphProp2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcGraphProp2,definition,
% 2.78/1.59 ( funcGraphProp2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.78/1.59 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.78/1.59 = Xy ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(funcextLem_type,type,
% 2.78/1.59 funcextLem: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcextLem,definition,
% 2.78/1.59 ( funcextLem
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ! [Xg: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xg )
% 2.78/1.59 => ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.78/1.59 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(funcGraphProp4_type,type,
% 2.78/1.59 funcGraphProp4: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcGraphProp4,definition,
% 2.78/1.59 ( funcGraphProp4
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.78/1.59 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.78/1.59 = Xy ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(subbreln_type,type,
% 2.78/1.59 subbreln: $o ).
% 2.78/1.59
% 2.78/1.59 thf(subbreln,definition,
% 2.78/1.59 ( subbreln
% 2.78/1.59 = ( ! [A: $i,B: $i,R: $i] :
% 2.78/1.59 ( ( breln @ A @ B @ R )
% 2.78/1.59 => ! [S: $i] :
% 2.78/1.59 ( ( breln @ A @ B @ S )
% 2.78/1.59 => ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.78/1.59 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(eqbreln_type,type,
% 2.78/1.59 eqbreln: $o ).
% 2.78/1.59
% 2.78/1.59 thf(eqbreln,definition,
% 2.78/1.59 ( eqbreln
% 2.78/1.59 = ( ! [A: $i,B: $i,R: $i] :
% 2.78/1.59 ( ( breln @ A @ B @ R )
% 2.78/1.59 => ! [S: $i] :
% 2.78/1.59 ( ( breln @ A @ B @ S )
% 2.78/1.59 => ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.78/1.59 => ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ! [Xy: $i] :
% 2.78/1.59 ( ( in @ Xy @ B )
% 2.78/1.59 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.78/1.59 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.78/1.59 => ( R = S ) ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(funcext_type,type,
% 2.78/1.59 funcext: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcext,definition,
% 2.78/1.59 ( funcext
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ! [Xg: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xg )
% 2.78/1.59 => ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.78/1.59 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.78/1.59 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(funcext2_type,type,
% 2.78/1.59 funcext2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(funcext2,definition,
% 2.78/1.59 ( funcext2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.78/1.59 => ! [Xg: $i] :
% 2.78/1.59 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.78/1.59 => ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.78/1.59 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.78/1.59 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ap2apEq1_type,type,
% 2.78/1.59 ap2apEq1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ap2apEq1,definition,
% 2.78/1.59 ( ap2apEq1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.78/1.59 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(ap2apEq2_type,type,
% 2.78/1.59 ap2apEq2: $o ).
% 2.78/1.59
% 2.78/1.59 thf(ap2apEq2,definition,
% 2.78/1.59 ( ap2apEq2
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.78/1.59 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(beta1_type,type,
% 2.78/1.59 beta1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(beta1,definition,
% 2.78/1.59 ( beta1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.78/1.59 ( ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.78/1.59 => ! [Xx: $i] :
% 2.78/1.59 ( ( in @ Xx @ A )
% 2.78/1.59 => ( ( ap @ A @ B
% 2.78/1.59 @ ( lam @ A @ B
% 2.78/1.59 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.78/1.59 @ Xx )
% 2.78/1.59 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.78/1.59
% 2.78/1.59 thf(eta1_type,type,
% 2.78/1.59 eta1: $o ).
% 2.78/1.59
% 2.78/1.59 thf(eta1,definition,
% 2.78/1.59 ( eta1
% 2.78/1.59 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.59 ( ( func @ A @ B @ Xf )
% 2.78/1.60 => ( ( lam @ A @ B
% 2.78/1.60 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.78/1.60 = Xf ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(lam2lamEq_type,type,
% 2.78/1.60 lam2lamEq: $o ).
% 2.78/1.60
% 2.78/1.60 thf(lam2lamEq,definition,
% 2.78/1.60 ( lam2lamEq
% 2.78/1.60 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.78/1.60 ( ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.78/1.60 => ( ( lam @ A @ B
% 2.78/1.60 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.78/1.60 = ( lam @ A @ B
% 2.78/1.60 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(beta2_type,type,
% 2.78/1.60 beta2: $o ).
% 2.78/1.60
% 2.78/1.60 thf(beta2,definition,
% 2.78/1.60 ( beta2
% 2.78/1.60 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.78/1.60 ( ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( ap @ A @ B
% 2.78/1.60 @ ( lam @ A @ B
% 2.78/1.60 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.78/1.60 @ Xx )
% 2.78/1.60 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(eta2_type,type,
% 2.78/1.60 eta2: $o ).
% 2.78/1.60
% 2.78/1.60 thf(eta2,definition,
% 2.78/1.60 ( eta2
% 2.78/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.78/1.60 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.78/1.60 => ( ( lam @ A @ B
% 2.78/1.60 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.78/1.60 = Xf ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(iffalseProp1_type,type,
% 2.78/1.60 iffalseProp1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(iffalseProp1,definition,
% 2.78/1.60 ( iffalseProp1
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( ~ Xphi
% 2.78/1.60 => ( in @ Xy
% 2.78/1.60 @ ( dsetconstr @ A
% 2.78/1.60 @ ^ [Xz: $i] :
% 2.78/1.60 ( ( Xphi
% 2.78/1.60 & ( Xz = Xx ) )
% 2.78/1.60 | ( ~ Xphi
% 2.78/1.60 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(iffalseProp2_type,type,
% 2.78/1.60 iffalseProp2: $o ).
% 2.78/1.60
% 2.78/1.60 thf(iffalseProp2,definition,
% 2.78/1.60 ( iffalseProp2
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( ~ Xphi
% 2.78/1.60 => ( ( dsetconstr @ A
% 2.78/1.60 @ ^ [Xz: $i] :
% 2.78/1.60 ( ( Xphi
% 2.78/1.60 & ( Xz = Xx ) )
% 2.78/1.60 | ( ~ Xphi
% 2.78/1.60 & ( Xz = Xy ) ) ) )
% 2.78/1.60 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(iftrueProp1_type,type,
% 2.78/1.60 iftrueProp1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(iftrueProp1,definition,
% 2.78/1.60 ( iftrueProp1
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( Xphi
% 2.78/1.60 => ( in @ Xx
% 2.78/1.60 @ ( dsetconstr @ A
% 2.78/1.60 @ ^ [Xz: $i] :
% 2.78/1.60 ( ( Xphi
% 2.78/1.60 & ( Xz = Xx ) )
% 2.78/1.60 | ( ~ Xphi
% 2.78/1.60 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(iftrueProp2_type,type,
% 2.78/1.60 iftrueProp2: $o ).
% 2.78/1.60
% 2.78/1.60 thf(iftrueProp2,definition,
% 2.78/1.60 ( iftrueProp2
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( Xphi
% 2.78/1.60 => ( ( dsetconstr @ A
% 2.78/1.60 @ ^ [Xz: $i] :
% 2.78/1.60 ( ( Xphi
% 2.78/1.60 & ( Xz = Xx ) )
% 2.78/1.60 | ( ~ Xphi
% 2.78/1.60 & ( Xz = Xy ) ) ) )
% 2.78/1.60 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(ifSingleton_type,type,
% 2.78/1.60 ifSingleton: $o ).
% 2.78/1.60
% 2.78/1.60 thf(ifSingleton,definition,
% 2.78/1.60 ( ifSingleton
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( singleton
% 2.78/1.60 @ ( dsetconstr @ A
% 2.78/1.60 @ ^ [Xz: $i] :
% 2.78/1.60 ( ( Xphi
% 2.78/1.60 & ( Xz = Xx ) )
% 2.78/1.60 | ( ~ Xphi
% 2.78/1.60 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(if_type,type,
% 2.78/1.60 if: $i > $o > $i > $i > $i ).
% 2.78/1.60
% 2.78/1.60 thf(ifp_type,type,
% 2.78/1.60 ifp: $o ).
% 2.78/1.60
% 2.78/1.60 thf(ifp,definition,
% 2.78/1.60 ( ifp
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(theeq_type,type,
% 2.78/1.60 theeq: $o ).
% 2.78/1.60
% 2.78/1.60 thf(theeq,definition,
% 2.78/1.60 ( theeq
% 2.78/1.60 = ( ! [X: $i] :
% 2.78/1.60 ( ( singleton @ X )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ X )
% 2.78/1.60 => ( ( setunion @ X )
% 2.78/1.60 = Xx ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(iftrue_type,type,
% 2.78/1.60 iftrue: $o ).
% 2.78/1.60
% 2.78/1.60 thf(iftrue,definition,
% 2.78/1.60 ( iftrue
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( Xphi
% 2.78/1.60 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.78/1.60 = Xx ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(iffalse_type,type,
% 2.78/1.60 iffalse: $o ).
% 2.78/1.60
% 2.78/1.60 thf(iffalse,definition,
% 2.78/1.60 ( iffalse
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( ~ Xphi
% 2.78/1.60 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.78/1.60 = Xy ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(iftrueorfalse_type,type,
% 2.78/1.60 iftrueorfalse: $o ).
% 2.78/1.60
% 2.78/1.60 thf(iftrueorfalse,definition,
% 2.78/1.60 ( iftrueorfalse
% 2.78/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ! [Xy: $i] :
% 2.78/1.60 ( ( in @ Xy @ A )
% 2.78/1.60 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(binintersectT_lem_type,type,
% 2.78/1.60 binintersectT_lem: $o ).
% 2.78/1.60
% 2.78/1.60 thf(binintersectT_lem,definition,
% 2.78/1.60 ( binintersectT_lem
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(binunionT_lem_type,type,
% 2.78/1.60 binunionT_lem: $o ).
% 2.78/1.60
% 2.78/1.60 thf(binunionT_lem,definition,
% 2.78/1.60 ( binunionT_lem
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(powersetT_lem_type,type,
% 2.78/1.60 powersetT_lem: $o ).
% 2.78/1.60
% 2.78/1.60 thf(powersetT_lem,definition,
% 2.78/1.60 ( powersetT_lem
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(setminusT_lem_type,type,
% 2.78/1.60 setminusT_lem: $o ).
% 2.78/1.60
% 2.78/1.60 thf(setminusT_lem,definition,
% 2.78/1.60 ( setminusT_lem
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(complementT_lem_type,type,
% 2.78/1.60 complementT_lem: $o ).
% 2.78/1.60
% 2.78/1.60 thf(complementT_lem,definition,
% 2.78/1.60 ( complementT_lem
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(setextT_type,type,
% 2.78/1.60 setextT: $o ).
% 2.78/1.60
% 2.78/1.60 thf(setextT,definition,
% 2.78/1.60 ( setextT
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ( ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( in @ Xx @ X )
% 2.78/1.60 => ( in @ Xx @ Y ) ) )
% 2.78/1.60 => ( ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( in @ Xx @ Y )
% 2.78/1.60 => ( in @ Xx @ X ) ) )
% 2.78/1.60 => ( X = Y ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(subsetTI_type,type,
% 2.78/1.60 subsetTI: $o ).
% 2.78/1.60
% 2.78/1.60 thf(subsetTI,definition,
% 2.78/1.60 ( subsetTI
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ( ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( in @ Xx @ X )
% 2.78/1.60 => ( in @ Xx @ Y ) ) )
% 2.78/1.60 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(powersetTI1_type,type,
% 2.78/1.60 powersetTI1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(powersetTI1,definition,
% 2.78/1.60 ( powersetTI1
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ( ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( in @ Xx @ X )
% 2.78/1.60 => ( in @ Xx @ Y ) ) )
% 2.78/1.60 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(powersetTE1_type,type,
% 2.78/1.60 powersetTE1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(powersetTE1,definition,
% 2.78/1.60 ( powersetTE1
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( in @ X @ ( powerset @ Y ) )
% 2.78/1.60 => ( ( in @ Xx @ X )
% 2.78/1.60 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(complementTI1_type,type,
% 2.78/1.60 complementTI1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(complementTI1,definition,
% 2.78/1.60 ( complementTI1
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( in @ Xx @ X )
% 2.78/1.60 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(complementTE1_type,type,
% 2.78/1.60 complementTE1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(complementTE1,definition,
% 2.78/1.60 ( complementTE1
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.78/1.60 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(binintersectTELcontra_type,type,
% 2.78/1.60 binintersectTELcontra: $o ).
% 2.78/1.60
% 2.78/1.60 thf(binintersectTELcontra,definition,
% 2.78/1.60 ( binintersectTELcontra
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ~ ( in @ Xx @ X )
% 2.78/1.60 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(binintersectTERcontra_type,type,
% 2.78/1.60 binintersectTERcontra: $o ).
% 2.78/1.60
% 2.78/1.60 thf(binintersectTERcontra,definition,
% 2.78/1.60 ( binintersectTERcontra
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ~ ( in @ Xx @ Y )
% 2.78/1.60 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(contrasubsetT_type,type,
% 2.78/1.60 contrasubsetT: $o ).
% 2.78/1.60
% 2.78/1.60 thf(contrasubsetT,definition,
% 2.78/1.60 ( contrasubsetT
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.78/1.60 => ( ( in @ Xx @ Y )
% 2.78/1.60 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(contrasubsetT1_type,type,
% 2.78/1.60 contrasubsetT1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(contrasubsetT1,definition,
% 2.78/1.60 ( contrasubsetT1
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( subset @ X @ Y )
% 2.78/1.60 => ( ~ ( in @ Xx @ Y )
% 2.78/1.60 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(contrasubsetT2_type,type,
% 2.78/1.60 contrasubsetT2: $o ).
% 2.78/1.60
% 2.78/1.60 thf(contrasubsetT2,definition,
% 2.78/1.60 ( contrasubsetT2
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ( ( subset @ X @ Y )
% 2.78/1.60 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(contrasubsetT3_type,type,
% 2.78/1.60 contrasubsetT3: $o ).
% 2.78/1.60
% 2.78/1.60 thf(contrasubsetT3,definition,
% 2.78/1.60 ( contrasubsetT3
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Y: $i] :
% 2.78/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.78/1.60 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.78/1.60 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementI1_type,type,
% 2.78/1.60 doubleComplementI1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementI1,definition,
% 2.78/1.60 ( doubleComplementI1
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( in @ Xx @ X )
% 2.78/1.60 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementE1_type,type,
% 2.78/1.60 doubleComplementE1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementE1,definition,
% 2.78/1.60 ( doubleComplementE1
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ! [Xx: $i] :
% 2.78/1.60 ( ( in @ Xx @ A )
% 2.78/1.60 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.78/1.60 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementSub1_type,type,
% 2.78/1.60 doubleComplementSub1: $o ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementSub1,definition,
% 2.78/1.60 ( doubleComplementSub1
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementSub2_type,type,
% 2.78/1.60 doubleComplementSub2: $o ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementSub2,definition,
% 2.78/1.60 ( doubleComplementSub2
% 2.78/1.60 = ( ! [A: $i,X: $i] :
% 2.78/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.78/1.60 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.78/1.60
% 2.78/1.60 thf(doubleComplementEq_type,type,
% 2.93/1.60 doubleComplementEq: $o ).
% 2.93/1.60
% 2.93/1.60 thf(doubleComplementEq,definition,
% 2.93/1.60 ( doubleComplementEq
% 2.93/1.60 = ( ! [A: $i,X: $i] :
% 2.93/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.60 => ( X
% 2.93/1.60 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.93/1.60
% 2.93/1.60 thf(complementTnotintersectT_type,type,
% 2.93/1.60 complementTnotintersectT: $o ).
% 2.93/1.60
% 2.93/1.60 thf(complementTnotintersectT,definition,
% 2.93/1.60 ( complementTnotintersectT
% 2.93/1.60 = ( ! [A: $i,X: $i] :
% 2.93/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.60 => ! [Y: $i] :
% 2.93/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.60 => ! [Xx: $i] :
% 2.93/1.60 ( ( in @ Xx @ A )
% 2.93/1.60 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.60 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.60
% 2.93/1.60 thf(complementImpComplementIntersect_type,type,
% 2.93/1.60 complementImpComplementIntersect: $o ).
% 2.93/1.60
% 2.93/1.60 thf(complementImpComplementIntersect,definition,
% 2.93/1.60 ( complementImpComplementIntersect
% 2.93/1.60 = ( ! [A: $i,X: $i] :
% 2.93/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.60 => ! [Y: $i] :
% 2.93/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.60 => ! [Xx: $i] :
% 2.93/1.60 ( ( in @ Xx @ A )
% 2.93/1.60 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.60 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.60
% 2.93/1.60 thf(complementSubsetComplementIntersect_type,type,
% 2.93/1.60 complementSubsetComplementIntersect: $o ).
% 2.93/1.60
% 2.93/1.60 thf(complementSubsetComplementIntersect,definition,
% 2.93/1.60 ( complementSubsetComplementIntersect
% 2.93/1.60 = ( ! [A: $i,X: $i] :
% 2.93/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.60 => ! [Y: $i] :
% 2.93/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.60 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.93/1.60
% 2.93/1.60 thf(complementInPowersetComplementIntersect_type,type,
% 2.93/1.60 complementInPowersetComplementIntersect: $o ).
% 2.93/1.60
% 2.93/1.60 thf(complementInPowersetComplementIntersect,definition,
% 2.93/1.60 ( complementInPowersetComplementIntersect
% 2.93/1.60 = ( ! [A: $i,X: $i] :
% 2.93/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.60 => ! [Y: $i] :
% 2.93/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.60 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.60
% 2.93/1.60 thf(contraSubsetComplement_type,type,
% 2.93/1.60 contraSubsetComplement: $o ).
% 2.93/1.60
% 2.93/1.60 thf(contraSubsetComplement,definition,
% 2.93/1.60 ( contraSubsetComplement
% 2.93/1.60 = ( ! [A: $i,X: $i] :
% 2.93/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.60 => ! [Y: $i] :
% 2.93/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.60 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.60 => ! [Xx: $i] :
% 2.93/1.60 ( ( in @ Xx @ A )
% 2.93/1.60 => ( ( in @ Xx @ Y )
% 2.93/1.60 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.93/1.60
% 2.93/1.60 thf(complementTcontraSubset_type,type,
% 2.93/1.60 complementTcontraSubset: $o ).
% 2.93/1.60
% 2.93/1.60 thf(complementTcontraSubset,definition,
% 2.93/1.60 ( complementTcontraSubset
% 2.93/1.60 = ( ! [A: $i,X: $i] :
% 2.93/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.60 => ! [Y: $i] :
% 2.93/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.60 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.60 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.60
% 2.93/1.60 thf(binunionTILcontra,conjecture,
% 2.93/1.60 ( setextAx
% 2.93/1.60 => ( emptysetAx
% 2.93/1.60 => ( setadjoinAx
% 2.93/1.60 => ( powersetAx
% 2.93/1.60 => ( setunionAx
% 2.93/1.60 => ( omega0Ax
% 2.93/1.60 => ( omegaSAx
% 2.93/1.60 => ( omegaIndAx
% 2.93/1.60 => ( replAx
% 2.93/1.60 => ( foundationAx
% 2.93/1.60 => ( wellorderingAx
% 2.93/1.60 => ( descrp
% 2.93/1.60 => ( dsetconstrI
% 2.93/1.60 => ( dsetconstrEL
% 2.93/1.60 => ( dsetconstrER
% 2.93/1.60 => ( exuE1
% 2.93/1.60 => ( prop2setE
% 2.93/1.60 => ( emptysetE
% 2.93/1.60 => ( emptysetimpfalse
% 2.93/1.60 => ( notinemptyset
% 2.93/1.60 => ( exuE3e
% 2.93/1.60 => ( setext
% 2.93/1.60 => ( emptyI
% 2.93/1.60 => ( noeltsimpempty
% 2.93/1.60 => ( setbeta
% 2.93/1.60 => ( nonemptyE1
% 2.93/1.60 => ( nonemptyI
% 2.93/1.60 => ( nonemptyI1
% 2.93/1.60 => ( setadjoinIL
% 2.93/1.60 => ( emptyinunitempty
% 2.93/1.60 => ( setadjoinIR
% 2.93/1.60 => ( setadjoinE
% 2.93/1.60 => ( setadjoinOr
% 2.93/1.60 => ( setoftrueEq
% 2.93/1.60 => ( powersetI
% 2.93/1.60 => ( emptyinPowerset
% 2.93/1.60 => ( emptyInPowerset
% 2.93/1.60 => ( powersetE
% 2.93/1.60 => ( setunionI
% 2.93/1.60 => ( setunionE
% 2.93/1.60 => ( subPowSU
% 2.93/1.60 => ( exuE2
% 2.93/1.60 => ( nonemptyImpWitness
% 2.93/1.60 => ( uniqinunit
% 2.93/1.60 => ( notinsingleton
% 2.93/1.60 => ( eqinunit
% 2.93/1.60 => ( singletonsswitch
% 2.93/1.60 => ( upairsetE
% 2.93/1.60 => ( upairsetIL
% 2.93/1.60 => ( upairsetIR
% 2.93/1.60 => ( emptyE1
% 2.93/1.60 => ( vacuousDall
% 2.93/1.60 => ( quantDeMorgan1
% 2.93/1.60 => ( quantDeMorgan2
% 2.93/1.60 => ( quantDeMorgan3
% 2.93/1.60 => ( quantDeMorgan4
% 2.93/1.60 => ( prop2setI
% 2.93/1.60 => ( prop2set2propI
% 2.93/1.60 => ( notdexE
% 2.93/1.60 => ( notdallE
% 2.93/1.60 => ( exuI1
% 2.93/1.60 => ( exuI3
% 2.93/1.60 => ( exuI2
% 2.93/1.60 => ( inCongP
% 2.93/1.60 => ( in__Cong
% 2.93/1.60 => ( exuE3u
% 2.93/1.60 => ( exu__Cong
% 2.93/1.60 => ( emptyset__Cong
% 2.93/1.60 => ( setadjoin__Cong
% 2.93/1.60 => ( powerset__Cong
% 2.93/1.60 => ( setunion__Cong
% 2.93/1.60 => ( omega__Cong
% 2.93/1.60 => ( exuEu
% 2.93/1.60 => ( descr__Cong
% 2.93/1.60 => ( dsetconstr__Cong
% 2.93/1.60 => ( subsetI1
% 2.93/1.60 => ( eqimpsubset2
% 2.93/1.60 => ( eqimpsubset1
% 2.93/1.60 => ( subsetI2
% 2.93/1.60 => ( emptysetsubset
% 2.93/1.60 => ( subsetE
% 2.93/1.60 => ( subsetE2
% 2.93/1.60 => ( notsubsetI
% 2.93/1.60 => ( notequalI1
% 2.93/1.60 => ( notequalI2
% 2.93/1.60 => ( subsetRefl
% 2.93/1.60 => ( subsetTrans
% 2.93/1.60 => ( setadjoinSub
% 2.93/1.60 => ( setadjoinSub2
% 2.93/1.60 => ( subset2powerset
% 2.93/1.60 => ( setextsub
% 2.93/1.60 => ( subsetemptysetimpeq
% 2.93/1.60 => ( powersetI1
% 2.93/1.60 => ( powersetE1
% 2.93/1.60 => ( inPowerset
% 2.93/1.60 => ( powersetsubset
% 2.93/1.60 => ( sepInPowerset
% 2.93/1.60 => ( sepSubset
% 2.93/1.60 => ( binunionIL
% 2.93/1.60 => ( upairset2IR
% 2.93/1.60 => ( binunionIR
% 2.93/1.60 => ( binunionEcases
% 2.93/1.60 => ( binunionE
% 2.93/1.60 => ( binunionLsub
% 2.93/1.60 => ( binunionRsub
% 2.93/1.60 => ( binintersectI
% 2.93/1.60 => ( binintersectSubset5
% 2.93/1.60 => ( binintersectEL
% 2.93/1.60 => ( binintersectLsub
% 2.93/1.60 => ( binintersectSubset2
% 2.93/1.60 => ( binintersectSubset3
% 2.93/1.60 => ( binintersectER
% 2.93/1.60 => ( disjointsetsI1
% 2.93/1.60 => ( binintersectRsub
% 2.93/1.60 => ( binintersectSubset4
% 2.93/1.60 => ( binintersectSubset1
% 2.93/1.60 => ( bs114d
% 2.93/1.60 => ( setminusI
% 2.93/1.60 => ( setminusEL
% 2.93/1.60 => ( setminusER
% 2.93/1.60 => ( setminusSubset2
% 2.93/1.60 => ( setminusERneg
% 2.93/1.60 => ( setminusELneg
% 2.93/1.60 => ( setminusILneg
% 2.93/1.60 => ( setminusIRneg
% 2.93/1.60 => ( setminusLsub
% 2.93/1.60 => ( setminusSubset1
% 2.93/1.60 => ( symdiffE
% 2.93/1.60 => ( symdiffI1
% 2.93/1.60 => ( symdiffI2
% 2.93/1.60 => ( symdiffIneg1
% 2.93/1.60 => ( symdiffIneg2
% 2.93/1.60 => ( secondinupair
% 2.93/1.60 => ( setukpairIL
% 2.93/1.60 => ( setukpairIR
% 2.93/1.60 => ( kpairiskpair
% 2.93/1.60 => ( kpairp
% 2.93/1.60 => ( singletonsubset
% 2.93/1.60 => ( singletoninpowerset
% 2.93/1.60 => ( singletoninpowunion
% 2.93/1.60 => ( upairset2E
% 2.93/1.60 => ( upairsubunion
% 2.93/1.60 => ( upairinpowunion
% 2.93/1.60 => ( ubforcartprodlem1
% 2.93/1.60 => ( ubforcartprodlem2
% 2.93/1.60 => ( ubforcartprodlem3
% 2.93/1.60 => ( cartprodpairin
% 2.93/1.60 => ( cartprodmempair1
% 2.93/1.60 => ( cartprodmempair
% 2.93/1.60 => ( setunionE2
% 2.93/1.60 => ( setunionsingleton1
% 2.93/1.60 => ( setunionsingleton2
% 2.93/1.60 => ( setunionsingleton
% 2.93/1.60 => ( singletonprop
% 2.93/1.60 => ( ex1E1
% 2.93/1.60 => ( ex1I
% 2.93/1.60 => ( ex1I2
% 2.93/1.60 => ( singletonsuniq
% 2.93/1.60 => ( setukpairinjL1
% 2.93/1.60 => ( kfstsingleton
% 2.93/1.60 => ( theprop
% 2.93/1.60 => ( kfstpairEq
% 2.93/1.60 => ( cartprodfstin
% 2.93/1.60 => ( setukpairinjL2
% 2.93/1.60 => ( setukpairinjL
% 2.93/1.60 => ( setukpairinjR11
% 2.93/1.60 => ( setukpairinjR12
% 2.93/1.60 => ( setukpairinjR1
% 2.93/1.60 => ( upairequniteq
% 2.93/1.60 => ( setukpairinjR2
% 2.93/1.60 => ( setukpairinjR
% 2.93/1.60 => ( ksndsingleton
% 2.93/1.60 => ( ksndpairEq
% 2.93/1.60 => ( kpairsurjEq
% 2.93/1.60 => ( cartprodsndin
% 2.93/1.60 => ( cartprodpairmemEL
% 2.93/1.60 => ( cartprodpairmemER
% 2.93/1.60 => ( cartprodmempaircEq
% 2.93/1.60 => ( cartprodfstpairEq
% 2.93/1.60 => ( cartprodsndpairEq
% 2.93/1.60 => ( cartprodpairsurjEq
% 2.93/1.60 => ( dpsetconstrI
% 2.93/1.60 => ( dpsetconstrSub
% 2.93/1.60 => ( setOfPairsIsBReln
% 2.93/1.60 => ( dpsetconstrERa
% 2.93/1.60 => ( dpsetconstrEL1
% 2.93/1.60 => ( dpsetconstrEL2
% 2.93/1.60 => ( dpsetconstrER
% 2.93/1.60 => ( funcImageSingleton
% 2.93/1.60 => ( apProp
% 2.93/1.60 => ( app
% 2.93/1.60 => ( infuncsetfunc
% 2.93/1.60 => ( ap2p
% 2.93/1.60 => ( funcinfuncset
% 2.93/1.60 => ( lamProp
% 2.93/1.60 => ( lamp
% 2.93/1.60 => ( lam2p
% 2.93/1.60 => ( brelnall1
% 2.93/1.60 => ( brelnall2
% 2.93/1.60 => ( ex1E2
% 2.93/1.60 => ( funcGraphProp1
% 2.93/1.60 => ( funcGraphProp3
% 2.93/1.60 => ( funcGraphProp2
% 2.93/1.60 => ( funcextLem
% 2.93/1.60 => ( funcGraphProp4
% 2.93/1.60 => ( subbreln
% 2.93/1.60 => ( eqbreln
% 2.93/1.60 => ( funcext
% 2.93/1.60 => ( funcext2
% 2.93/1.60 => ( ap2apEq1
% 2.93/1.60 => ( ap2apEq2
% 2.93/1.60 => ( beta1
% 2.93/1.60 => ( eta1
% 2.93/1.60 => ( lam2lamEq
% 2.93/1.60 => ( beta2
% 2.93/1.60 => ( eta2
% 2.93/1.60 => ( iffalseProp1
% 2.93/1.60 => ( iffalseProp2
% 2.93/1.60 => ( iftrueProp1
% 2.93/1.60 => ( iftrueProp2
% 2.93/1.60 => ( ifSingleton
% 2.93/1.60 => ( ifp
% 2.93/1.60 => ( theeq
% 2.93/1.60 => ( iftrue
% 2.93/1.60 => ( iffalse
% 2.93/1.60 => ( iftrueorfalse
% 2.93/1.60 => ( binintersectT_lem
% 2.93/1.60 => ( binunionT_lem
% 2.93/1.60 => ( powersetT_lem
% 2.93/1.60 => ( setminusT_lem
% 2.93/1.60 => ( complementT_lem
% 2.93/1.60 => ( setextT
% 2.93/1.60 => ( subsetTI
% 2.93/1.60 => ( powersetTI1
% 2.93/1.60 => ( powersetTE1
% 2.93/1.60 => ( complementTI1
% 2.93/1.60 => ( complementTE1
% 2.93/1.60 => ( binintersectTELcontra
% 2.93/1.60 => ( binintersectTERcontra
% 2.93/1.60 => ( contrasubsetT
% 2.93/1.60 => ( contrasubsetT1
% 2.93/1.60 => ( contrasubsetT2
% 2.93/1.60 => ( contrasubsetT3
% 2.93/1.60 => ( doubleComplementI1
% 2.93/1.60 => ( doubleComplementE1
% 2.93/1.60 => ( doubleComplementSub1
% 2.93/1.60 => ( doubleComplementSub2
% 2.93/1.60 => ( doubleComplementEq
% 2.93/1.60 => ( complementTnotintersectT
% 2.93/1.60 => ( complementImpComplementIntersect
% 2.93/1.60 => ( complementSubsetComplementIntersect
% 2.93/1.60 => ( complementInPowersetComplementIntersect
% 2.93/1.60 => ( contraSubsetComplement
% 2.93/1.60 => ( complementTcontraSubset
% 2.93/1.60 => ! [A: $i,X: $i] :
% 2.93/1.60 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.60 => ! [Y: $i] :
% 2.93/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.60 => ! [Xx: $i] :
% 2.93/1.60 ( ( in @ Xx @ A )
% 2.93/1.60 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.93/1.60 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.60 ------------------------
% 2.93/1.62 % (11994)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/183Mi)
% 2.93/1.63 % (11995)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_11747 for (2999ds/4Mi)
% 2.93/1.63 % (11996)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_11747 for (2999ds/27Mi)
% 2.93/1.63 % (11997)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/2Mi)
% 2.93/1.63 % (11999)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_11747 for (2999ds/275Mi)
% 2.93/1.63 % (11998)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_11747 for (2999ds/2Mi)
% 2.93/1.63 % (12000)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/18Mi)
% 2.93/1.63 % (11997)Instruction limit reached!
% 2.93/1.63 % (11997)------------------------------
% 2.93/1.63 % (11997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.63 % (11997)Termination reason: Unknown
% 2.93/1.63 % (11997)Termination phase: shuffling
% 2.93/1.63
% 2.93/1.63 % (11997)Memory used [KB]: 1407
% 2.93/1.63 % (11997)Time elapsed: 0.003 s
% 2.93/1.63 % (11997)Instructions burned: 2 (million)
% 2.93/1.63 % (11997)------------------------------
% 2.93/1.63 % (11997)------------------------------
% 2.93/1.63 % (11998)Instruction limit reached!
% 2.93/1.63 % (11998)------------------------------
% 2.93/1.63 % (11998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.63 % (11998)Termination reason: Unknown
% 2.93/1.63 % (11998)Termination phase: shuffling
% 2.93/1.63
% 2.93/1.63 % (11998)Memory used [KB]: 1407
% 2.93/1.63 % (11998)Time elapsed: 0.003 s
% 2.93/1.63 % (11998)Instructions burned: 2 (million)
% 2.93/1.63 % (11998)------------------------------
% 2.93/1.63 % (11998)------------------------------
% 2.93/1.63 % (11995)Instruction limit reached!
% 2.93/1.63 % (11995)------------------------------
% 2.93/1.63 % (11995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.63 % (11995)Termination reason: Unknown
% 2.93/1.63 % (11995)Termination phase: shuffling
% 2.93/1.63
% 2.93/1.63 % (11995)Memory used [KB]: 1535
% 2.93/1.63 % (11995)Time elapsed: 0.004 s
% 2.93/1.63 % (11995)Instructions burned: 4 (million)
% 2.93/1.63 % (11995)------------------------------
% 2.93/1.63 % (11995)------------------------------
% 2.93/1.64 % (12000)Instruction limit reached!
% 2.93/1.64 % (12000)------------------------------
% 2.93/1.64 % (12000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.64 % (12000)Termination reason: Unknown
% 2.93/1.64 % (12000)Termination phase: shuffling
% 2.93/1.64
% 2.93/1.64 % (12000)Memory used [KB]: 1791
% 2.93/1.64 % (12000)Time elapsed: 0.012 s
% 2.93/1.64 % (12000)Instructions burned: 18 (million)
% 2.93/1.64 % (12000)------------------------------
% 2.93/1.64 % (12000)------------------------------
% 2.93/1.64 % (11996)Instruction limit reached!
% 2.93/1.64 % (11996)------------------------------
% 2.93/1.64 % (11996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.64 % (11996)Termination reason: Unknown
% 2.93/1.64 % (11996)Termination phase: shuffling
% 2.93/1.64
% 2.93/1.64 % (11996)Memory used [KB]: 2046
% 2.93/1.64 % (11996)Time elapsed: 0.016 s
% 2.93/1.64 % (11996)Instructions burned: 28 (million)
% 2.93/1.64 % (11996)------------------------------
% 2.93/1.64 % (11996)------------------------------
% 2.93/1.65 % (12017)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/3Mi)
% 2.93/1.65 % (12018)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_11747 for (2999ds/37Mi)
% 2.93/1.65 % (12019)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_11747 for (2999ds/15Mi)
% 2.93/1.65 % (12017)Instruction limit reached!
% 2.93/1.65 % (12017)------------------------------
% 2.93/1.65 % (12017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.65 % (12017)Termination reason: Unknown
% 2.93/1.65 % (12017)Termination phase: shuffling
% 2.93/1.65
% 2.93/1.65 % (12017)Memory used [KB]: 1535
% 2.93/1.65 % (12017)Time elapsed: 0.004 s
% 2.93/1.65 % (12017)Instructions burned: 3 (million)
% 2.93/1.65 % (12017)------------------------------
% 2.93/1.65 % (12017)------------------------------
% 2.93/1.65 % (12026)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_11747 for (2999ds/3Mi)
% 2.93/1.65 % (12026)Instruction limit reached!
% 2.93/1.65 % (12026)------------------------------
% 2.93/1.65 % (12026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.65 % (12026)Termination reason: Unknown
% 2.93/1.65 % (12026)Termination phase: shuffling
% 2.93/1.65
% 2.93/1.65 % (12026)Memory used [KB]: 1535
% 2.93/1.65 % (12026)Time elapsed: 0.003 s
% 2.93/1.65 % (12026)Instructions burned: 5 (million)
% 2.93/1.65 % (12026)------------------------------
% 2.93/1.65 % (12026)------------------------------
% 2.93/1.66 % (12019)Instruction limit reached!
% 2.93/1.66 % (12019)------------------------------
% 2.93/1.66 % (12019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.66 % (12019)Termination reason: Unknown
% 2.93/1.66 % (12019)Termination phase: shuffling
% 2.93/1.66
% 2.93/1.66 % (12019)Memory used [KB]: 1791
% 2.93/1.66 % (12019)Time elapsed: 0.011 s
% 2.93/1.66 % (12019)Instructions burned: 15 (million)
% 2.93/1.66 % (12019)------------------------------
% 2.93/1.66 % (12019)------------------------------
% 2.93/1.66 % (12029)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_11747 for (2999ds/1041Mi)
% 2.93/1.66 % (12018)Instruction limit reached!
% 2.93/1.66 % (12018)------------------------------
% 2.93/1.66 % (12018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.66 % (12018)Termination reason: Unknown
% 2.93/1.66 % (12018)Termination phase: shuffling
% 2.93/1.66
% 2.93/1.66 % (12018)Memory used [KB]: 2302
% 2.93/1.66 % (12018)Time elapsed: 0.020 s
% 2.93/1.66 % (12018)Instructions burned: 38 (million)
% 2.93/1.66 % (12018)------------------------------
% 2.93/1.66 % (12018)------------------------------
% 2.93/1.67 % (12031)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/16Mi)
% 2.93/1.67 % (12030)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/7Mi)
% 2.93/1.67 % (11994)Instruction limit reached!
% 2.93/1.67 % (11994)------------------------------
% 2.93/1.67 % (11994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.67 % (11994)Termination reason: Unknown
% 2.93/1.67 % (11994)Termination phase: Saturation
% 2.93/1.67
% 2.93/1.67 % (11994)Memory used [KB]: 7675
% 2.93/1.67 % (11994)Time elapsed: 0.050 s
% 2.93/1.67 % (11994)Instructions burned: 184 (million)
% 2.93/1.67 % (11994)------------------------------
% 2.93/1.67 % (11994)------------------------------
% 2.93/1.67 % (12031)Instruction limit reached!
% 2.93/1.67 % (12031)------------------------------
% 2.93/1.67 % (12031)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.67 % (12031)Termination reason: Unknown
% 2.93/1.67 % (12031)Termination phase: shuffling
% 2.93/1.67
% 2.93/1.67 % (12031)Memory used [KB]: 1791
% 2.93/1.67 % (12031)Time elapsed: 0.006 s
% 2.93/1.67 % (12031)Instructions burned: 16 (million)
% 2.93/1.67 % (12031)------------------------------
% 2.93/1.67 % (12031)------------------------------
% 2.93/1.67 % (12030)Instruction limit reached!
% 2.93/1.67 % (12030)------------------------------
% 2.93/1.67 % (12030)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.67 % (12030)Termination reason: Unknown
% 2.93/1.67 % (12030)Termination phase: shuffling
% 2.93/1.67
% 2.93/1.67 % (12030)Memory used [KB]: 1535
% 2.93/1.67 % (12030)Time elapsed: 0.006 s
% 2.93/1.67 % (12030)Instructions burned: 7 (million)
% 2.93/1.67 % (12030)------------------------------
% 2.93/1.67 % (12030)------------------------------
% 2.93/1.67 % (12032)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_11747 for (2999ds/3Mi)
% 2.93/1.68 % (12032)Instruction limit reached!
% 2.93/1.68 % (12032)------------------------------
% 2.93/1.68 % (12032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.68 % (12032)Termination reason: Unknown
% 2.93/1.68 % (12032)Termination phase: shuffling
% 2.93/1.68
% 2.93/1.68 % (12032)Memory used [KB]: 1535
% 2.93/1.68 % (12032)Time elapsed: 0.004 s
% 2.93/1.68 % (12032)Instructions burned: 4 (million)
% 2.93/1.68 % (12032)------------------------------
% 2.93/1.68 % (12032)------------------------------
% 2.93/1.68 % (12045)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/7Mi)
% 2.93/1.68 % (12043)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_11747 for (2999ds/3Mi)
% 2.93/1.68 % (12045)Instruction limit reached!
% 2.93/1.68 % (12045)------------------------------
% 2.93/1.68 % (12045)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.68 % (12045)Termination reason: Unknown
% 2.93/1.68 % (12045)Termination phase: shuffling
% 2.93/1.68
% 2.93/1.68 % (12045)Memory used [KB]: 1535
% 2.93/1.68 % (12045)Time elapsed: 0.003 s
% 2.93/1.68 % (12045)Instructions burned: 7 (million)
% 2.93/1.68 % (12045)------------------------------
% 2.93/1.68 % (12045)------------------------------
% 2.93/1.68 % (12043)Instruction limit reached!
% 2.93/1.68 % (12043)------------------------------
% 2.93/1.68 % (12043)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.68 % (12043)Termination reason: Unknown
% 2.93/1.68 % (12043)Termination phase: shuffling
% 2.93/1.68
% 2.93/1.68 % (12043)Memory used [KB]: 1535
% 2.93/1.68 % (12043)Time elapsed: 0.003 s
% 2.93/1.68 % (12043)Instructions burned: 4 (million)
% 2.93/1.68 % (12043)------------------------------
% 2.93/1.68 % (12043)------------------------------
% 2.93/1.68 % (12048)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/3Mi)
% 2.93/1.68 % (12048)Instruction limit reached!
% 2.93/1.68 % (12048)------------------------------
% 2.93/1.68 % (12048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.68 % (12048)Termination reason: Unknown
% 2.93/1.68 % (12048)Termination phase: shuffling
% 2.93/1.68
% 2.93/1.68 % (12048)Memory used [KB]: 1535
% 2.93/1.68 % (12048)Time elapsed: 0.002 s
% 2.93/1.68 % (12048)Instructions burned: 4 (million)
% 2.93/1.68 % (12048)------------------------------
% 2.93/1.68 % (12048)------------------------------
% 2.93/1.69 % (12049)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/4Mi)
% 2.93/1.69 % (12053)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/710Mi)
% 2.93/1.69 % (12054)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/6Mi)
% 2.93/1.69 % (12049)Instruction limit reached!
% 2.93/1.69 % (12049)------------------------------
% 2.93/1.69 % (12049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.69 % (12049)Termination reason: Unknown
% 2.93/1.69 % (12049)Termination phase: shuffling
% 2.93/1.69
% 2.93/1.69 % (12049)Memory used [KB]: 1535
% 2.93/1.69 % (12049)Time elapsed: 0.005 s
% 2.93/1.69 % (12049)Instructions burned: 5 (million)
% 2.93/1.69 % (12049)------------------------------
% 2.93/1.69 % (12049)------------------------------
% 2.93/1.69 % (12055)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_11747 for (2999ds/902Mi)
% 2.93/1.69 % (12054)Instruction limit reached!
% 2.93/1.69 % (12054)------------------------------
% 2.93/1.69 % (12054)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.69 % (12054)Termination reason: Unknown
% 2.93/1.69 % (12054)Termination phase: shuffling
% 2.93/1.69
% 2.93/1.69 % (12054)Memory used [KB]: 1535
% 2.93/1.69 % (12054)Time elapsed: 0.003 s
% 2.93/1.69 % (12054)Instructions burned: 6 (million)
% 2.93/1.69 % (12054)------------------------------
% 2.93/1.69 % (12054)------------------------------
% 2.93/1.69 % (12052)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/18Mi)
% 2.93/1.70 % (12052)Instruction limit reached!
% 2.93/1.70 % (12052)------------------------------
% 2.93/1.70 % (12052)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.70 % (12052)Termination reason: Unknown
% 2.93/1.70 % (12052)Termination phase: shuffling
% 2.93/1.70
% 2.93/1.70 % (12052)Memory used [KB]: 1791
% 2.93/1.70 % (12052)Time elapsed: 0.009 s
% 2.93/1.70 % (12052)Instructions burned: 20 (million)
% 2.93/1.70 % (12052)------------------------------
% 2.93/1.70 % (12052)------------------------------
% 2.93/1.70 % (12061)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/5Mi)
% 2.93/1.70 % (12061)Instruction limit reached!
% 2.93/1.70 % (12061)------------------------------
% 2.93/1.70 % (12061)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.70 % (12061)Termination reason: Unknown
% 2.93/1.70 % (12061)Termination phase: shuffling
% 2.93/1.70
% 2.93/1.70 % (12061)Memory used [KB]: 1535
% 2.93/1.70 % (12061)Time elapsed: 0.003 s
% 2.93/1.70 % (12061)Instructions burned: 6 (million)
% 2.93/1.70 % (12061)------------------------------
% 2.93/1.70 % (12061)------------------------------
% 2.93/1.71 % (12060)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_11747 for (2999ds/21Mi)
% 2.93/1.71 % (12067)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_11747 for (2999ds/6Mi)
% 2.93/1.71 % (12070)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_11747 for (2999ds/377Mi)
% 2.93/1.71 % (12067)Instruction limit reached!
% 2.93/1.71 % (12067)------------------------------
% 2.93/1.71 % (12067)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.71 % (12067)Termination reason: Unknown
% 2.93/1.71 % (12067)Termination phase: shuffling
% 2.93/1.71
% 2.93/1.71 % (12067)Memory used [KB]: 1535
% 2.93/1.71 % (12067)Time elapsed: 0.004 s
% 2.93/1.71 % (12067)Instructions burned: 8 (million)
% 2.93/1.71 % (12067)------------------------------
% 2.93/1.71 % (12067)------------------------------
% 2.93/1.72 % (12060)Instruction limit reached!
% 2.93/1.72 % (12060)------------------------------
% 2.93/1.72 % (12060)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.72 % (12060)Termination reason: Unknown
% 2.93/1.72 % (12060)Termination phase: shuffling
% 2.93/1.72
% 2.93/1.72 % (12060)Memory used [KB]: 1918
% 2.93/1.72 % (12060)Time elapsed: 0.013 s
% 2.93/1.72 % (12060)Instructions burned: 22 (million)
% 2.93/1.72 % (12060)------------------------------
% 2.93/1.72 % (12060)------------------------------
% 2.93/1.72 % (12075)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_11747 for (2999ds/779Mi)
% 2.93/1.74 % (12077)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_11747 for (2998ds/19Mi)
% 2.93/1.75 % (12077)Instruction limit reached!
% 2.93/1.75 % (12077)------------------------------
% 2.93/1.75 % (12077)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.75 % (12077)Termination reason: Unknown
% 2.93/1.75 % (12077)Termination phase: shuffling
% 2.93/1.75
% 2.93/1.75 % (12077)Memory used [KB]: 1791
% 2.93/1.75 % (12077)Time elapsed: 0.010 s
% 2.93/1.75 % (12077)Instructions burned: 20 (million)
% 2.93/1.75 % (12077)------------------------------
% 2.93/1.75 % (12077)------------------------------
% 2.93/1.75 % (11999)Instruction limit reached!
% 2.93/1.75 % (11999)------------------------------
% 2.93/1.75 % (11999)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.75 % (11999)Termination reason: Unknown
% 2.93/1.75 % (11999)Termination phase: Saturation
% 2.93/1.75
% 2.93/1.75 % (11999)Memory used [KB]: 9594
% 2.93/1.75 % (11999)Time elapsed: 0.127 s
% 2.93/1.75 % (11999)Instructions burned: 275 (million)
% 2.93/1.75 % (11999)------------------------------
% 2.93/1.75 % (11999)------------------------------
% 2.93/1.76 % (12087)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_11747 for (2998ds/879Mi)
% 2.93/1.76 % (12091)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_11747 for (2998ds/17Mi)
% 2.93/1.77 % (12091)Instruction limit reached!
% 2.93/1.77 % (12091)------------------------------
% 2.93/1.77 % (12091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.77 % (12091)Termination reason: Unknown
% 2.93/1.77 % (12091)Termination phase: shuffling
% 2.93/1.77
% 2.93/1.77 % (12091)Memory used [KB]: 1791
% 2.93/1.77 % (12091)Time elapsed: 0.006 s
% 2.93/1.77 % (12091)Instructions burned: 17 (million)
% 2.93/1.77 % (12091)------------------------------
% 2.93/1.77 % (12091)------------------------------
% 2.93/1.78 % (12097)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_11747 for (2998ds/3Mi)
% 2.93/1.78 % (12097)Instruction limit reached!
% 2.93/1.78 % (12097)------------------------------
% 2.93/1.78 % (12097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.78 % (12097)Termination reason: Unknown
% 2.93/1.78 % (12097)Termination phase: shuffling
% 2.93/1.78
% 2.93/1.78 % (12097)Memory used [KB]: 1535
% 2.93/1.78 % (12097)Time elapsed: 0.002 s
% 2.93/1.78 % (12097)Instructions burned: 4 (million)
% 2.93/1.78 % (12097)------------------------------
% 2.93/1.78 % (12097)------------------------------
% 2.93/1.79 % (12103)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_11747 for (2998ds/30Mi)
% 2.93/1.80 % (12103)Instruction limit reached!
% 2.93/1.80 % (12103)------------------------------
% 2.93/1.80 % (12103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.80 % (12103)Termination reason: Unknown
% 2.93/1.80 % (12103)Termination phase: shuffling
% 2.93/1.80
% 2.93/1.80 % (12103)Memory used [KB]: 2046
% 2.93/1.80 % (12103)Time elapsed: 0.010 s
% 2.93/1.80 % (12103)Instructions burned: 31 (million)
% 2.93/1.80 % (12103)------------------------------
% 2.93/1.80 % (12103)------------------------------
% 2.93/1.80 % (12055)First to succeed.
% 2.93/1.81 % (12055)Refutation found. Thanks to Tanya!
% 2.93/1.81 % SZS status Theorem for DTF2THF_11747
% 2.93/1.81 % SZS output start Proof for DTF2THF_11747
% 2.93/1.81 thf(func_def_0, type, in: $i > $i > $o).
% 2.93/1.81 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.93/1.81 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.93/1.81 thf(func_def_8, type, powerset: $i > $i).
% 2.93/1.81 thf(func_def_10, type, setunion: $i > $i).
% 2.93/1.81 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.93/1.81 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_26, type, prop2set: $o > $i).
% 2.93/1.81 thf(func_def_36, type, nonempty: $i > $o).
% 2.93/1.81 thf(func_def_69, type, set2prop: $i > $o).
% 2.93/1.81 thf(func_def_88, type, subset: $i > $i > $o).
% 2.93/1.81 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.93/1.81 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.93/1.81 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.93/1.81 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.93/1.81 thf(func_def_135, type, regular: $i > $o).
% 2.93/1.81 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.93/1.81 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.93/1.81 thf(func_def_153, type, iskpair: $i > $o).
% 2.93/1.81 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.93/1.81 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.93/1.81 thf(func_def_177, type, singleton: $i > $o).
% 2.93/1.81 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.93/1.81 thf(func_def_184, type, atmost1p: $i > $o).
% 2.93/1.81 thf(func_def_185, type, atleast2p: $i > $o).
% 2.93/1.81 thf(func_def_186, type, atmost2p: $i > $o).
% 2.93/1.81 thf(func_def_187, type, upairsetp: $i > $o).
% 2.93/1.81 thf(func_def_191, type, kfst: $i > $i).
% 2.93/1.81 thf(func_def_203, type, ksnd: $i > $i).
% 2.93/1.81 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.93/1.81 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.93/1.81 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.93/1.81 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.93/1.81 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.93/1.81 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.93/1.81 thf(func_def_306, type, sP0: $i > $o > $i > $i > $o).
% 2.93/1.81 thf(func_def_307, type, sP1: $i > $i > $o).
% 2.93/1.81 thf(func_def_308, type, sP2: $i > $o).
% 2.93/1.81 thf(func_def_309, type, sP3: $i > $i > $o).
% 2.93/1.81 thf(func_def_310, type, sP4: $i > $i > $o).
% 2.93/1.81 thf(func_def_311, type, sP5: $i > $i > $i > $i > $o).
% 2.93/1.81 thf(func_def_317, type, sP11: $i > $i > $i > $o).
% 2.93/1.81 thf(func_def_360, type, sK54: $i > $o).
% 2.93/1.81 thf(func_def_362, type, sK56: ($i > $o) > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_363, type, sK57: ($i > $o) > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_370, type, sK64: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_371, type, sK65: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_377, type, sK71: $i > $o).
% 2.93/1.81 thf(func_def_390, type, sK84: $i > $i > $i).
% 2.93/1.81 thf(func_def_398, type, sK92: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_407, type, sK101: $i > $i > $o).
% 2.93/1.81 thf(func_def_408, type, sK102: $i > $i > $i).
% 2.93/1.81 thf(func_def_409, type, sK103: $i > $i > $i).
% 2.93/1.81 thf(func_def_410, type, sK104: $i > $i > $i).
% 2.93/1.81 thf(func_def_411, type, sK105: $i > $i > $i).
% 2.93/1.81 thf(func_def_412, type, sK106: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_413, type, sK107: $i > $i).
% 2.93/1.81 thf(func_def_414, type, sK108: $i > $i).
% 2.93/1.81 thf(func_def_415, type, sK109: $i > $i).
% 2.93/1.81 thf(func_def_416, type, sK110: $i > $i).
% 2.93/1.81 thf(func_def_417, type, sK111: $i > $i > $i).
% 2.93/1.81 thf(func_def_418, type, sK112: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_419, type, sK113: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_420, type, sK114: $i > $i > $i).
% 2.93/1.81 thf(func_def_421, type, sK115: $i > $i > $i).
% 2.93/1.81 thf(func_def_423, type, sK117: $i > $i).
% 2.93/1.81 thf(func_def_424, type, sK118: $i > $i).
% 2.93/1.81 thf(func_def_425, type, sK119: $i > $i).
% 2.93/1.81 thf(func_def_426, type, sK120: $i > $i > $i).
% 2.93/1.81 thf(func_def_428, type, sK122: $i > $i).
% 2.93/1.81 thf(func_def_431, type, sK125: $i > ($i > $i) > $i > $i).
% 2.93/1.81 thf(func_def_447, type, sK141: $i > $i).
% 2.93/1.81 thf(func_def_459, type, sK153: ($i > $o) > $i).
% 2.93/1.81 thf(func_def_460, type, sK154: $i > $o).
% 2.93/1.81 thf(func_def_461, type, sK155: $i > $i).
% 2.93/1.81 thf(func_def_472, type, sK166: $i > $o).
% 2.93/1.81 thf(func_def_474, type, sK168: ($i > $o) > $i > $i).
% 2.93/1.81 thf(func_def_479, type, sK173: $i > $o).
% 2.93/1.81 thf(func_def_484, type, sK178: $o > $i > $i > $i).
% 2.93/1.81 thf(func_def_485, type, sK179: $i > $i).
% 2.93/1.81 thf(func_def_497, type, sK191: $i > $o).
% 2.93/1.81 thf(func_def_498, type, sK192: $i > $o).
% 2.93/1.81 thf(func_def_499, type, sK193: $i > $o).
% 2.93/1.81 thf(func_def_500, type, sK194: ($i > $o) > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_501, type, sK195: ($i > $o) > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_530, type, sK224: $i > $i).
% 2.93/1.81 thf(func_def_531, type, sK225: ($i > $i) > $i > $i > $i).
% 2.93/1.81 thf(func_def_541, type, sK235: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_542, type, sK236: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_543, type, sK237: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_544, type, sK238: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_548, type, sK242: $i > $i > $i).
% 2.93/1.81 thf(func_def_553, type, sK247: $i > $o).
% 2.93/1.81 thf(func_def_555, type, sK249: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_569, type, sK263: $i > $i > ($i > $i) > $i).
% 2.93/1.81 thf(func_def_570, type, sK264: $i > $i).
% 2.93/1.81 thf(func_def_577, type, sK271: $i > $i > $o).
% 2.93/1.81 thf(func_def_580, type, sK274: $i > $o).
% 2.93/1.81 thf(func_def_585, type, sK279: $i > $i > $o).
% 2.93/1.81 thf(func_def_595, type, sK289: $i > $i > $i).
% 2.93/1.81 thf(func_def_600, type, sK294: $i > $o).
% 2.93/1.81 thf(func_def_609, type, sK303: $i > $i > $i).
% 2.93/1.81 thf(func_def_635, type, sK329: $i > $i).
% 2.93/1.81 thf(func_def_637, type, sK331: ($i > $i) > $i > $i > $i).
% 2.93/1.81 thf(func_def_643, type, sK337: $i > $i > $o).
% 2.93/1.81 thf(func_def_647, type, sK341: $i > $o).
% 2.93/1.81 thf(func_def_650, type, sK344: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_651, type, sK345: $i > $o).
% 2.93/1.81 thf(func_def_655, type, sK349: $i > $o).
% 2.93/1.81 thf(func_def_675, type, sK369: $i > $i > $o).
% 2.93/1.81 thf(func_def_684, type, sK378: $i > $i).
% 2.93/1.81 thf(func_def_687, type, sK381: $i > $i > $o).
% 2.93/1.81 thf(func_def_688, type, sK382: $i > $i).
% 2.93/1.81 thf(func_def_689, type, sK383: $i > $i).
% 2.93/1.81 thf(func_def_690, type, sK384: ($i > $i > $o) > $i > $i).
% 2.93/1.81 thf(func_def_691, type, sK385: ($i > $i > $o) > $i > $i).
% 2.93/1.81 thf(func_def_692, type, sK386: $i > ($i > $i > $o) > $i > $i).
% 2.93/1.81 thf(func_def_717, type, sK411: $i > $i > $i).
% 2.93/1.81 thf(func_def_719, type, sK413: $i > $o).
% 2.93/1.81 thf(func_def_750, type, sK444: $i > ($i > $i) > $i > $i).
% 2.93/1.81 thf(func_def_752, type, sK446: $i > $i).
% 2.93/1.81 thf(func_def_754, type, sK448: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_759, type, sK453: ($i > $o) > $i).
% 2.93/1.81 thf(func_def_760, type, sK454: $i > $o).
% 2.93/1.81 thf(func_def_767, type, sK461: $i > $i > $i).
% 2.93/1.81 thf(func_def_781, type, sK475: $i > $i > $o).
% 2.93/1.81 thf(func_def_783, type, sK477: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_784, type, sK478: $i > $o).
% 2.93/1.81 thf(func_def_801, type, sK495: $i > $o).
% 2.93/1.81 thf(func_def_804, type, sK498: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_805, type, sK499: $i > $o).
% 2.93/1.81 thf(func_def_806, type, sK500: $i > $o).
% 2.93/1.81 thf(func_def_807, type, sK501: ($i > $o) > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_808, type, sK502: ($i > $o) > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_815, type, sK509: $i > $i).
% 2.93/1.81 thf(func_def_827, type, sK521: $i > $o).
% 2.93/1.81 thf(func_def_829, type, sK523: ($i > $o) > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_830, type, sK524: ($i > $o) > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_837, type, sK531: $i > $i).
% 2.93/1.81 thf(func_def_849, type, sK543: $i > $o).
% 2.93/1.81 thf(func_def_878, type, sK572: ($i > $o) > $i > $i > $i).
% 2.93/1.81 thf(func_def_881, type, sK575: $i > $o).
% 2.93/1.81 thf(func_def_887, type, sK581: $i > $o).
% 2.93/1.81 thf(func_def_888, type, sK582: $i > $o).
% 2.93/1.81 thf(func_def_901, type, sK595: ($i > $i) > $i > $i > $i).
% 2.93/1.81 thf(func_def_904, type, sK598: $i > $i).
% 2.93/1.81 thf(func_def_912, type, sK606: $i > $o).
% 2.93/1.81 thf(func_def_933, type, sK627: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.93/1.81 thf(func_def_934, type, sK628: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.93/1.81 thf(func_def_937, type, sK631: $i > $o).
% 2.93/1.81 thf(func_def_938, type, sK632: $i > $o).
% 2.93/1.81 thf(func_def_953, type, sK647: $i > $i > $i).
% 2.93/1.81 thf(func_def_956, type, sK650: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_961, type, sK655: $i > $o).
% 2.93/1.81 thf(func_def_962, type, sK656: $i > $i).
% 2.93/1.81 thf(func_def_963, type, sK657: ($i > $o) > $i).
% 2.93/1.81 thf(func_def_965, type, sK659: $i > $o).
% 2.93/1.81 thf(func_def_967, type, sK661: ($i > $o) > $i > $i).
% 2.93/1.81 thf(func_def_976, type, sK670: $i > $o).
% 2.93/1.81 thf(func_def_979, type, sK673: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_980, type, sK674: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_984, type, sK678: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_989, type, sK683: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_990, type, sK684: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_999, type, sK693: $i > $i > $o).
% 2.93/1.81 thf(func_def_1008, type, sK702: $i > $i > $i).
% 2.93/1.81 thf(func_def_1009, type, sK703: $i > $i > $i).
% 2.93/1.81 thf(func_def_1010, type, sK704: $i > $o).
% 2.93/1.81 thf(func_def_1013, type, sK707: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_1014, type, sK708: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_1015, type, sK709: $i > $o).
% 2.93/1.81 thf(func_def_1020, type, sK714: $i > $i).
% 2.93/1.81 thf(func_def_1023, type, sK717: $i > $i).
% 2.93/1.81 thf(func_def_1024, type, sK718: ($i > $o) > $i).
% 2.93/1.81 thf(func_def_1025, type, sK719: ($i > $o) > $i).
% 2.93/1.81 thf(func_def_1026, type, sK720: $i > $o).
% 2.93/1.81 thf(func_def_1036, type, sK730: $i > $o).
% 2.93/1.81 thf(func_def_1039, type, sK733: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_1040, type, sK734: $i > ($i > $o) > $i).
% 2.93/1.81 thf(func_def_1044, type, sK738: $i > $i > $i).
% 2.93/1.81 thf(func_def_1067, type, sK761: $i > $o).
% 2.93/1.81 thf(func_def_1083, type, sK777: $i > $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_1088, type, sK782: $i > $o).
% 2.93/1.81 thf(func_def_1089, type, sK783: ($i > $o) > $i > $i).
% 2.93/1.81 thf(func_def_1097, type, sK791: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_1101, type, sK795: $i > $i > $i > $i).
% 2.93/1.81 thf(func_def_1113, type, ph807: !>[X0: $tType]:(X0)).
% 2.93/1.81 thf(f4014,plain,(
% 2.93/1.81 $false),
% 2.93/1.81 inference(trivial_inequality_removal,[status(thm)],[f4006])).
% 2.93/1.81 thf(f4006,plain,(
% 2.93/1.81 ($true != $true)),
% 2.93/1.81 inference(superposition,[status(thm)],[f3975,f2567])).
% 2.93/1.81 thf(f2567,plain,(
% 2.93/1.81 ((in @ sK134 @ sK132) = $true)),
% 2.93/1.81 inference(cnf_transformation,[status(thm)],[f1322])).
% 2.93/1.81 thf(f1322,plain,(
% 2.93/1.81 (sepInPowerset = $true) & (brelnall2 = $true) & (binunionIL = $true) & (exuE3e = $true) & (dpsetconstrER = $true) & (subbreln = $true) & (setukpairIL = $true) & (vacuousDall = $true) & (inCongP = $true) & (setadjoinIR = $true) & (dpsetconstrEL2 = $true) & (apProp = $true) & (ex1E2 = $true) & (ex1I2 = $true) & (cartprodpairsurjEq = $true) & (quantDeMorgan4 = $true) & (setukpairinjL1 = $true) & (eqinunit = $true) & (dpsetconstrSub = $true) & (complementTI1 = $true) & (disjointsetsI1 = $true) & (ifp = $true) & (cartprodsndpairEq = $true) & (binintersectTERcontra = $true) & (descr__Cong = $true) & (exuI1 = $true) & (binintersectLsub = $true) & (funcGraphProp3 = $true) & (setminusLsub = $true) & (contrasubsetT = $true) & (subsetI1 = $true) & (descrp = $true) & (iffalse = $true) & (uniqinunit = $true) & (dsetconstrI = $true) & (ex1E1 = $true) & (omegaSAx = $true) & (cartprodpairmemEL = $true) & (brelnall1 = $true) & (binintersectEL = $true) & (singletonsubset = $true) & (setoftrueEq = $true) & (complementImpComplementIntersect = $true) & (ksndpairEq = $true) & (dsetconstr__Cong = $true) & (subsetTrans = $true) & (foundationAx = $true) & (binunionIR = $true) & (notsubsetI = $true) & (nonemptyI = $true) & (symdiffIneg1 = $true) & (binintersectSubset3 = $true) & (eqimpsubset2 = $true) & (doubleComplementSub1 = $true) & (exuE3u = $true) & (setukpairinjR12 = $true) & (symdiffE = $true) & (notinemptyset = $true) & (kpairp = $true) & (powersetAx = $true) & (beta1 = $true) & (upairsetIL = $true) & (noeltsimpempty = $true) & (exuE2 = $true) & (complementSubsetComplementIntersect = $true) & (setukpairinjR2 = $true) & (dsetconstrER = $true) & (upairsubunion = $true) & (cartprodmempair = $true) & (subsetRefl = $true) & (powersetsubset = $true) & (emptysetE = $true) & (iffalseProp2 = $true) & (binintersectSubset4 = $true) & (iftrue = $true) & (setminusELneg = $true) & (ap2apEq1 = $true) & (setunionsingleton1 = $true) & (subset2powerset = $true) & (setOfPairsIsBReln = $true) & (ap2apEq2 = $true) & (binintersectSubset1 = $true) & (eta1 = $true) & (setminusIRneg = $true) & (contraSubsetComplement = $true) & (setunionE = $true) & (notequalI2 = $true) & (funcGraphProp4 = $true) & (kpairsurjEq = $true) & (funcext = $true) & (omega__Cong = $true) & (setextT = $true) & (subPowSU = $true) & (setadjoinE = $true) & (emptyI = $true) & (kfstsingleton = $true) & (kpairiskpair = $true) & (ex1I = $true) & (setukpairinjL = $true) & (complementTcontraSubset = $true) & (powersetE = $true) & (binintersectSubset2 = $true) & (theeq = $true) & (setunionsingleton = $true) & (in__Cong = $true) & (subsetemptysetimpeq = $true) & (eqimpsubset1 = $true) & (emptysetAx = $true) & (bs114d = $true) & (notinsingleton = $true) & (dpsetconstrI = $true) & (upairset2E = $true) & (kfstpairEq = $true) & (funcGraphProp1 = $true) & (upairinpowunion = $true) & (quantDeMorgan3 = $true) & (quantDeMorgan2 = $true) & (emptysetimpfalse = $true) & (setextsub = $true) & (emptyE1 = $true) & (doubleComplementSub2 = $true) & (subsetI2 = $true) & (lam2lamEq = $true) & (complementT_lem = $true) & (infuncsetfunc = $true) & (funcext2 = $true) & (beta2 = $true) & (binintersectSubset5 = $true) & (setminusSubset2 = $true) & (subsetE2 = $true) & (upairset2IR = $true) & (prop2set2propI = $true) & (notequalI1 = $true) & (funcextLem = $true) & (ap2p = $true) & (binintersectTELcontra = $true) & (omegaIndAx = $true) & (app = $true) & (exuI2 = $true) & (binunionLsub = $true) & (upairsetIR = $true) & (lamp = $true) & (setadjoinSub2 = $true) & (setukpairinjR1 = $true) & (prop2setI = $true) & (nonemptyE1 = $true) & (setminusT_lem = $true) & (wellorderingAx = $true) & (setminusI = $true) & (setadjoinSub = $true) & (iftrueProp1 = $true) & (omega0Ax = $true) & (setukpairinjR = $true) & (emptyinPowerset = $true) & (binintersectER = $true) & (singletoninpowerset = $true) & (emptysetsubset = $true) & (sepSubset = $true) & (upairequniteq = $true) & (funcGraphProp2 = $true) & (powersetI1 = $true) & (setunionAx = $true) & (nonemptyI1 = $true) & (cartprodpairmemER = $true) & (binintersectRsub = $true) & (emptyset__Cong = $true) & (powersetTE1 = $true) & (powerset__Cong = $true) & (setadjoinOr = $true) & (subsetE = $true) & (contrasubsetT1 = $true) & (setadjoinAx = $true) & (setunionsingleton2 = $true) & (symdiffIneg2 = $true) & (singletonsswitch = $true) & (symdiffI2 = $true) & (symdiffI1 = $true) & (binunionEcases = $true) & (doubleComplementE1 = $true) & (setminusEL = $true) & (cartprodfstpairEq = $true) & (ksndsingleton = $true) & (setminusSubset1 = $true) & (iffalseProp1 = $true) & (subsetTI = $true) & (dpsetconstrERa = $true) & (contrasubsetT2 = $true) & (binintersectI = $true) & (doubleComplementI1 = $true) & (cartprodmempair1 = $true) & (cartprodpairin = $true) & (complementTnotintersectT = $true) & (theprop = $true) & (singletonprop = $true) & (upairsetE = $true) & (iftrueorfalse = $true) & (powersetTI1 = $true) & (funcImageSingleton = $true) & (setext = $true) & (prop2setE = $true) & (notdexE = $true) & (complementInPowersetComplementIntersect = $true) & (cartprodsndin = $true) & (exu__Cong = $true) & (singletonsuniq = $true) & (setunionI = $true) & (inPowerset = $true) & (setunion__Cong = $true) & (nonemptyImpWitness = $true) & (ifSingleton = $true) & (binunionRsub = $true) & (exuEu = $true) & (contrasubsetT3 = $true) & (replAx = $true) & (setadjoin__Cong = $true) & (ubforcartprodlem2 = $true) & (setminusERneg = $true) & (setukpairinjL2 = $true) & (exuI3 = $true) & (lamProp = $true) & (emptyInPowerset = $true) & (setukpairinjR11 = $true) & (setukpairIR = $true) & (((in @ sK132 @ (powerset @ sK131)) = $true) & ((((in @ sK134 @ sK132) = $true) & ((in @ sK134 @ sK131) = $true) & ((in @ sK134 @ (binunion @ sK132 @ sK133)) != $true)) & ($true = (in @ sK133 @ (powerset @ sK131))))) & (complementTE1 = $true) & (eqbreln = $true) & (setunionE2 = $true) & (funcinfuncset = $true) & (binunionT_lem = $true) & (quantDeMorgan1 = $true) & (cartprodfstin = $true) & (setminusER = $true) & (emptyinunitempty = $true) & (setminusILneg = $true) & (setextAx = $true) & (cartprodmempaircEq = $true) & (secondinupair = $true) & (ubforcartprodlem3 = $true) & (setadjoinIL = $true) & (binintersectT_lem = $true) & (powersetT_lem = $true) & (notdallE = $true) & (iftrueProp2 = $true) & (setbeta = $true) & (powersetE1 = $true) & (powersetI = $true) & (singletoninpowunion = $true) & (binunionE = $true) & (dsetconstrEL = $true) & (lam2p = $true) & (dpsetconstrEL1 = $true) & (doubleComplementEq = $true) & (eta2 = $true) & (exuE1 = $true) & (ubforcartprodlem1 = $true)),
% 2.93/1.81 inference(skolemisation,[status(esa),new_symbols(skolem,[sK131,sK132,sK133,sK134])],[f851,f1321,f1320,f1319])).
% 2.93/1.81 thf(f1319,plain,(
% 2.93/1.81 ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ($true = (in @ X3 @ X0)) & ((in @ X3 @ (binunion @ X1 @ X2)) != $true)) & ($true = (in @ X2 @ (powerset @ X0))))) => (((in @ sK132 @ (powerset @ sK131)) = $true) & ? [X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ sK132)) & ((in @ X3 @ sK131) = $true) & ((in @ X3 @ (binunion @ sK132 @ X2)) != $true)) & ((in @ X2 @ (powerset @ sK131)) = $true)))),
% 2.93/1.81 introduced(definition,[],[choice_axiom])).
% 2.93/1.81 thf(f1320,plain,(
% 2.93/1.81 ? [X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ sK132)) & ((in @ X3 @ sK131) = $true) & ((in @ X3 @ (binunion @ sK132 @ X2)) != $true)) & ((in @ X2 @ (powerset @ sK131)) = $true)) => (? [X3 : $i] : (($true = (in @ X3 @ sK132)) & ((in @ X3 @ sK131) = $true) & ((in @ X3 @ (binunion @ sK132 @ sK133)) != $true)) & ($true = (in @ sK133 @ (powerset @ sK131))))),
% 2.93/1.81 introduced(definition,[],[choice_axiom])).
% 2.93/1.81 thf(f1321,plain,(
% 2.93/1.81 ? [X3 : $i] : (($true = (in @ X3 @ sK132)) & ((in @ X3 @ sK131) = $true) & ((in @ X3 @ (binunion @ sK132 @ sK133)) != $true)) => (((in @ sK134 @ sK132) = $true) & ((in @ sK134 @ sK131) = $true) & ((in @ sK134 @ (binunion @ sK132 @ sK133)) != $true))),
% 2.93/1.81 introduced(definition,[],[choice_axiom])).
% 2.93/1.81 thf(f851,plain,(
% 2.93/1.81 (sepInPowerset = $true) & (brelnall2 = $true) & (binunionIL = $true) & (exuE3e = $true) & (dpsetconstrER = $true) & (subbreln = $true) & (setukpairIL = $true) & (vacuousDall = $true) & (inCongP = $true) & (setadjoinIR = $true) & (dpsetconstrEL2 = $true) & (apProp = $true) & (ex1E2 = $true) & (ex1I2 = $true) & (cartprodpairsurjEq = $true) & (quantDeMorgan4 = $true) & (setukpairinjL1 = $true) & (eqinunit = $true) & (dpsetconstrSub = $true) & (complementTI1 = $true) & (disjointsetsI1 = $true) & (ifp = $true) & (cartprodsndpairEq = $true) & (binintersectTERcontra = $true) & (descr__Cong = $true) & (exuI1 = $true) & (binintersectLsub = $true) & (funcGraphProp3 = $true) & (setminusLsub = $true) & (contrasubsetT = $true) & (subsetI1 = $true) & (descrp = $true) & (iffalse = $true) & (uniqinunit = $true) & (dsetconstrI = $true) & (ex1E1 = $true) & (omegaSAx = $true) & (cartprodpairmemEL = $true) & (brelnall1 = $true) & (binintersectEL = $true) & (singletonsubset = $true) & (setoftrueEq = $true) & (complementImpComplementIntersect = $true) & (ksndpairEq = $true) & (dsetconstr__Cong = $true) & (subsetTrans = $true) & (foundationAx = $true) & (binunionIR = $true) & (notsubsetI = $true) & (nonemptyI = $true) & (symdiffIneg1 = $true) & (binintersectSubset3 = $true) & (eqimpsubset2 = $true) & (doubleComplementSub1 = $true) & (exuE3u = $true) & (setukpairinjR12 = $true) & (symdiffE = $true) & (notinemptyset = $true) & (kpairp = $true) & (powersetAx = $true) & (beta1 = $true) & (upairsetIL = $true) & (noeltsimpempty = $true) & (exuE2 = $true) & (complementSubsetComplementIntersect = $true) & (setukpairinjR2 = $true) & (dsetconstrER = $true) & (upairsubunion = $true) & (cartprodmempair = $true) & (subsetRefl = $true) & (powersetsubset = $true) & (emptysetE = $true) & (iffalseProp2 = $true) & (binintersectSubset4 = $true) & (iftrue = $true) & (setminusELneg = $true) & (ap2apEq1 = $true) & (setunionsingleton1 = $true) & (subset2powerset = $true) & (setOfPairsIsBReln = $true) & (ap2apEq2 = $true) & (binintersectSubset1 = $true) & (eta1 = $true) & (setminusIRneg = $true) & (contraSubsetComplement = $true) & (setunionE = $true) & (notequalI2 = $true) & (funcGraphProp4 = $true) & (kpairsurjEq = $true) & (funcext = $true) & (omega__Cong = $true) & (setextT = $true) & (subPowSU = $true) & (setadjoinE = $true) & (emptyI = $true) & (kfstsingleton = $true) & (kpairiskpair = $true) & (ex1I = $true) & (setukpairinjL = $true) & (complementTcontraSubset = $true) & (powersetE = $true) & (binintersectSubset2 = $true) & (theeq = $true) & (setunionsingleton = $true) & (in__Cong = $true) & (subsetemptysetimpeq = $true) & (eqimpsubset1 = $true) & (emptysetAx = $true) & (bs114d = $true) & (notinsingleton = $true) & (dpsetconstrI = $true) & (upairset2E = $true) & (kfstpairEq = $true) & (funcGraphProp1 = $true) & (upairinpowunion = $true) & (quantDeMorgan3 = $true) & (quantDeMorgan2 = $true) & (emptysetimpfalse = $true) & (setextsub = $true) & (emptyE1 = $true) & (doubleComplementSub2 = $true) & (subsetI2 = $true) & (lam2lamEq = $true) & (complementT_lem = $true) & (infuncsetfunc = $true) & (funcext2 = $true) & (beta2 = $true) & (binintersectSubset5 = $true) & (setminusSubset2 = $true) & (subsetE2 = $true) & (upairset2IR = $true) & (prop2set2propI = $true) & (notequalI1 = $true) & (funcextLem = $true) & (ap2p = $true) & (binintersectTELcontra = $true) & (omegaIndAx = $true) & (app = $true) & (exuI2 = $true) & (binunionLsub = $true) & (upairsetIR = $true) & (lamp = $true) & (setadjoinSub2 = $true) & (setukpairinjR1 = $true) & (prop2setI = $true) & (nonemptyE1 = $true) & (setminusT_lem = $true) & (wellorderingAx = $true) & (setminusI = $true) & (setadjoinSub = $true) & (iftrueProp1 = $true) & (omega0Ax = $true) & (setukpairinjR = $true) & (emptyinPowerset = $true) & (binintersectER = $true) & (singletoninpowerset = $true) & (emptysetsubset = $true) & (sepSubset = $true) & (upairequniteq = $true) & (funcGraphProp2 = $true) & (powersetI1 = $true) & (setunionAx = $true) & (nonemptyI1 = $true) & (cartprodpairmemER = $true) & (binintersectRsub = $true) & (emptyset__Cong = $true) & (powersetTE1 = $true) & (powerset__Cong = $true) & (setadjoinOr = $true) & (subsetE = $true) & (contrasubsetT1 = $true) & (setadjoinAx = $true) & (setunionsingleton2 = $true) & (symdiffIneg2 = $true) & (singletonsswitch = $true) & (symdiffI2 = $true) & (symdiffI1 = $true) & (binunionEcases = $true) & (doubleComplementE1 = $true) & (setminusEL = $true) & (cartprodfstpairEq = $true) & (ksndsingleton = $true) & (setminusSubset1 = $true) & (iffalseProp1 = $true) & (subsetTI = $true) & (dpsetconstrERa = $true) & (contrasubsetT2 = $true) & (binintersectI = $true) & (doubleComplementI1 = $true) & (cartprodmempair1 = $true) & (cartprodpairin = $true) & (complementTnotintersectT = $true) & (theprop = $true) & (singletonprop = $true) & (upairsetE = $true) & (iftrueorfalse = $true) & (powersetTI1 = $true) & (funcImageSingleton = $true) & (setext = $true) & (prop2setE = $true) & (notdexE = $true) & (complementInPowersetComplementIntersect = $true) & (cartprodsndin = $true) & (exu__Cong = $true) & (singletonsuniq = $true) & (setunionI = $true) & (inPowerset = $true) & (setunion__Cong = $true) & (nonemptyImpWitness = $true) & (ifSingleton = $true) & (binunionRsub = $true) & (exuEu = $true) & (contrasubsetT3 = $true) & (replAx = $true) & (setadjoin__Cong = $true) & (ubforcartprodlem2 = $true) & (setminusERneg = $true) & (setukpairinjL2 = $true) & (exuI3 = $true) & (lamProp = $true) & (emptyInPowerset = $true) & (setukpairinjR11 = $true) & (setukpairIR = $true) & ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ($true = (in @ X3 @ X0)) & ((in @ X3 @ (binunion @ X1 @ X2)) != $true)) & ($true = (in @ X2 @ (powerset @ X0))))) & (complementTE1 = $true) & (eqbreln = $true) & (setunionE2 = $true) & (funcinfuncset = $true) & (binunionT_lem = $true) & (quantDeMorgan1 = $true) & (cartprodfstin = $true) & (setminusER = $true) & (emptyinunitempty = $true) & (setminusILneg = $true) & (setextAx = $true) & (cartprodmempaircEq = $true) & (secondinupair = $true) & (ubforcartprodlem3 = $true) & (setadjoinIL = $true) & (binintersectT_lem = $true) & (powersetT_lem = $true) & (notdallE = $true) & (iftrueProp2 = $true) & (setbeta = $true) & (powersetE1 = $true) & (powersetI = $true) & (singletoninpowunion = $true) & (binunionE = $true) & (dsetconstrEL = $true) & (lam2p = $true) & (dpsetconstrEL1 = $true) & (doubleComplementEq = $true) & (eta2 = $true) & (exuE1 = $true) & (ubforcartprodlem1 = $true)),
% 2.93/1.81 inference(flattening,[status(thm)],[f850])).
% 2.93/1.81 thf(f850,plain,(
% 2.93/1.81 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : ((((in @ X3 @ X1) = $true) & ((in @ X3 @ (binunion @ X1 @ X2)) != $true)) & ($true = (in @ X3 @ X0))) & ($true = (in @ X2 @ (powerset @ X0)))) & ($true = (in @ X1 @ (powerset @ X0)))) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.93/1.81 inference(ennf_transformation,[status(thm)],[f794])).
% 2.93/1.81 thf(f794,plain,(
% 2.93/1.81 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ! [X3 : $i] : (($true = (in @ X3 @ X0)) => (((in @ X3 @ (binunion @ X1 @ X2)) != $true) => ((in @ X3 @ X1) != $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.93/1.81 inference(flattening,[status(thm)],[f435])).
% 2.93/1.81 thf(f435,plain,(
% 2.93/1.81 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ! [X3 : $i] : (($true = (in @ X3 @ X0)) => (~((in @ X3 @ (binunion @ X1 @ X2)) = $true) => ~((in @ X3 @ X1) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.93/1.81 inference(fool_elimination,[status(thm)],[f434])).
% 2.93/1.81 thf(f434,plain,(
% 2.93/1.81 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ! [X3 : $i] : ((in @ X3 @ X0) => (~(in @ X3 @ (binunion @ X1 @ X2)) => ~(in @ X3 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.93/1.81 inference(rectify,[status(thm)],[f260])).
% 2.93/1.81 thf(f260,negated_conjecture,(
% 2.93/1.81 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ (binunion @ X11 @ X16)) => ~(in @ X1 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.93/1.81 inference(negated_conjecture,[status(cth)],[f259])).
% 2.93/1.81 thf(f259,conjecture,(
% 2.93/1.81 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ (binunion @ X11 @ X16)) => ~(in @ X1 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.93/1.81 file('/export/starexec/sandbox2/tmp/tmp.ebZAxzcvYS/DTF2THF_11747.p',binunionTILcontra)).
% 2.93/1.81 thf(f3975,plain,(
% 2.93/1.81 ((in @ sK134 @ sK132) != $true)),
% 2.93/1.81 inference(trivial_inequality_removal,[status(thm)],[f3974])).
% 2.93/1.81 thf(f3974,plain,(
% 2.93/1.81 ($true != $true) | ((in @ sK134 @ sK132) != $true)),
% 2.93/1.81 inference(forward_demodulation,[status(thm)],[f3966,f2789])).
% 2.93/1.81 thf(f2789,plain,(
% 2.93/1.81 (binunionIL = $true)),
% 2.93/1.81 inference(cnf_transformation,[status(thm)],[f1322])).
% 2.93/1.81 thf(f3966,plain,(
% 2.93/1.81 (binunionIL != $true) | ((in @ sK134 @ sK132) != $true)),
% 2.93/1.81 inference(trivial_inequality_removal,[status(thm)],[f3957])).
% 2.93/1.81 thf(f3957,plain,(
% 2.93/1.81 ((in @ sK134 @ sK132) != $true) | (binunionIL != $true) | ($true != $true)),
% 2.93/1.81 inference(superposition,[status(thm)],[f2565,f2973])).
% 2.93/1.81 thf(f2973,plain,(
% 2.93/1.81 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (binunion @ X5 @ X4)) = $true) | ((in @ X3 @ X5) != $true) | (binunionIL != $true)) )),
% 2.93/1.81 inference(cnf_transformation,[status(thm)],[f1543])).
% 2.93/1.81 thf(f1543,plain,(
% 2.93/1.81 ((binunionIL = $true) | (((in @ sK267 @ (binunion @ sK269 @ sK268)) != $true) & ((in @ sK267 @ sK269) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (binunion @ X5 @ X4)) = $true) | ((in @ X3 @ X5) != $true)) | (binunionIL != $true))),
% 2.93/1.81 inference(skolemisation,[status(esa),new_symbols(skolem,[sK267,sK268,sK269])],[f1541,f1542])).
% 2.93/1.81 thf(f1542,plain,(
% 2.93/1.81 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) != $true) & ($true = (in @ X0 @ X2))) => (((in @ sK267 @ (binunion @ sK269 @ sK268)) != $true) & ((in @ sK267 @ sK269) = $true))),
% 2.93/1.81 introduced(definition,[],[choice_axiom])).
% 2.93/1.81 thf(f1541,plain,(
% 2.93/1.81 ((binunionIL = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) != $true) & ($true = (in @ X0 @ X2)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (binunion @ X5 @ X4)) = $true) | ((in @ X3 @ X5) != $true)) | (binunionIL != $true))),
% 2.93/1.81 inference(rectify,[status(thm)],[f1540])).
% 2.93/1.81 thf(f1540,plain,(
% 2.93/1.81 ((binunionIL = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) != $true) & ($true = (in @ X0 @ X2)))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) = $true) | ($true != (in @ X0 @ X2))) | (binunionIL != $true))),
% 2.93/1.81 inference(nnf_transformation,[status(thm)],[f824])).
% 2.93/1.81 thf(f824,plain,(
% 2.93/1.81 (binunionIL = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) = $true) | ($true != (in @ X0 @ X2)))),
% 2.93/1.81 inference(ennf_transformation,[status(thm)],[f533])).
% 2.93/1.81 thf(f533,plain,(
% 2.93/1.81 (binunionIL = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X2)) => ((in @ X0 @ (binunion @ X2 @ X1)) = $true))),
% 2.93/1.81 inference(fool_elimination,[status(thm)],[f532])).
% 2.93/1.81 thf(f532,plain,(
% 2.93/1.81 (binunionIL = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => (in @ X0 @ (binunion @ X2 @ X1))))),
% 2.93/1.81 inference(rectify,[status(thm)],[f101])).
% 2.93/1.81 thf(f101,axiom,(
% 2.93/1.81 (binunionIL = ! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ X3) => (in @ X1 @ (binunion @ X3 @ X4))))),
% 2.93/1.81 file('/export/starexec/sandbox2/tmp/tmp.ebZAxzcvYS/DTF2THF_11747.p',binunionIL)).
% 2.93/1.81 thf(f2565,plain,(
% 2.93/1.81 ((in @ sK134 @ (binunion @ sK132 @ sK133)) != $true)),
% 2.93/1.81 inference(cnf_transformation,[status(thm)],[f1322])).
% 2.93/1.81 % SZS output end Proof for DTF2THF_11747
% 2.93/1.81 % (12055)------------------------------
% 2.93/1.81 % (12055)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.81 % (12055)Termination reason: Refutation
% 2.93/1.81
% 2.93/1.81 % (12055)Memory used [KB]: 10106
% 2.93/1.81 % (12055)Time elapsed: 0.117 s
% 2.93/1.81 % (12055)Instructions burned: 345 (million)
% 2.93/1.81 % (12055)------------------------------
% 2.93/1.81 % (12055)------------------------------
% 2.93/1.81 % (11993)Success in time 0.215 s
%------------------------------------------------------------------------------