%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU673^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:06 AM UTC 2026
% Result : Theorem 2.88s 2.03s
% Output : Refutation 5.42s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 14
% Syntax : Number of formulae : 83 ( 18 unt; 0 typ; 2 def)
% Number of atoms : 3771 ( 988 equ; 0 cnn)
% Maximal formula atoms : 193 ( 45 avg)
% Number of connectives : 2347 ( 133 ~; 97 |; 618 &; 700 @)
% ( 11 <=>; 788 =>; 0 <=; 0 <~>)
% Maximal formula depth : 206 ( 21 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 11 ( 11 >; 0 *; 0 +; 0 <<)
% Number of symbols : 358 ( 355 usr; 206 con; 0-4 aty)
% Number of variables : 211 ( 55 ^; 110 !; 46 ?; 211 :)
% 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_239,type,
sP1: $i > $i > $o ).
thf(func_def_240,type,
sP2: $i > $o ).
thf(func_def_241,type,
sP3: $i > $i > $o ).
thf(func_def_242,type,
sP4: $i > $i > $o ).
thf(func_def_244,type,
sP6: $i > $o > $i > $i > $o ).
thf(func_def_246,type,
sK8: $i > $o ).
thf(func_def_260,type,
sK22: $i > $o ).
thf(func_def_261,type,
sK23: ( $i > $o ) > $i ).
thf(func_def_272,type,
sK34: $i > $i ).
thf(func_def_276,type,
sK38: $i > $i > $o ).
thf(func_def_281,type,
sK43: $i > $i > $o ).
thf(func_def_283,type,
sK45: $i > $i ).
thf(func_def_284,type,
sK46: $i > $i ).
thf(func_def_285,type,
sK47: $i > ( $i > $i > $o ) > $i ).
thf(func_def_286,type,
sK48: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_287,type,
sK49: $i > ( $i > $i > $o ) > $i ).
thf(func_def_292,type,
sK54: $i > $i ).
thf(func_def_300,type,
sK62: $i > $o ).
thf(func_def_305,type,
sK67: $i > $i > $i > $i ).
thf(func_def_306,type,
sK68: $i > $i > $i > $i ).
thf(func_def_309,type,
sK71: $i > $i > $o ).
thf(func_def_329,type,
sK91: $i > $i > $i ).
thf(func_def_330,type,
sK92: $i > $i > $i ).
thf(func_def_336,type,
sK98: $i > $i ).
thf(func_def_337,type,
sK99: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_338,type,
sK100: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_339,type,
sK101: $i > $o ).
thf(func_def_340,type,
sK102: $i > $o ).
thf(func_def_343,type,
sK105: $i > $o ).
thf(func_def_344,type,
sK106: $i > $i ).
thf(func_def_345,type,
sK107: ( $i > $o ) > $i ).
thf(func_def_357,type,
sK119: $i > $i ).
thf(func_def_358,type,
sK120: $i > ( $i > $o ) > $i ).
thf(func_def_359,type,
sK121: $i > $o ).
thf(func_def_386,type,
sK148: $i > ( $i > $o ) > $i ).
thf(func_def_387,type,
sK149: $i > $o ).
thf(func_def_392,type,
sK154: $i > $i > $i ).
thf(func_def_393,type,
sK155: $i > $i > $i ).
thf(func_def_394,type,
sK156: $i > $i > $i ).
thf(func_def_395,type,
sK157: $i > $i > $i ).
thf(func_def_396,type,
sK158: $i > $i > $i > $i ).
thf(func_def_397,type,
sK159: $i > $i ).
thf(func_def_398,type,
sK160: $i > $i ).
thf(func_def_399,type,
sK161: $i > $i ).
thf(func_def_400,type,
sK162: $i > $i ).
thf(func_def_401,type,
sK163: $i > $i > $i ).
thf(func_def_402,type,
sK164: $i > $i > $i > $i ).
thf(func_def_403,type,
sK165: $i > $i > $i > $i ).
thf(func_def_404,type,
sK166: $i > $i > $i ).
thf(func_def_405,type,
sK167: $i > $i > $i ).
thf(func_def_407,type,
sK169: $i > $i ).
thf(func_def_408,type,
sK170: $i > $i ).
thf(func_def_409,type,
sK171: $i > $i ).
thf(func_def_410,type,
sK172: $i > $i > $i ).
thf(func_def_412,type,
sK174: $i > $o ).
thf(func_def_426,type,
sK188: $i > $i > $i ).
thf(func_def_437,type,
sK199: $i > $o ).
thf(func_def_439,type,
sK201: $i > ( $i > $o ) > $i ).
thf(func_def_440,type,
sK202: $i > $i > $i ).
thf(func_def_456,type,
sK218: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_457,type,
sK219: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_460,type,
sK222: $i > $o ).
thf(func_def_461,type,
sK223: $i > $o ).
thf(func_def_469,type,
sK231: $i > $o ).
thf(func_def_472,type,
sK234: $i > ( $i > $o ) > $i ).
thf(func_def_473,type,
sK235: $i > ( $i > $o ) > $i ).
thf(func_def_480,type,
sK242: $i > ( $i > $o ) > $i ).
thf(func_def_481,type,
sK243: $i > $o ).
thf(func_def_483,type,
sK245: $i > $o ).
thf(func_def_487,type,
sK249: $i > $o ).
thf(func_def_510,type,
sK272: $i > $o ).
thf(func_def_514,type,
sK276: $i > $o ).
thf(func_def_517,type,
sK279: $i > $o ).
thf(func_def_532,type,
sK294: $i > $i > $o ).
thf(func_def_536,type,
sK298: ( $i > $o ) > $i ).
thf(func_def_537,type,
sK299: ( $i > $o ) > $i ).
thf(func_def_538,type,
sK300: $i > $o ).
thf(func_def_547,type,
sK309: $i > $o ).
thf(func_def_548,type,
sK310: $i > $i ).
thf(func_def_549,type,
sK311: ( $i > $o ) > $i ).
thf(func_def_561,type,
sK323: $o > $i > $i > $i ).
thf(func_def_568,type,
sK330: $i > $i > $o ).
thf(func_def_593,type,
sK355: $i > $o ).
thf(func_def_595,type,
sK357: $i > ( $i > $o ) > $i ).
thf(func_def_596,type,
sK358: $i > $i ).
thf(func_def_605,type,
sK367: $i > $i > $i ).
thf(func_def_609,type,
sK371: $i > $i > $i ).
thf(func_def_617,type,
sK379: $i > $o ).
thf(func_def_622,type,
sK384: $i > $i > $i ).
thf(func_def_646,type,
sK408: $i > $o ).
thf(func_def_648,type,
sK410: ( $i > $o ) > $i > $i ).
thf(func_def_657,type,
sK419: $i > $i > $o ).
thf(func_def_662,type,
sK424: $i > $o ).
thf(func_def_664,type,
sK426: $i > $o ).
thf(func_def_668,type,
sK430: ( $i > $o ) > $i > $i ).
thf(func_def_670,type,
sK432: $i > $o ).
thf(func_def_681,type,
sK443: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_682,type,
sK444: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_683,type,
sK445: $i > $o ).
thf(func_def_684,type,
sK446: $i > $o ).
thf(func_def_686,type,
sK448: $i > $o ).
thf(func_def_688,type,
sK450: ( $i > $o ) > $i > $i ).
thf(func_def_689,type,
sK451: ( $i > $o ) > $i > $i ).
thf(func_def_702,type,
sK464: $i > $i > $i ).
thf(func_def_709,type,
sK471: $i > $i > $o ).
thf(func_def_710,type,
sK472: $i > $o ).
thf(func_def_723,type,
sK485: $i > $i > $i ).
thf(func_def_724,type,
sK486: $i > $i > $i ).
thf(func_def_725,type,
sK487: ( $i > $o ) > $i > $i > $i ).
thf(func_def_728,type,
sK490: $i > $o ).
thf(func_def_750,type,
sK512: $i > $i > $o ).
thf(func_def_771,type,
sK533: $i > $o ).
thf(func_def_773,type,
sK535: ( $i > $o ) > $i > $i ).
thf(func_def_776,type,
sK538: $i > $i ).
thf(func_def_777,type,
sK539: $i > $i ).
thf(func_def_779,type,
sK541: $i > $o ).
thf(func_def_782,type,
sK544: $i > ( $i > $o ) > $i ).
thf(func_def_787,type,
ph549:
!>[X0: $tType] : X0 ).
thf(f5580,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3500,f5169,f5573]) ).
thf(f5573,plain,
spl548_10,
inference(avatar_contradiction_clause,[status(thm)],[f5572]) ).
thf(f5572,plain,
( $false
| spl548_10 ),
inference(trivial_inequality_removal,[status(thm)],[f5561]) ).
thf(f5561,plain,
( ( $true != $true )
| spl548_10 ),
inference(superposition,[status(thm)],[f5357,f2347]) ).
thf(f2347,plain,
( ( in @ sK399 @ sK397 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1457]) ).
thf(f1457,plain,
( ( omegaIndAx = $true )
& ( omega__Cong = $true )
& ( emptyset__Cong = $true )
& ( ubforcartprodlem1 = $true )
& ( binunionIL = $true )
& ( binintersectSubset1 = $true )
& ( binunionE = $true )
& ( kfstsingleton = $true )
& ( kpairsurjEq = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset5 = $true )
& ( binintersectER = $true )
& ( binintersectRsub = $true )
& ( in__Cong = $true )
& ( upairset2IR = $true )
& ( upairsubunion = $true )
& ( cartprodmempair1 = $true )
& ( ksndpairEq = $true )
& ( setukpairinjR1 = $true )
& ( powersetsubset = $true )
& ( singletonsuniq = $true )
& ( setadjoinIL = $true )
& ( eqimpsubset2 = $true )
& ( setunionI = $true )
& ( setminusERneg = $true )
& ( nonemptyI1 = $true )
& ( setminusSubset1 = $true )
& ( setunionsingleton2 = $true )
& ( upairset2E = $true )
& ( inCongP = $true )
& ( notequalI1 = $true )
& ( setadjoinAx = $true )
& ( kpairp = $true )
& ( symdiffI1 = $true )
& ( setukpairIR = $true )
& ( exuE2 = $true )
& ( exu__Cong = $true )
& ( upairsetIR = $true )
& ( setminusIRneg = $true )
& ( dpsetconstrER = $true )
& ( notinsingleton = $true )
& ( replAx = $true )
& ( ubforcartprodlem2 = $true )
& ( bs114d = $true )
& ( theprop = $true )
& ( dpsetconstrI = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( emptysetsubset = $true )
& ( emptyinPowerset = $true )
& ( ex1I2 = $true )
& ( ksndsingleton = $true )
& ( dpsetconstrEL1 = $true )
& ( setminusLsub = $true )
& ( upairequniteq = $true )
& ( exuI2 = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusSubset2 = $true )
& ( ( func @ sK397 @ sK396 @ sK398 )
= $true )
& ( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
@ sK396 )
!= $true )
& ( ( in @ sK399 @ sK397 )
= $true )
& ( sepInPowerset = $true )
& ( dpsetconstrSub = $true )
& ( notdexE = $true )
& ( emptysetE = $true )
& ( emptyE1 = $true )
& ( prop2set2propI = $true )
& ( subPowSU = $true )
& ( setunionAx = $true )
& ( powersetE1 = $true )
& ( setextAx = $true )
& ( uniqinunit = $true )
& ( powersetE = $true )
& ( setextsub = $true )
& ( setukpairinjL2 = $true )
& ( exuI3 = $true )
& ( powersetAx = $true )
& ( singletonsswitch = $true )
& ( notsubsetI = $true )
& ( vacuousDall = $true )
& ( setukpairinjL = $true )
& ( subsetE2 = $true )
& ( ubforcartprodlem3 = $true )
& ( symdiffE = $true )
& ( emptyinunitempty = $true )
& ( powersetI1 = $true )
& ( binintersectI = $true )
& ( powersetI = $true )
& ( foundationAx = $true )
& ( setunion__Cong = $true )
& ( setbeta = $true )
& ( setunionE = $true )
& ( cartprodpairsurjEq = $true )
& ( symdiffI2 = $true )
& ( setminusER = $true )
& ( eqinunit = $true )
& ( setunionE2 = $true )
& ( cartprodpairmemEL = $true )
& ( upairsetIL = $true )
& ( cartprodfstin = $true )
& ( subsetI2 = $true )
& ( singletonsubset = $true )
& ( setminusILneg = $true )
& ( eqimpsubset1 = $true )
& ( setukpairinjR2 = $true )
& ( setukpairIL = $true )
& ( quantDeMorgan4 = $true )
& ( sepSubset = $true )
& ( dsetconstrEL = $true )
& ( singletoninpowerset = $true )
& ( binintersectSubset4 = $true )
& ( exuE1 = $true )
& ( setukpairinjR12 = $true )
& ( dsetconstrI = $true )
& ( ex1E1 = $true )
& ( exuI1 = $true )
& ( subsetE = $true )
& ( singletonprop = $true )
& ( setadjoinSub = $true )
& ( binunionRsub = $true )
& ( setukpairinjR11 = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodmempaircEq = $true )
& ( setadjoinSub2 = $true )
& ( ex1I = $true )
& ( descr__Cong = $true )
& ( binunionEcases = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodsndin = $true )
& ( prop2setE = $true )
& ( setadjoinOr = $true )
& ( binintersectLsub = $true )
& ( setunionsingleton = $true )
& ( kpairiskpair = $true )
& ( setadjoin__Cong = $true )
& ( setminusI = $true )
& ( cartprodpairin = $true )
& ( setext = $true )
& ( binunionLsub = $true )
& ( cartprodpairmemER = $true )
& ( disjointsetsI1 = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodsndpairEq = $true )
& ( singletoninpowunion = $true )
& ( binintersectSubset2 = $true )
& ( cartprodmempair = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( nonemptyI = $true )
& ( setunionsingleton1 = $true )
& ( emptysetimpfalse = $true )
& ( subsetRefl = $true )
& ( exuE3u = $true )
& ( cartprodfstpairEq = $true )
& ( omegaSAx = $true )
& ( setukpairinjR = $true )
& ( notinemptyset = $true )
& ( kfstpairEq = $true )
& ( dsetconstrER = $true )
& ( emptysetAx = $true )
& ( subsetI1 = $true )
& ( setadjoinIR = $true )
& ( notequalI2 = $true )
& ( wellorderingAx = $true )
& ( prop2setI = $true )
& ( subset2powerset = $true )
& ( binintersectSubset3 = $true )
& ( exuE3e = $true )
& ( emptyI = $true )
& ( symdiffIneg2 = $true )
& ( setminusEL = $true )
& ( emptyInPowerset = $true )
& ( setadjoinE = $true )
& ( notdallE = $true )
& ( exuEu = $true )
& ( setukpairinjL1 = $true )
& ( setminusELneg = $true )
& ( descrp = $true )
& ( funcImageSingleton = $true )
& ( binunionIR = $true )
& ( subsetTrans = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectEL = $true )
& ( noeltsimpempty = $true )
& ( secondinupair = $true )
& ( symdiffIneg1 = $true )
& ( upairsetE = $true )
& ( omega0Ax = $true )
& ( upairinpowunion = $true )
& ( inPowerset = $true )
& ( nonemptyE1 = $true )
& ( powerset__Cong = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK396,sK397,sK398,sK399])],[f785,f1456,f1455]) ).
thf(f1455,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
= $true )
& ? [X3: $i] :
( ( ( in
@ ( setunion
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X2 ) ) )
@ X0 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) ) )
=> ( ( ( func @ sK397 @ sK396 @ sK398 )
= $true )
& ? [X3: $i] :
( ( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ sK398 ) ) )
@ sK396 )
!= $true )
& ( $true
= ( in @ X3 @ sK397 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1456,plain,
( ? [X3: $i] :
( ( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ sK398 ) ) )
@ sK396 )
!= $true )
& ( $true
= ( in @ X3 @ sK397 ) ) )
=> ( ( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
@ sK396 )
!= $true )
& ( ( in @ sK399 @ sK397 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f785,plain,
( ( omegaIndAx = $true )
& ( omega__Cong = $true )
& ( emptyset__Cong = $true )
& ( ubforcartprodlem1 = $true )
& ( binunionIL = $true )
& ( binintersectSubset1 = $true )
& ( binunionE = $true )
& ( kfstsingleton = $true )
& ( kpairsurjEq = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset5 = $true )
& ( binintersectER = $true )
& ( binintersectRsub = $true )
& ( in__Cong = $true )
& ( upairset2IR = $true )
& ( upairsubunion = $true )
& ( cartprodmempair1 = $true )
& ( ksndpairEq = $true )
& ( setukpairinjR1 = $true )
& ( powersetsubset = $true )
& ( singletonsuniq = $true )
& ( setadjoinIL = $true )
& ( eqimpsubset2 = $true )
& ( setunionI = $true )
& ( setminusERneg = $true )
& ( nonemptyI1 = $true )
& ( setminusSubset1 = $true )
& ( setunionsingleton2 = $true )
& ( upairset2E = $true )
& ( inCongP = $true )
& ( notequalI1 = $true )
& ( setadjoinAx = $true )
& ( kpairp = $true )
& ( symdiffI1 = $true )
& ( setukpairIR = $true )
& ( exuE2 = $true )
& ( exu__Cong = $true )
& ( upairsetIR = $true )
& ( setminusIRneg = $true )
& ( dpsetconstrER = $true )
& ( notinsingleton = $true )
& ( replAx = $true )
& ( ubforcartprodlem2 = $true )
& ( bs114d = $true )
& ( theprop = $true )
& ( dpsetconstrI = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( emptysetsubset = $true )
& ( emptyinPowerset = $true )
& ( ex1I2 = $true )
& ( ksndsingleton = $true )
& ( dpsetconstrEL1 = $true )
& ( setminusLsub = $true )
& ( upairequniteq = $true )
& ( exuI2 = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusSubset2 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
= $true )
& ? [X3: $i] :
( ( ( in
@ ( setunion
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X2 ) ) )
@ X0 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) ) )
& ( sepInPowerset = $true )
& ( dpsetconstrSub = $true )
& ( notdexE = $true )
& ( emptysetE = $true )
& ( emptyE1 = $true )
& ( prop2set2propI = $true )
& ( subPowSU = $true )
& ( setunionAx = $true )
& ( powersetE1 = $true )
& ( setextAx = $true )
& ( uniqinunit = $true )
& ( powersetE = $true )
& ( setextsub = $true )
& ( setukpairinjL2 = $true )
& ( exuI3 = $true )
& ( powersetAx = $true )
& ( singletonsswitch = $true )
& ( notsubsetI = $true )
& ( vacuousDall = $true )
& ( setukpairinjL = $true )
& ( subsetE2 = $true )
& ( ubforcartprodlem3 = $true )
& ( symdiffE = $true )
& ( emptyinunitempty = $true )
& ( powersetI1 = $true )
& ( binintersectI = $true )
& ( powersetI = $true )
& ( foundationAx = $true )
& ( setunion__Cong = $true )
& ( setbeta = $true )
& ( setunionE = $true )
& ( cartprodpairsurjEq = $true )
& ( symdiffI2 = $true )
& ( setminusER = $true )
& ( eqinunit = $true )
& ( setunionE2 = $true )
& ( cartprodpairmemEL = $true )
& ( upairsetIL = $true )
& ( cartprodfstin = $true )
& ( subsetI2 = $true )
& ( singletonsubset = $true )
& ( setminusILneg = $true )
& ( eqimpsubset1 = $true )
& ( setukpairinjR2 = $true )
& ( setukpairIL = $true )
& ( quantDeMorgan4 = $true )
& ( sepSubset = $true )
& ( dsetconstrEL = $true )
& ( singletoninpowerset = $true )
& ( binintersectSubset4 = $true )
& ( exuE1 = $true )
& ( setukpairinjR12 = $true )
& ( dsetconstrI = $true )
& ( ex1E1 = $true )
& ( exuI1 = $true )
& ( subsetE = $true )
& ( singletonprop = $true )
& ( setadjoinSub = $true )
& ( binunionRsub = $true )
& ( setukpairinjR11 = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodmempaircEq = $true )
& ( setadjoinSub2 = $true )
& ( ex1I = $true )
& ( descr__Cong = $true )
& ( binunionEcases = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodsndin = $true )
& ( prop2setE = $true )
& ( setadjoinOr = $true )
& ( binintersectLsub = $true )
& ( setunionsingleton = $true )
& ( kpairiskpair = $true )
& ( setadjoin__Cong = $true )
& ( setminusI = $true )
& ( cartprodpairin = $true )
& ( setext = $true )
& ( binunionLsub = $true )
& ( cartprodpairmemER = $true )
& ( disjointsetsI1 = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodsndpairEq = $true )
& ( singletoninpowunion = $true )
& ( binintersectSubset2 = $true )
& ( cartprodmempair = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( nonemptyI = $true )
& ( setunionsingleton1 = $true )
& ( emptysetimpfalse = $true )
& ( subsetRefl = $true )
& ( exuE3u = $true )
& ( cartprodfstpairEq = $true )
& ( omegaSAx = $true )
& ( setukpairinjR = $true )
& ( notinemptyset = $true )
& ( kfstpairEq = $true )
& ( dsetconstrER = $true )
& ( emptysetAx = $true )
& ( subsetI1 = $true )
& ( setadjoinIR = $true )
& ( notequalI2 = $true )
& ( wellorderingAx = $true )
& ( prop2setI = $true )
& ( subset2powerset = $true )
& ( binintersectSubset3 = $true )
& ( exuE3e = $true )
& ( emptyI = $true )
& ( symdiffIneg2 = $true )
& ( setminusEL = $true )
& ( emptyInPowerset = $true )
& ( setadjoinE = $true )
& ( notdallE = $true )
& ( exuEu = $true )
& ( setukpairinjL1 = $true )
& ( setminusELneg = $true )
& ( descrp = $true )
& ( funcImageSingleton = $true )
& ( binunionIR = $true )
& ( subsetTrans = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectEL = $true )
& ( noeltsimpempty = $true )
& ( secondinupair = $true )
& ( symdiffIneg1 = $true )
& ( upairsetE = $true )
& ( omega0Ax = $true )
& ( upairinpowunion = $true )
& ( inPowerset = $true )
& ( nonemptyE1 = $true )
& ( powerset__Cong = $true ) ),
inference(flattening,[status(thm)],[f784]) ).
thf(f784,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
= $true )
& ? [X3: $i] :
( ( ( in
@ ( setunion
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X2 ) ) )
@ X0 )
!= $true )
& ( ( in @ X3 @ X1 )
= $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)],[f581]) ).
thf(f581,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( in
@ ( setunion
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X2 ) ) )
@ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f580]) ).
thf(f580,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X1 @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X0
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) )
@ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f198]) ).
thf(f198,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
=> ! [X4: $i,X3: $i,X13: $i] :
( ( func @ X3 @ X4 @ X13 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X4
@ ^ [X2: $i] : ( in @ ( kpair @ X1 @ X2 ) @ X13 ) ) )
@ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f197]) ).
thf(f197,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
=> ! [X4: $i,X3: $i,X13: $i] :
( ( func @ X3 @ X4 @ X13 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X4
@ ^ [X2: $i] : ( in @ ( kpair @ X1 @ X2 ) @ X13 ) ) )
@ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',apProp) ).
thf(f5357,plain,
( ( ( in @ sK399 @ sK397 )
!= $true )
| spl548_10 ),
inference(trivial_inequality_removal,[status(thm)],[f5356]) ).
thf(f5356,plain,
( ( ( in @ sK399 @ sK397 )
!= $true )
| ( $true != $true )
| spl548_10 ),
inference(superposition,[status(thm)],[f4946,f2349]) ).
thf(f2349,plain,
( ( func @ sK397 @ sK396 @ sK398 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1457]) ).
thf(f4946,plain,
( ! [X0: $i] :
( ( ( func @ X0 @ sK396 @ sK398 )
!= $true )
| ( ( in @ sK399 @ X0 )
!= $true ) )
| spl548_10 ),
inference(trivial_inequality_removal,[status(thm)],[f4945]) ).
thf(f4945,plain,
( ! [X0: $i] :
( ( ( in @ sK399 @ X0 )
!= $true )
| ( ( func @ X0 @ sK396 @ sK398 )
!= $true )
| ( $true != $true ) )
| spl548_10 ),
inference(forward_demodulation,[status(thm)],[f4939,f2229]) ).
thf(f2229,plain,
funcImageSingleton = $true,
inference(cnf_transformation,[status(thm)],[f1457]) ).
thf(f4939,plain,
( ! [X0: $i] :
( ( ( in @ sK399 @ X0 )
!= $true )
| ( funcImageSingleton != $true )
| ( ( func @ X0 @ sK396 @ sK398 )
!= $true ) )
| spl548_10 ),
inference(trivial_inequality_removal,[status(thm)],[f4932]) ).
thf(f4932,plain,
( ! [X0: $i] :
( ( funcImageSingleton != $true )
| ( $true != $true )
| ( ( in @ sK399 @ X0 )
!= $true )
| ( ( func @ X0 @ sK396 @ sK398 )
!= $true ) )
| spl548_10 ),
inference(superposition,[status(thm)],[f2998,f1839]) ).
thf(f1839,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( singleton
@ ( dsetconstr @ X6
@ ^ [Y0: $i] : ( in @ ( kpair @ X7 @ Y0 ) @ X5 ) ) )
= $true )
| ( ( in @ X7 @ X4 )
!= $true )
| ( $true
!= ( func @ X4 @ X6 @ X5 ) )
| ( funcImageSingleton != $true ) ),
inference(cnf_transformation,[status(thm)],[f1014]) ).
thf(f1014,plain,
( ( ( funcImageSingleton = $true )
| ( ( ( func @ sK111 @ sK113 @ sK112 )
= $true )
& ( $true
= ( in @ sK114 @ sK111 ) )
& ( $true
!= ( singleton
@ ( dsetconstr @ sK113
@ ^ [Y0: $i] : ( in @ ( kpair @ sK114 @ Y0 ) @ sK112 ) ) ) ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ( $true
!= ( func @ X4 @ X6 @ X5 ) )
| ! [X7: $i] :
( ( ( in @ X7 @ X4 )
!= $true )
| ( ( singleton
@ ( dsetconstr @ X6
@ ^ [Y0: $i] : ( in @ ( kpair @ X7 @ Y0 ) @ X5 ) ) )
= $true ) ) )
| ( funcImageSingleton != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK111,sK112,sK113,sK114])],[f1011,f1013,f1012]) ).
thf(f1012,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( ( singleton
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) )
!= $true ) ) )
=> ( ( ( func @ sK111 @ sK113 @ sK112 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ sK111 )
= $true )
& ( ( singleton
@ ( dsetconstr @ sK113
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ sK112 ) ) )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1013,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK111 )
= $true )
& ( ( singleton
@ ( dsetconstr @ sK113
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ sK112 ) ) )
!= $true ) )
=> ( ( $true
= ( in @ sK114 @ sK111 ) )
& ( $true
!= ( singleton
@ ( dsetconstr @ sK113
@ ^ [Y0: $i] : ( in @ ( kpair @ sK114 @ Y0 ) @ sK112 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1011,plain,
( ( ( funcImageSingleton = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( ( singleton
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) )
!= $true ) ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ( $true
!= ( func @ X4 @ X6 @ X5 ) )
| ! [X7: $i] :
( ( ( in @ X7 @ X4 )
!= $true )
| ( ( singleton
@ ( dsetconstr @ X6
@ ^ [Y0: $i] : ( in @ ( kpair @ X7 @ Y0 ) @ X5 ) ) )
= $true ) ) )
| ( funcImageSingleton != $true ) ) ),
inference(rectify,[status(thm)],[f1010]) ).
thf(f1010,plain,
( ( ( funcImageSingleton = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( ( singleton
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) )
!= $true ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( func @ X0 @ X2 @ X1 ) )
| ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X0 ) )
| ( ( singleton
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) )
= $true ) ) )
| ( funcImageSingleton != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f679]) ).
thf(f679,plain,
( ( funcImageSingleton = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( func @ X0 @ X2 @ X1 ) )
| ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X0 ) )
| ( ( singleton
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) )
= $true ) ) ) ),
inference(ennf_transformation,[status(thm)],[f374]) ).
thf(f374,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ( ( singleton
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) )
= $true ) ) )
<=> ( funcImageSingleton = $true ) ),
inference(fool_elimination,[status(thm)],[f373]) ).
thf(f373,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X1 ) ) ) ) )
= funcImageSingleton ),
inference(rectify,[status(thm)],[f196]) ).
thf(f196,axiom,
! [X3: $i,X13: $i,X4: $i] :
( ( ( func @ X3 @ X4 @ X13 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( singleton
@ ( dsetconstr @ X4
@ ^ [X2: $i] : ( in @ ( kpair @ X1 @ X2 ) @ X13 ) ) ) ) )
= funcImageSingleton ),
file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',funcImageSingleton) ).
thf(f2998,plain,
( ( ( singleton
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
!= $true )
| spl548_10 ),
inference(avatar_component_clause,[status(thm)],[f2996]) ).
thf(f2996,definition,
( spl548_10
<=> ( ( singleton
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl548_10])],[avatar_definition]) ).
thf(f5169,plain,
spl548_9,
inference(avatar_contradiction_clause,[status(thm)],[f5168]) ).
thf(f5168,plain,
( $false
| spl548_9 ),
inference(trivial_inequality_removal,[status(thm)],[f5167]) ).
thf(f5167,plain,
( ( $true != $true )
| spl548_9 ),
inference(superposition,[status(thm)],[f4585,f2300]) ).
thf(f2300,plain,
sepSubset = $true,
inference(cnf_transformation,[status(thm)],[f1457]) ).
thf(f4585,plain,
( ( sepSubset != $true )
| spl548_9 ),
inference(trivial_inequality_removal,[status(thm)],[f4577]) ).
thf(f4577,plain,
( ( $true != $true )
| ( sepSubset != $true )
| spl548_9 ),
inference(superposition,[status(thm)],[f2994,f2659]) ).
thf(f2659,plain,
! [X0: $i > $o,X1: $i] :
( ( ( subset @ ( dsetconstr @ X1 @ X0 ) @ X1 )
= $true )
| ( sepSubset != $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1700]) ).
thf(f1700,plain,
! [X0: $i > $o,X1: $i] :
( ( ( subset
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
@ X1 )
= $true )
| ( sepSubset != $true ) ),
inference(cnf_transformation,[status(thm)],[f835]) ).
thf(f835,plain,
( ( ! [X0: $i > $o,X1: $i] :
( ( subset
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
@ X1 )
= $true )
| ( sepSubset != $true ) )
& ( ( sepSubset = $true )
| ( ( subset
@ ( dsetconstr @ sK9
@ ^ [Y0: $i] : ( sK8 @ Y0 ) )
@ sK9 )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9])],[f833,f834]) ).
thf(f834,plain,
( ? [X2: $i > $o,X3: $i] :
( ( subset
@ ( dsetconstr @ X3
@ ^ [Y0: $i] : ( X2 @ Y0 ) )
@ X3 )
!= $true )
=> ( ( subset
@ ( dsetconstr @ sK9
@ ^ [Y0: $i] : ( sK8 @ Y0 ) )
@ sK9 )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f833,plain,
( ( ! [X0: $i > $o,X1: $i] :
( ( subset
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
@ X1 )
= $true )
| ( sepSubset != $true ) )
& ( ( sepSubset = $true )
| ? [X2: $i > $o,X3: $i] :
( ( subset
@ ( dsetconstr @ X3
@ ^ [Y0: $i] : ( X2 @ Y0 ) )
@ X3 )
!= $true ) ) ),
inference(rectify,[status(thm)],[f832]) ).
thf(f832,plain,
( ( ! [X0: $i > $o,X1: $i] :
( ( subset
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
@ X1 )
= $true )
| ( sepSubset != $true ) )
& ( ( sepSubset = $true )
| ? [X0: $i > $o,X1: $i] :
( ( subset
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
@ X1 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f441]) ).
thf(f441,plain,
( ! [X0: $i > $o,X1: $i] :
( ( subset
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
@ X1 )
= $true )
<=> ( sepSubset = $true ) ),
inference(fool_elimination,[status(thm)],[f440]) ).
thf(f440,plain,
( sepSubset
= ( ! [X0: $i > $o,X1: $i] :
( subset
@ ( dsetconstr @ X1
@ ^ [X2: $i] : ( X0 @ X2 ) )
@ X1 ) ) ),
inference(rectify,[status(thm)],[f100]) ).
thf(f100,axiom,
( sepSubset
= ( ! [X0: $i > $o,X3: $i] :
( subset
@ ( dsetconstr @ X3
@ ^ [X1: $i] : ( X0 @ X1 ) )
@ X3 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',sepSubset) ).
thf(f2994,plain,
( ( ( subset
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) )
@ sK396 )
!= $true )
| spl548_9 ),
inference(avatar_component_clause,[status(thm)],[f2992]) ).
thf(f2992,definition,
( spl548_9
<=> ( ( subset
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) )
@ sK396 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl548_9])],[avatar_definition]) ).
thf(f3500,plain,
( ~ spl548_10
| ~ spl548_9 ),
inference(avatar_split_clause,[status(thm)],[f3499,f2992,f2996]) ).
thf(f3499,plain,
( ( ( subset
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) )
@ sK396 )
!= $true )
| ( ( singleton
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3498]) ).
thf(f3498,plain,
( ( ( singleton
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
!= $true )
| ( ( subset
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) )
@ sK396 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3422,f2363]) ).
thf(f2363,plain,
theprop = $true,
inference(cnf_transformation,[status(thm)],[f1457]) ).
thf(f3422,plain,
( ( ( singleton
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
!= $true )
| ( theprop != $true )
| ( ( subset
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) )
@ sK396 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3327]) ).
thf(f3327,plain,
( ( ( singleton
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
!= $true )
| ( ( subset
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) )
@ sK396 )
!= $true )
| ( $true != $true )
| ( theprop != $true ) ),
inference(superposition,[status(thm)],[f2782,f2541]) ).
thf(f2541,plain,
! [X1: $i] :
( ( ( in @ ( setunion @ X1 ) @ X1 )
= $true )
| ( theprop != $true )
| ( $true
!= ( singleton @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1615]) ).
thf(f1615,plain,
( ( ( theprop = $true )
| ( ( ( singleton @ sK495 )
= $true )
& ( $true
!= ( in @ ( setunion @ sK495 ) @ sK495 ) ) ) )
& ( ! [X1: $i] :
( ( $true
!= ( singleton @ X1 ) )
| ( ( in @ ( setunion @ X1 ) @ X1 )
= $true ) )
| ( theprop != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK495])],[f1613,f1614]) ).
thf(f1614,plain,
( ? [X0: $i] :
( ( ( singleton @ X0 )
= $true )
& ( ( in @ ( setunion @ X0 ) @ X0 )
!= $true ) )
=> ( ( ( singleton @ sK495 )
= $true )
& ( $true
!= ( in @ ( setunion @ sK495 ) @ sK495 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1613,plain,
( ( ( theprop = $true )
| ? [X0: $i] :
( ( ( singleton @ X0 )
= $true )
& ( ( in @ ( setunion @ X0 ) @ X0 )
!= $true ) ) )
& ( ! [X1: $i] :
( ( $true
!= ( singleton @ X1 ) )
| ( ( in @ ( setunion @ X1 ) @ X1 )
= $true ) )
| ( theprop != $true ) ) ),
inference(rectify,[status(thm)],[f1612]) ).
thf(f1612,plain,
( ( ( theprop = $true )
| ? [X0: $i] :
( ( ( singleton @ X0 )
= $true )
& ( ( in @ ( setunion @ X0 ) @ X0 )
!= $true ) ) )
& ( ! [X0: $i] :
( ( ( singleton @ X0 )
!= $true )
| ( ( in @ ( setunion @ X0 ) @ X0 )
= $true ) )
| ( theprop != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f805]) ).
thf(f805,plain,
( ( theprop = $true )
<=> ! [X0: $i] :
( ( ( singleton @ X0 )
!= $true )
| ( ( in @ ( setunion @ X0 ) @ X0 )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f262]) ).
thf(f262,plain,
( ! [X0: $i] :
( ( ( singleton @ X0 )
= $true )
=> ( ( in @ ( setunion @ X0 ) @ X0 )
= $true ) )
<=> ( theprop = $true ) ),
inference(fool_elimination,[status(thm)],[f261]) ).
thf(f261,plain,
( theprop
= ( ! [X0: $i] :
( ( singleton @ X0 )
=> ( in @ ( setunion @ X0 ) @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f165]) ).
thf(f165,axiom,
( theprop
= ( ! [X11: $i] :
( ( singleton @ X11 )
=> ( in @ ( setunion @ X11 ) @ X11 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',theprop) ).
thf(f2782,plain,
! [X0: $i] :
( ( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
@ X0 )
!= $true )
| ( $true
!= ( subset @ X0 @ sK396 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2781]) ).
thf(f2781,plain,
! [X0: $i] :
( ( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
@ X0 )
!= $true )
| ( $true
!= ( subset @ X0 @ sK396 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2759,f2326]) ).
thf(f2326,plain,
subsetE2 = $true,
inference(cnf_transformation,[status(thm)],[f1457]) ).
thf(f2759,plain,
! [X0: $i] :
( ( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
@ X0 )
!= $true )
| ( $true
!= ( subset @ X0 @ sK396 ) )
| ( subsetE2 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2757]) ).
thf(f2757,plain,
! [X0: $i] :
( ( $true != $true )
| ( subsetE2 != $true )
| ( $true
!= ( subset @ X0 @ sK396 ) )
| ( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
@ X0 )
!= $true ) ),
inference(superposition,[status(thm)],[f2348,f2011]) ).
thf(f2011,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X5 ) )
| ( subsetE2 != $true )
| ( ( in @ X3 @ X4 )
!= $true )
| ( ( subset @ X4 @ X5 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1211]) ).
thf(f1211,plain,
( ( ( subsetE2 = $true )
| ( ( ( in @ sK236 @ sK237 )
= $true )
& ( ( in @ sK236 @ sK238 )
!= $true )
& ( $true
= ( subset @ sK237 @ sK238 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X4 )
!= $true )
| ( $true
= ( in @ X3 @ X5 ) )
| ( ( subset @ X4 @ X5 )
!= $true ) )
| ( subsetE2 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK236,sK237,sK238])],[f1209,f1210]) ).
thf(f1210,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ X2 ) )
& ( ( subset @ X1 @ X2 )
= $true ) )
=> ( ( ( in @ sK236 @ sK237 )
= $true )
& ( ( in @ sK236 @ sK238 )
!= $true )
& ( $true
= ( subset @ sK237 @ sK238 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1209,plain,
( ( ( subsetE2 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ X2 ) )
& ( ( subset @ X1 @ X2 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X4 )
!= $true )
| ( $true
= ( in @ X3 @ X5 ) )
| ( ( subset @ X4 @ X5 )
!= $true ) )
| ( subsetE2 != $true ) ) ),
inference(rectify,[status(thm)],[f1208]) ).
thf(f1208,plain,
( ( ( subsetE2 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ X2 ) )
& ( ( subset @ X1 @ X2 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ X2 ) )
| ( ( subset @ X1 @ X2 )
!= $true ) )
| ( subsetE2 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f634]) ).
thf(f634,plain,
( ( subsetE2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ X2 ) )
| ( ( subset @ X1 @ X2 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f633]) ).
thf(f633,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ X2 ) )
| ( ( subset @ X1 @ X2 )
!= $true ) )
<=> ( subsetE2 = $true ) ),
inference(ennf_transformation,[status(thm)],[f615]) ).
thf(f615,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ X2 )
= $true )
=> ( ( $true
!= ( in @ X0 @ X2 ) )
=> ( $true
!= ( in @ X0 @ X1 ) ) ) )
<=> ( subsetE2 = $true ) ),
inference(flattening,[status(thm)],[f426]) ).
thf(f426,plain,
( ( subsetE2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ X2 )
= $true )
=> ( ( $true
!= ( in @ X0 @ X2 ) )
=> ( $true
!= ( in @ X0 @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f425]) ).
thf(f425,plain,
( subsetE2
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X0 @ X2 )
=> ~ ( in @ X0 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f84]) ).
thf(f84,axiom,
( subsetE2
= ( ! [X1: $i,X3: $i,X4: $i] :
( ( subset @ X3 @ X4 )
=> ( ~ ( in @ X1 @ X4 )
=> ~ ( in @ X1 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',subsetE2) ).
thf(f2348,plain,
( ( in
@ ( setunion
@ ( dsetconstr @ sK396
@ ^ [Y0: $i] : ( in @ ( kpair @ sK399 @ Y0 ) @ sK398 ) ) )
@ sK396 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1457]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU673^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n011.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:30:17 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.39 ---- Original DTF file ---
% 0.16/0.39 thf(spec,logic,$$dhol).
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 % File : SEU673^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.39 % Domain : Set Theory
% 0.16/0.39 % Problem : Functions - Application
% 0.16/0.39 % Version : Especial.
% 0.16/0.39 % English : (! A:i.! B:i.! f:i.func A B f -> (! x:i.in x A -> in (setunion
% 0.16/0.39 % (dsetconstr B (^ y:i.in (kpair x y) f))) B))
% 0.16/0.39
% 0.16/0.39 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.39 % Source : [Bro08]
% 0.16/0.39 % Names : ZFC175g [Bro08]
% 0.16/0.39
% 0.16/0.39 % Status : Theorem
% 0.16/0.39 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.70 v8.2.0, 0.77 v8.1.0, 0.82 v7.5.0, 0.71 v7.4.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.16/0.39 % Syntax : Number of formulae : 421 ( 196 unt; 224 typ; 196 def)
% 0.16/0.39 % Number of atoms : 1038 ( 289 equ; 0 cnn)
% 0.16/0.39 % Maximal formula atoms : 193 ( 5 avg)
% 0.16/0.39 % Number of connectives : 1871 ( 47 ~; 7 |; 36 &;1311 @)
% 0.16/0.39 % ( 14 <=>; 456 =>; 0 <=; 0 <~>)
% 0.16/0.39 % Maximal formula depth : 205 ( 2 avg)
% 0.16/0.39 % Number of types : 2 ( 0 usr)
% 0.16/0.39 % Number of type conns : 111 ( 111 >; 0 *; 0 +; 0 <<)
% 0.16/0.39 % Number of symbols : 227 ( 224 usr; 193 con; 0-3 aty)
% 0.16/0.39 % Number of variables : 607 ( 66 ^; 505 !; 36 ?; 607 :)
% 0.16/0.39 % SPC : TH0_THM_EQU_NAR
% 0.16/0.39
% 0.16/0.39 % Comments : http://mathgate.info/detsetitem.php?id=224
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 thf(in_type,type,
% 0.16/0.39 in: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(exu_type,type,
% 0.16/0.39 exu: ( $i > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(exu,definition,
% 0.16/0.39 ( exu
% 0.16/0.39 = ( ^ [Xphi: $i > $o] :
% 0.16/0.39 ? [Xx: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 & ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setextAx_type,type,
% 0.16/0.39 setextAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setextAx,definition,
% 0.16/0.39 ( setextAx
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 <=> ( in @ Xx @ B ) )
% 0.16/0.39 => ( A = B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyset_type,type,
% 0.16/0.39 emptyset: $i ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetAx_type,type,
% 0.16/0.39 emptysetAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetAx,definition,
% 0.16/0.39 ( emptysetAx
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoin_type,type,
% 0.16/0.39 setadjoin: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinAx_type,type,
% 0.16/0.39 setadjoinAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinAx,definition,
% 0.16/0.39 ( setadjoinAx
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 <=> ( ( Xy = Xx )
% 0.16/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powerset_type,type,
% 0.16/0.39 powerset: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(powersetAx_type,type,
% 0.16/0.39 powersetAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetAx,definition,
% 0.16/0.39 ( powersetAx
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.39 <=> ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunion_type,type,
% 0.16/0.39 setunion: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(setunionAx_type,type,
% 0.16/0.39 setunionAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionAx,definition,
% 0.16/0.39 ( setunionAx
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.39 <=> ? [B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omega_type,type,
% 0.16/0.39 omega: $i ).
% 0.16/0.39
% 0.16/0.39 thf(omega0Ax_type,type,
% 0.16/0.39 omega0Ax: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omega0Ax,definition,
% 0.16/0.39 ( omega0Ax
% 0.16/0.39 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omegaSAx_type,type,
% 0.16/0.39 omegaSAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omegaSAx,definition,
% 0.16/0.39 ( omegaSAx
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ omega )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omegaIndAx_type,type,
% 0.16/0.39 omegaIndAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omegaIndAx,definition,
% 0.16/0.39 ( omegaIndAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( ( in @ emptyset @ A )
% 0.16/0.39 & ! [Xx: $i] :
% 0.16/0.39 ( ( ( in @ Xx @ omega )
% 0.16/0.39 & ( in @ Xx @ A ) )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ omega )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(replAx_type,type,
% 0.16/0.39 replAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(replAx,definition,
% 0.16/0.39 ( replAx
% 0.16/0.39 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.39 => ? [B: $i] :
% 0.16/0.39 ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 <=> ? [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(foundationAx_type,type,
% 0.16/0.39 foundationAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(foundationAx,definition,
% 0.16/0.39 ( foundationAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.39 => ? [B: $i] :
% 0.16/0.39 ( ( in @ B @ A )
% 0.16/0.39 & ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(wellorderingAx_type,type,
% 0.16/0.39 wellorderingAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(wellorderingAx,definition,
% 0.16/0.39 ( wellorderingAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ? [B: $i] :
% 0.16/0.39 ( ! [C: $i] :
% 0.16/0.39 ( ( in @ C @ B )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ A ) ) )
% 0.16/0.39 & ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( ( in @ Xx @ A )
% 0.16/0.39 & ( in @ Xy @ A ) )
% 0.16/0.39 => ( ! [C: $i] :
% 0.16/0.39 ( ( in @ C @ B )
% 0.16/0.39 => ( ( in @ Xx @ C )
% 0.16/0.39 <=> ( in @ Xy @ C ) ) )
% 0.16/0.39 => ( Xx = Xy ) ) )
% 0.16/0.39 & ! [C: $i,D: $i] :
% 0.16/0.39 ( ( ( in @ C @ B )
% 0.16/0.39 & ( in @ D @ B ) )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ D ) )
% 0.16/0.39 | ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ D )
% 0.16/0.39 => ( in @ Xx @ C ) ) ) )
% 0.16/0.39 & ! [C: $i] :
% 0.16/0.39 ( ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.39 => ? [D: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ D @ B )
% 0.16/0.39 & ( in @ Xx @ C )
% 0.16/0.39 & ~ ? [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ D )
% 0.16/0.39 & ( in @ Xy @ C ) )
% 0.16/0.39 & ! [E: $i] :
% 0.16/0.39 ( ( in @ E @ B )
% 0.16/0.39 => ( ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ E )
% 0.16/0.39 => ( in @ Xy @ D ) )
% 0.16/0.39 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(descr_type,type,
% 0.16/0.39 descr: ( $i > $o ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(descrp_type,type,
% 0.16/0.39 descrp: $o ).
% 0.16/0.39
% 0.16/0.39 thf(descrp,definition,
% 0.16/0.39 ( descrp
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ( Xphi
% 0.16/0.39 @ ( descr
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstr_type,type,
% 0.16/0.39 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrI_type,type,
% 0.16/0.39 dsetconstrI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrI,definition,
% 0.16/0.39 ( dsetconstrI
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrEL_type,type,
% 0.16/0.39 dsetconstrEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrEL,definition,
% 0.16/0.39 ( dsetconstrEL
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrER_type,type,
% 0.16/0.39 dsetconstrER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrER,definition,
% 0.16/0.39 ( dsetconstrER
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE1_type,type,
% 0.16/0.39 exuE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE1,definition,
% 0.16/0.39 ( exuE1
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 & ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(prop2set_type,type,
% 0.16/0.39 prop2set: $o > $i ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setE_type,type,
% 0.16/0.39 prop2setE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setE,definition,
% 0.16/0.39 ( prop2setE
% 0.16/0.39 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.39 => Xphi ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetE_type,type,
% 0.16/0.39 emptysetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetE,definition,
% 0.16/0.39 ( emptysetE
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ emptyset )
% 0.16/0.39 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetimpfalse_type,type,
% 0.16/0.39 emptysetimpfalse: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetimpfalse,definition,
% 0.16/0.39 ( emptysetimpfalse
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ emptyset )
% 0.16/0.39 => $false ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notinemptyset_type,type,
% 0.16/0.39 notinemptyset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notinemptyset,definition,
% 0.16/0.39 ( notinemptyset
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3e_type,type,
% 0.16/0.39 exuE3e: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3e,definition,
% 0.16/0.39 ( exuE3e
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setext_type,type,
% 0.16/0.39 setext: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setext,definition,
% 0.16/0.39 ( setext
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 => ( A = B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyI_type,type,
% 0.16/0.39 emptyI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyI,definition,
% 0.16/0.39 ( emptyI
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ A )
% 0.16/0.39 => ( A = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(noeltsimpempty_type,type,
% 0.16/0.39 noeltsimpempty: $o ).
% 0.16/0.39
% 0.16/0.39 thf(noeltsimpempty,definition,
% 0.16/0.39 ( noeltsimpempty
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ A )
% 0.16/0.39 => ( A = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setbeta_type,type,
% 0.16/0.39 setbeta: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setbeta,definition,
% 0.16/0.39 ( setbeta
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonempty_type,type,
% 0.16/0.39 nonempty: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonempty,definition,
% 0.16/0.39 ( nonempty
% 0.16/0.39 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyE1_type,type,
% 0.16/0.39 nonemptyE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyE1,definition,
% 0.16/0.39 ( nonemptyE1
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( nonempty @ A )
% 0.16/0.39 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI_type,type,
% 0.16/0.39 nonemptyI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI,definition,
% 0.16/0.39 ( nonemptyI
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( nonempty
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI1_type,type,
% 0.16/0.39 nonemptyI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI1,definition,
% 0.16/0.39 ( nonemptyI1
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.39 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIL_type,type,
% 0.16/0.39 setadjoinIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIL,definition,
% 0.16/0.39 ( setadjoinIL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinunitempty_type,type,
% 0.16/0.39 emptyinunitempty: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinunitempty,definition,
% 0.16/0.39 ( emptyinunitempty
% 0.16/0.39 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIR_type,type,
% 0.16/0.39 setadjoinIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIR,definition,
% 0.16/0.39 ( setadjoinIR
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinE_type,type,
% 0.16/0.39 setadjoinE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinE,definition,
% 0.16/0.39 ( setadjoinE
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 => ! [Xphi: $o] :
% 0.16/0.39 ( ( ( Xy = Xx )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => ( ( ( in @ Xy @ A )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinOr_type,type,
% 0.16/0.39 setadjoinOr: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinOr,definition,
% 0.16/0.39 ( setadjoinOr
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 => ( ( Xy = Xx )
% 0.16/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setoftrueEq_type,type,
% 0.16/0.39 setoftrueEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setoftrueEq,definition,
% 0.16/0.39 ( setoftrueEq
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : $true )
% 0.16/0.39 = A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI_type,type,
% 0.16/0.39 powersetI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI,definition,
% 0.16/0.39 ( powersetI
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinPowerset_type,type,
% 0.16/0.39 emptyinPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinPowerset,definition,
% 0.16/0.39 ( emptyinPowerset
% 0.16/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyInPowerset_type,type,
% 0.16/0.39 emptyInPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyInPowerset,definition,
% 0.16/0.39 ( emptyInPowerset
% 0.16/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE_type,type,
% 0.16/0.39 powersetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE,definition,
% 0.16/0.39 ( powersetE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionI_type,type,
% 0.16/0.39 setunionI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionI,definition,
% 0.16/0.39 ( setunionI
% 0.16/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( ( in @ B @ A )
% 0.16/0.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE_type,type,
% 0.16/0.39 setunionE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE,definition,
% 0.16/0.39 ( setunionE
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.39 => ! [Xphi: $o] :
% 0.16/0.39 ( ! [B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( ( in @ B @ A )
% 0.16/0.39 => Xphi ) )
% 0.16/0.39 => Xphi ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subPowSU_type,type,
% 0.16/0.39 subPowSU: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subPowSU,definition,
% 0.16/0.39 ( subPowSU
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE2_type,type,
% 0.16/0.39 exuE2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE2,definition,
% 0.16/0.39 ( exuE2
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyImpWitness_type,type,
% 0.16/0.39 nonemptyImpWitness: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyImpWitness,definition,
% 0.16/0.39 ( nonemptyImpWitness
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( nonempty @ A )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & $true ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(uniqinunit_type,type,
% 0.16/0.39 uniqinunit: $o ).
% 0.16/0.39
% 0.16/0.39 thf(uniqinunit,definition,
% 0.16/0.39 ( uniqinunit
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notinsingleton_type,type,
% 0.16/0.39 notinsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notinsingleton,definition,
% 0.16/0.39 ( notinsingleton
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx != Xy )
% 0.16/0.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eqinunit_type,type,
% 0.16/0.39 eqinunit: $o ).
% 0.16/0.39
% 0.16/0.39 thf(eqinunit,definition,
% 0.16/0.39 ( eqinunit
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsswitch_type,type,
% 0.16/0.39 singletonsswitch: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsswitch,definition,
% 0.16/0.39 ( singletonsswitch
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetE_type,type,
% 0.16/0.39 upairsetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetE,definition,
% 0.16/0.39 ( upairsetE
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.39 => ( ( Xz = Xx )
% 0.16/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetIL_type,type,
% 0.16/0.39 upairsetIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetIL,definition,
% 0.16/0.39 ( upairsetIL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetIR_type,type,
% 0.16/0.39 upairsetIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetIR,definition,
% 0.16/0.39 ( upairsetIR
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyE1_type,type,
% 0.16/0.39 emptyE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyE1,definition,
% 0.16/0.39 ( emptyE1
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ( ( ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 = emptyset )
% 0.16/0.39 => $false ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(vacuousDall_type,type,
% 0.16/0.39 vacuousDall: $o ).
% 0.16/0.39
% 0.16/0.39 thf(vacuousDall,definition,
% 0.16/0.39 ( vacuousDall
% 0.16/0.39 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ emptyset )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan1_type,type,
% 0.16/0.39 quantDeMorgan1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan1,definition,
% 0.16/0.39 ( quantDeMorgan1
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ~ ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan2_type,type,
% 0.16/0.39 quantDeMorgan2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan2,definition,
% 0.16/0.39 ( quantDeMorgan2
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ~ ( Xphi @ Xx ) )
% 0.16/0.39 => ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan3_type,type,
% 0.16/0.39 quantDeMorgan3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan3,definition,
% 0.16/0.39 ( quantDeMorgan3
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan4_type,type,
% 0.16/0.39 quantDeMorgan4: $o ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan4,definition,
% 0.16/0.39 ( quantDeMorgan4
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ~ ( Xphi @ Xx ) )
% 0.16/0.39 => ~ ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setI_type,type,
% 0.16/0.39 prop2setI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setI,definition,
% 0.16/0.39 ( prop2setI
% 0.16/0.39 = ( ! [Xphi: $o] :
% 0.16/0.39 ( Xphi
% 0.16/0.39 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(set2prop_type,type,
% 0.16/0.39 set2prop: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2set2propI_type,type,
% 0.16/0.39 prop2set2propI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2set2propI,definition,
% 0.16/0.39 ( prop2set2propI
% 0.16/0.39 = ( ! [Xphi: $o] :
% 0.16/0.39 ( Xphi
% 0.16/0.39 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notdexE_type,type,
% 0.16/0.39 notdexE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notdexE,definition,
% 0.16/0.39 ( notdexE
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notdallE_type,type,
% 0.16/0.39 notdallE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notdallE,definition,
% 0.16/0.39 ( notdallE
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ~ ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuI1_type,type,
% 0.16/0.39 exuI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuI1,definition,
% 0.16/0.39 ( exuI1
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 & ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuI3_type,type,
% 0.16/0.39 exuI3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuI3,definition,
% 0.16/0.39 ( exuI3
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.39 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuI2_type,type,
% 0.16/0.39 exuI2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuI2,definition,
% 0.16/0.39 ( exuI2
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] :
% 0.16/0.39 ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 <=> ( Xy = Xx ) )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(inCongP_type,type,
% 0.16/0.39 inCongP: $o ).
% 0.16/0.39
% 0.16/0.39 thf(inCongP,definition,
% 0.16/0.39 ( inCongP
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(in__Cong_type,type,
% 0.16/0.39 in__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(in__Cong,definition,
% 0.16/0.39 ( in__Cong
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3u_type,type,
% 0.16/0.39 exuE3u: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3u,definition,
% 0.16/0.39 ( exuE3u
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exu__Cong_type,type,
% 0.16/0.39 exu__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exu__Cong,definition,
% 0.16/0.39 ( exu__Cong
% 0.16/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.39 => ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 <=> ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyset__Cong_type,type,
% 0.16/0.39 emptyset__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyset__Cong,definition,
% 0.16/0.39 ( emptyset__Cong
% 0.16/0.39 = ( emptyset = emptyset ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoin__Cong_type,type,
% 0.16/0.39 setadjoin__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoin__Cong,definition,
% 0.16/0.39 ( setadjoin__Cong
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ! [Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( Xz = Xu )
% 0.16/0.39 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.39 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powerset__Cong_type,type,
% 0.16/0.39 powerset__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powerset__Cong,definition,
% 0.16/0.39 ( powerset__Cong
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ( ( powerset @ A )
% 0.16/0.39 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunion__Cong_type,type,
% 0.16/0.39 setunion__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunion__Cong,definition,
% 0.16/0.39 ( setunion__Cong
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ( ( setunion @ A )
% 0.16/0.39 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omega__Cong_type,type,
% 0.16/0.39 omega__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omega__Cong,definition,
% 0.16/0.39 ( omega__Cong
% 0.16/0.39 = ( omega = omega ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuEu_type,type,
% 0.16/0.39 exuEu: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuEu,definition,
% 0.16/0.39 ( exuEu
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(descr__Cong_type,type,
% 0.16/0.39 descr__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(descr__Cong,definition,
% 0.16/0.39 ( descr__Cong
% 0.16/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.39 => ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.39 => ( ( descr
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 = ( descr
% 0.16/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstr__Cong_type,type,
% 0.16/0.39 dsetconstr__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstr__Cong,definition,
% 0.16/0.39 ( dsetconstr__Cong
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( Xx = Xy )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.39 => ( ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 = ( dsetconstr @ B
% 0.16/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subset_type,type,
% 0.16/0.39 subset: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(disjoint_type,type,
% 0.16/0.39 disjoint: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(setsmeet_type,type,
% 0.16/0.39 setsmeet: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetI1_type,type,
% 0.16/0.39 subsetI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetI1,definition,
% 0.16/0.39 ( subsetI1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eqimpsubset2_type,type,
% 0.16/0.39 eqimpsubset2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(eqimpsubset2,definition,
% 0.16/0.39 ( eqimpsubset2
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eqimpsubset1_type,type,
% 0.16/0.39 eqimpsubset1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(eqimpsubset1,definition,
% 0.16/0.39 ( eqimpsubset1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetI2_type,type,
% 0.16/0.39 subsetI2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetI2,definition,
% 0.16/0.39 ( subsetI2
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetsubset_type,type,
% 0.16/0.39 emptysetsubset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetsubset,definition,
% 0.16/0.39 ( emptysetsubset
% 0.16/0.39 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetE_type,type,
% 0.16/0.39 subsetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetE,definition,
% 0.16/0.39 ( subsetE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetE2_type,type,
% 0.16/0.39 subsetE2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetE2,definition,
% 0.16/0.39 ( subsetE2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notsubsetI_type,type,
% 0.16/0.39 notsubsetI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notsubsetI,definition,
% 0.16/0.39 ( notsubsetI
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notequalI1_type,type,
% 0.16/0.39 notequalI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notequalI1,definition,
% 0.16/0.39 ( notequalI1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ~ ( subset @ A @ B )
% 0.16/0.39 => ( A != B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notequalI2_type,type,
% 0.16/0.39 notequalI2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notequalI2,definition,
% 0.16/0.39 ( notequalI2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ( A != B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetRefl_type,type,
% 0.16/0.39 subsetRefl: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetRefl,definition,
% 0.16/0.39 ( subsetRefl
% 0.16/0.39 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetTrans_type,type,
% 0.16/0.39 subsetTrans: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetTrans,definition,
% 0.16/0.39 ( subsetTrans
% 0.16/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( subset @ B @ C )
% 0.16/0.39 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinSub_type,type,
% 0.16/0.39 setadjoinSub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinSub,definition,
% 0.16/0.39 ( setadjoinSub
% 0.16/0.39 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinSub2_type,type,
% 0.16/0.39 setadjoinSub2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinSub2,definition,
% 0.16/0.39 ( setadjoinSub2
% 0.16/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subset2powerset_type,type,
% 0.16/0.39 subset2powerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subset2powerset,definition,
% 0.16/0.39 ( subset2powerset
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setextsub_type,type,
% 0.16/0.39 setextsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setextsub,definition,
% 0.16/0.39 ( setextsub
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( subset @ B @ A )
% 0.16/0.39 => ( A = B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetemptysetimpeq_type,type,
% 0.16/0.39 subsetemptysetimpeq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetemptysetimpeq,definition,
% 0.16/0.39 ( subsetemptysetimpeq
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( subset @ A @ emptyset )
% 0.16/0.39 => ( A = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI1_type,type,
% 0.16/0.39 powersetI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI1,definition,
% 0.16/0.39 ( powersetI1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ B @ A )
% 0.16/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE1_type,type,
% 0.16/0.39 powersetE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE1,definition,
% 0.16/0.39 ( powersetE1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(inPowerset_type,type,
% 0.16/0.39 inPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(inPowerset,definition,
% 0.16/0.39 ( inPowerset
% 0.16/0.39 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetsubset_type,type,
% 0.16/0.39 powersetsubset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetsubset,definition,
% 0.16/0.39 ( powersetsubset
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(sepInPowerset_type,type,
% 0.16/0.39 sepInPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(sepInPowerset,definition,
% 0.16/0.39 ( sepInPowerset
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( in
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(sepSubset_type,type,
% 0.16/0.39 sepSubset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(sepSubset,definition,
% 0.16/0.39 ( sepSubset
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( subset
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunion_type,type,
% 0.16/0.39 binunion: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(binunionIL_type,type,
% 0.16/0.39 binunionIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionIL,definition,
% 0.16/0.39 ( binunionIL
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairset2IR_type,type,
% 0.16/0.39 upairset2IR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairset2IR,definition,
% 0.16/0.39 ( upairset2IR
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionIR_type,type,
% 0.16/0.39 binunionIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionIR,definition,
% 0.16/0.39 ( binunionIR
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionEcases_type,type,
% 0.16/0.39 binunionEcases: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionEcases,definition,
% 0.16/0.39 ( binunionEcases
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.39 => ( ( ( in @ Xx @ A )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => ( ( ( in @ Xx @ B )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionE_type,type,
% 0.16/0.39 binunionE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionE,definition,
% 0.16/0.39 ( binunionE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionLsub_type,type,
% 0.16/0.39 binunionLsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionLsub,definition,
% 0.16/0.39 ( binunionLsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionRsub_type,type,
% 0.16/0.39 binunionRsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionRsub,definition,
% 0.16/0.39 ( binunionRsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersect_type,type,
% 0.16/0.39 binintersect: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectI_type,type,
% 0.16/0.39 binintersectI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectI,definition,
% 0.16/0.39 ( binintersectI
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset5_type,type,
% 0.16/0.39 binintersectSubset5: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset5,definition,
% 0.16/0.39 ( binintersectSubset5
% 0.16/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.39 ( ( subset @ C @ A )
% 0.16/0.39 => ( ( subset @ C @ B )
% 0.16/0.39 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectEL_type,type,
% 0.16/0.39 binintersectEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectEL,definition,
% 0.16/0.39 ( binintersectEL
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectLsub_type,type,
% 0.16/0.39 binintersectLsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectLsub,definition,
% 0.16/0.39 ( binintersectLsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset2_type,type,
% 0.16/0.39 binintersectSubset2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset2,definition,
% 0.16/0.39 ( binintersectSubset2
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( binintersect @ A @ B )
% 0.16/0.39 = A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset3_type,type,
% 0.16/0.39 binintersectSubset3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset3,definition,
% 0.16/0.39 ( binintersectSubset3
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( ( binintersect @ A @ B )
% 0.16/0.39 = B )
% 0.16/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectER_type,type,
% 0.16/0.39 binintersectER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectER,definition,
% 0.16/0.39 ( binintersectER
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.39 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(disjointsetsI1_type,type,
% 0.16/0.39 disjointsetsI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(disjointsetsI1,definition,
% 0.16/0.39 ( disjointsetsI1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( in @ Xx @ B ) )
% 0.16/0.39 => ( ( binintersect @ A @ B )
% 0.16/0.39 = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectRsub_type,type,
% 0.16/0.39 binintersectRsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectRsub,definition,
% 0.16/0.39 ( binintersectRsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset4_type,type,
% 0.16/0.39 binintersectSubset4: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset4,definition,
% 0.16/0.39 ( binintersectSubset4
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ B @ A )
% 0.16/0.39 => ( ( binintersect @ A @ B )
% 0.16/0.39 = B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset1_type,type,
% 0.16/0.39 binintersectSubset1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset1,definition,
% 0.16/0.39 ( binintersectSubset1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( ( binintersect @ A @ B )
% 0.16/0.39 = A )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(bs114d_type,type,
% 0.16/0.39 bs114d: $o ).
% 0.16/0.39
% 0.16/0.39 thf(bs114d,definition,
% 0.16/0.39 ( bs114d
% 0.16/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.39 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.39 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(regular_type,type,
% 0.16/0.39 regular: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminus_type,type,
% 0.16/0.39 setminus: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(setminusI_type,type,
% 0.16/0.39 setminusI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusI,definition,
% 0.16/0.39 ( setminusI
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusEL_type,type,
% 0.16/0.39 setminusEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusEL,definition,
% 0.16/0.39 ( setminusEL
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusER_type,type,
% 0.16/0.39 setminusER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusER,definition,
% 0.16/0.39 ( setminusER
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.39 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusSubset2_type,type,
% 0.16/0.39 setminusSubset2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusSubset2,definition,
% 0.16/0.39 ( setminusSubset2
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( setminus @ A @ B )
% 0.16/0.39 = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusERneg_type,type,
% 0.16/0.39 setminusERneg: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusERneg,definition,
% 0.16/0.39 ( setminusERneg
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusELneg_type,type,
% 0.16/0.39 setminusELneg: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusELneg,definition,
% 0.16/0.39 ( setminusELneg
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusILneg_type,type,
% 0.16/0.39 setminusILneg: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusILneg,definition,
% 0.16/0.39 ( setminusILneg
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ A )
% 0.16/0.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusIRneg_type,type,
% 0.16/0.39 setminusIRneg: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusIRneg,definition,
% 0.16/0.39 ( setminusIRneg
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusLsub_type,type,
% 0.16/0.39 setminusLsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusLsub,definition,
% 0.16/0.39 ( setminusLsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusSubset1_type,type,
% 0.16/0.39 setminusSubset1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusSubset1,definition,
% 0.16/0.39 ( setminusSubset1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( ( setminus @ A @ B )
% 0.16/0.39 = emptyset )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiff_type,type,
% 0.16/0.39 symdiff: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffE_type,type,
% 0.16/0.39 symdiffE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffE,definition,
% 0.16/0.39 ( symdiffE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.39 => ! [Xphi: $o] :
% 0.16/0.39 ( ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => Xphi ) )
% 0.16/0.39 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => Xphi ) )
% 0.16/0.39 => Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffI1_type,type,
% 0.16/0.39 symdiffI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffI1,definition,
% 0.16/0.39 ( symdiffI1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffI2_type,type,
% 0.16/0.39 symdiffI2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffI2,definition,
% 0.16/0.39 ( symdiffI2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffIneg1_type,type,
% 0.16/0.39 symdiffIneg1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffIneg1,definition,
% 0.16/0.39 ( symdiffIneg1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffIneg2_type,type,
% 0.16/0.39 symdiffIneg2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffIneg2,definition,
% 0.16/0.39 ( symdiffIneg2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(iskpair_type,type,
% 0.16/0.39 iskpair: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(secondinupair_type,type,
% 0.16/0.39 secondinupair: $o ).
% 0.16/0.39
% 0.16/0.39 thf(secondinupair,definition,
% 0.16/0.39 ( secondinupair
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairIL_type,type,
% 0.16/0.39 setukpairIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairIL,definition,
% 0.16/0.39 ( setukpairIL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairIR_type,type,
% 0.16/0.39 setukpairIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairIR,definition,
% 0.16/0.39 ( setukpairIR
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kpairiskpair_type,type,
% 0.16/0.39 kpairiskpair: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kpairiskpair,definition,
% 0.16/0.39 ( kpairiskpair
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kpair_type,type,
% 0.16/0.39 kpair: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(kpairp_type,type,
% 0.16/0.39 kpairp: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kpairp,definition,
% 0.16/0.39 ( kpairp
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprod_type,type,
% 0.16/0.39 cartprod: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsubset_type,type,
% 0.16/0.39 singletonsubset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsubset,definition,
% 0.16/0.39 ( singletonsubset
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletoninpowerset_type,type,
% 0.16/0.39 singletoninpowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletoninpowerset,definition,
% 0.16/0.39 ( singletoninpowerset
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletoninpowunion_type,type,
% 0.16/0.39 singletoninpowunion: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletoninpowunion,definition,
% 0.16/0.39 ( singletoninpowunion
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairset2E_type,type,
% 0.16/0.39 upairset2E: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairset2E,definition,
% 0.16/0.39 ( upairset2E
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.39 => ( ( Xz = Xx )
% 0.16/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairsubunion_type,type,
% 0.16/0.39 upairsubunion: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsubunion,definition,
% 0.16/0.39 ( upairsubunion
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairinpowunion_type,type,
% 0.16/0.39 upairinpowunion: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairinpowunion,definition,
% 0.16/0.39 ( upairinpowunion
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem1_type,type,
% 0.16/0.39 ubforcartprodlem1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem1,definition,
% 0.16/0.39 ( ubforcartprodlem1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem2_type,type,
% 0.16/0.39 ubforcartprodlem2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem2,definition,
% 0.16/0.39 ( ubforcartprodlem2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem3_type,type,
% 0.16/0.39 ubforcartprodlem3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem3,definition,
% 0.16/0.39 ( ubforcartprodlem3
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairin_type,type,
% 0.16/0.39 cartprodpairin: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairin,definition,
% 0.16/0.39 ( cartprodpairin
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempair1_type,type,
% 0.16/0.39 cartprodmempair1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempair1,definition,
% 0.16/0.39 ( cartprodmempair1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ? [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 & ( Xu
% 0.16/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempair_type,type,
% 0.16/0.39 cartprodmempair: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempair,definition,
% 0.16/0.39 ( cartprodmempair
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE2_type,type,
% 0.16/0.39 setunionE2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE2,definition,
% 0.16/0.39 ( setunionE2
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.39 => ? [X: $i] :
% 0.16/0.39 ( ( in @ X @ A )
% 0.16/0.39 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton1_type,type,
% 0.16/0.39 setunionsingleton1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton1,definition,
% 0.16/0.39 ( setunionsingleton1
% 0.16/0.39 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton2_type,type,
% 0.16/0.39 setunionsingleton2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton2,definition,
% 0.16/0.39 ( setunionsingleton2
% 0.16/0.39 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton_type,type,
% 0.16/0.39 setunionsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton,definition,
% 0.16/0.39 ( setunionsingleton
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.39 = Xx ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singleton_type,type,
% 0.16/0.39 singleton: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(singleton,definition,
% 0.16/0.39 ( singleton
% 0.16/0.39 = ( ^ [A: $i] :
% 0.16/0.39 ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( A
% 0.16/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletonprop_type,type,
% 0.16/0.39 singletonprop: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletonprop,definition,
% 0.16/0.39 ( singletonprop
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) )
% 0.16/0.39 => ( ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ( singleton
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1_type,type,
% 0.16/0.39 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1,definition,
% 0.16/0.39 ( ex1
% 0.16/0.39 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( singleton
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1E1_type,type,
% 0.16/0.39 ex1E1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1E1,definition,
% 0.16/0.39 ( ex1E1
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ( ex1 @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1I_type,type,
% 0.16/0.39 ex1I: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1I,definition,
% 0.16/0.39 ( ex1I
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xy = Xx ) ) )
% 0.16/0.39 => ( ex1 @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1I2_type,type,
% 0.16/0.39 ex1I2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1I2,definition,
% 0.16/0.39 ( ex1I2
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) )
% 0.16/0.39 => ( ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ( ex1 @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsuniq_type,type,
% 0.16/0.39 singletonsuniq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsuniq,definition,
% 0.16/0.39 ( singletonsuniq
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.39 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(atmost1p_type,type,
% 0.16/0.39 atmost1p: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(atleast2p_type,type,
% 0.16/0.39 atleast2p: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(atmost2p_type,type,
% 0.16/0.39 atmost2p: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetp_type,type,
% 0.16/0.39 upairsetp: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL1_type,type,
% 0.16/0.39 setukpairinjL1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL1,definition,
% 0.16/0.39 ( setukpairinjL1
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.39 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kfstsingleton_type,type,
% 0.16/0.39 kfstsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kfstsingleton,definition,
% 0.16/0.39 ( kfstsingleton
% 0.16/0.39 = ( ! [Xu: $i] :
% 0.16/0.39 ( ( iskpair @ Xu )
% 0.16/0.39 => ( singleton
% 0.16/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.39 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(theprop_type,type,
% 0.16/0.39 theprop: $o ).
% 0.16/0.39
% 0.16/0.39 thf(theprop,definition,
% 0.16/0.39 ( theprop
% 0.16/0.39 = ( ! [X: $i] :
% 0.16/0.39 ( ( singleton @ X )
% 0.16/0.39 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kfst_type,type,
% 0.16/0.39 kfst: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(kfstpairEq_type,type,
% 0.16/0.39 kfstpairEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kfstpairEq,definition,
% 0.16/0.39 ( kfstpairEq
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.39 = Xx ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodfstin_type,type,
% 0.16/0.39 cartprodfstin: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodfstin,definition,
% 0.16/0.39 ( cartprodfstin
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL2_type,type,
% 0.16/0.39 setukpairinjL2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL2,definition,
% 0.16/0.39 ( setukpairinjL2
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL_type,type,
% 0.16/0.39 setukpairinjL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL,definition,
% 0.16/0.39 ( setukpairinjL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.39 = ( kpair @ Xz @ Xu ) )
% 0.16/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR11_type,type,
% 0.16/0.39 setukpairinjR11: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR11,definition,
% 0.16/0.39 ( setukpairinjR11
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR12_type,type,
% 0.16/0.39 setukpairinjR12: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR12,definition,
% 0.16/0.39 ( setukpairinjR12
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR1_type,type,
% 0.16/0.39 setukpairinjR1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR1,definition,
% 0.16/0.39 ( setukpairinjR1
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.39 => ( ( Xz = Xu )
% 0.16/0.39 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairequniteq_type,type,
% 0.16/0.39 upairequniteq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairequniteq,definition,
% 0.16/0.39 ( upairequniteq
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.39 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR2_type,type,
% 0.16/0.39 setukpairinjR2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR2,definition,
% 0.16/0.39 ( setukpairinjR2
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR_type,type,
% 0.16/0.39 setukpairinjR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR,definition,
% 0.16/0.39 ( setukpairinjR
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.39 = ( kpair @ Xz @ Xu ) )
% 0.16/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ksndsingleton_type,type,
% 0.16/0.39 ksndsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ksndsingleton,definition,
% 0.16/0.39 ( ksndsingleton
% 0.16/0.39 = ( ! [Xu: $i] :
% 0.16/0.39 ( ( iskpair @ Xu )
% 0.16/0.39 => ( singleton
% 0.16/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.39 @ ^ [Xx: $i] :
% 0.16/0.39 ( Xu
% 0.16/0.39 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ksnd_type,type,
% 0.16/0.39 ksnd: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(ksndpairEq_type,type,
% 0.16/0.39 ksndpairEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ksndpairEq,definition,
% 0.16/0.39 ( ksndpairEq
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.39 = Xy ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kpairsurjEq_type,type,
% 0.16/0.39 kpairsurjEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kpairsurjEq,definition,
% 0.16/0.39 ( kpairsurjEq
% 0.16/0.39 = ( ! [Xu: $i] :
% 0.16/0.39 ( ( iskpair @ Xu )
% 0.16/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.39 = Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodsndin_type,type,
% 0.16/0.39 cartprodsndin: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodsndin,definition,
% 0.16/0.39 ( cartprodsndin
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairmemEL_type,type,
% 0.16/0.39 cartprodpairmemEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairmemEL,definition,
% 0.16/0.39 ( cartprodpairmemEL
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairmemER_type,type,
% 0.16/0.39 cartprodpairmemER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairmemER,definition,
% 0.16/0.39 ( cartprodpairmemER
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempaircEq_type,type,
% 0.16/0.39 cartprodmempaircEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempaircEq,definition,
% 0.16/0.39 ( cartprodmempaircEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( kpair @ Xx @ Xy )
% 0.16/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodfstpairEq_type,type,
% 0.16/0.39 cartprodfstpairEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodfstpairEq,definition,
% 0.16/0.39 ( cartprodfstpairEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.39 = Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodsndpairEq_type,type,
% 0.16/0.39 cartprodsndpairEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodsndpairEq,definition,
% 0.16/0.39 ( cartprodsndpairEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.39 = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairsurjEq_type,type,
% 0.16/0.39 cartprodpairsurjEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairsurjEq,definition,
% 0.16/0.39 ( cartprodpairsurjEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.39 = Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(breln_type,type,
% 0.16/0.39 breln: $i > $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(breln,definition,
% 0.16/0.39 ( breln
% 0.16/0.39 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstr_type,type,
% 0.16/0.39 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrI_type,type,
% 0.16/0.39 dpsetconstrI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrI,definition,
% 0.16/0.39 ( dpsetconstrI
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( Xphi @ Xx @ Xy )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrSub_type,type,
% 0.16/0.39 dpsetconstrSub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrSub,definition,
% 0.16/0.39 ( dpsetconstrSub
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.39 ( subset
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.16/0.39 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setOfPairsIsBReln_type,type,
% 0.16/0.39 setOfPairsIsBReln: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setOfPairsIsBReln,definition,
% 0.16/0.39 ( setOfPairsIsBReln
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.39 ( breln @ A @ B
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrERa_type,type,
% 0.16/0.39 dpsetconstrERa: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrERa,definition,
% 0.16/0.39 ( dpsetconstrERa
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrEL1_type,type,
% 0.16/0.39 dpsetconstrEL1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrEL1,definition,
% 0.16/0.39 ( dpsetconstrEL1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrEL2_type,type,
% 0.16/0.39 dpsetconstrEL2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrEL2,definition,
% 0.16/0.39 ( dpsetconstrEL2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrER_type,type,
% 0.16/0.39 dpsetconstrER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrER,definition,
% 0.16/0.39 ( dpsetconstrER
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(func_type,type,
% 0.16/0.39 func: $i > $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(func,definition,
% 0.16/0.39 ( func
% 0.16/0.39 = ( ^ [A: $i,B: $i,R: $i] :
% 0.16/0.39 ( ( breln @ A @ B @ R )
% 0.16/0.39 & ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ex1 @ B
% 0.16/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcSet_type,type,
% 0.16/0.39 funcSet: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(funcSet,definition,
% 0.16/0.39 ( funcSet
% 0.16/0.39 = ( ^ [A: $i,B: $i] :
% 0.16/0.39 ( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
% 0.16/0.39 @ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcImageSingleton_type,type,
% 0.16/0.39 funcImageSingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcImageSingleton,definition,
% 0.16/0.39 ( funcImageSingleton
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( singleton
% 0.16/0.39 @ ( dsetconstr @ B
% 0.16/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(apProp,conjecture,
% 0.16/0.39 ( setextAx
% 0.16/0.39 => ( emptysetAx
% 0.16/0.39 => ( setadjoinAx
% 0.16/0.39 => ( powersetAx
% 0.16/0.39 => ( setunionAx
% 0.16/0.39 => ( omega0Ax
% 0.16/0.39 => ( omegaSAx
% 0.16/0.39 => ( omegaIndAx
% 0.16/0.39 => ( replAx
% 0.16/0.39 => ( foundationAx
% 0.16/0.39 => ( wellorderingAx
% 0.16/0.39 => ( descrp
% 0.16/0.39 => ( dsetconstrI
% 0.16/0.39 => ( dsetconstrEL
% 0.16/0.39 => ( dsetconstrER
% 0.16/0.39 => ( exuE1
% 0.16/0.39 => ( prop2setE
% 0.16/0.39 => ( emptysetE
% 0.16/0.39 => ( emptysetimpfalse
% 0.16/0.39 => ( notinemptyset
% 0.16/0.39 => ( exuE3e
% 0.16/0.39 => ( setext
% 0.16/0.39 => ( emptyI
% 0.16/0.39 => ( noeltsimpempty
% 0.16/0.39 => ( setbeta
% 0.16/0.39 => ( nonemptyE1
% 0.16/0.39 => ( nonemptyI
% 0.16/0.39 => ( nonemptyI1
% 0.16/0.39 => ( setadjoinIL
% 0.16/0.39 => ( emptyinunitempty
% 0.16/0.39 => ( setadjoinIR
% 0.16/0.39 => ( setadjoinE
% 0.16/0.39 => ( setadjoinOr
% 0.16/0.39 => ( setoftrueEq
% 0.16/0.39 => ( powersetI
% 0.16/0.39 => ( emptyinPowerset
% 0.16/0.39 => ( emptyInPowerset
% 0.16/0.39 => ( powersetE
% 0.16/0.39 => ( setunionI
% 0.16/0.39 => ( setunionE
% 0.16/0.39 => ( subPowSU
% 0.16/0.39 => ( exuE2
% 0.16/0.39 => ( nonemptyImpWitness
% 0.16/0.39 => ( uniqinunit
% 0.16/0.39 => ( notinsingleton
% 0.16/0.39 => ( eqinunit
% 0.16/0.39 => ( singletonsswitch
% 0.16/0.39 => ( upairsetE
% 0.16/0.39 => ( upairsetIL
% 0.16/0.39 => ( upairsetIR
% 0.16/0.39 => ( emptyE1
% 0.16/0.39 => ( vacuousDall
% 0.16/0.39 => ( quantDeMorgan1
% 0.16/0.39 => ( quantDeMorgan2
% 0.16/0.39 => ( quantDeMorgan3
% 0.16/0.39 => ( quantDeMorgan4
% 0.16/0.39 => ( prop2setI
% 0.16/0.39 => ( prop2set2propI
% 0.16/0.39 => ( notdexE
% 0.16/0.39 => ( notdallE
% 0.16/0.39 => ( exuI1
% 0.16/0.39 => ( exuI3
% 0.16/0.39 => ( exuI2
% 0.16/0.39 => ( inCongP
% 0.16/0.39 => ( in__Cong
% 0.16/0.39 => ( exuE3u
% 0.16/0.39 => ( exu__Cong
% 0.16/0.39 => ( emptyset__Cong
% 0.16/0.39 => ( setadjoin__Cong
% 0.16/0.39 => ( powerset__Cong
% 0.16/0.39 => ( setunion__Cong
% 0.16/0.39 => ( omega__Cong
% 0.16/0.39 => ( exuEu
% 0.16/0.39 => ( descr__Cong
% 0.16/0.39 => ( dsetconstr__Cong
% 0.16/0.39 => ( subsetI1
% 0.16/0.39 => ( eqimpsubset2
% 0.16/0.39 => ( eqimpsubset1
% 0.16/0.39 => ( subsetI2
% 0.16/0.39 => ( emptysetsubset
% 0.16/0.39 => ( subsetE
% 0.16/0.39 => ( subsetE2
% 0.16/0.39 => ( notsubsetI
% 0.16/0.39 => ( notequalI1
% 0.16/0.39 => ( notequalI2
% 0.16/0.39 => ( subsetRefl
% 0.16/0.39 => ( subsetTrans
% 0.16/0.39 => ( setadjoinSub
% 0.16/0.39 => ( setadjoinSub2
% 0.16/0.39 => ( subset2powerset
% 0.16/0.39 => ( setextsub
% 0.16/0.39 => ( subsetemptysetimpeq
% 0.16/0.39 => ( powersetI1
% 0.16/0.39 => ( powersetE1
% 0.16/0.39 => ( inPowerset
% 0.16/0.39 => ( powersetsubset
% 0.16/0.39 => ( sepInPowerset
% 0.16/0.39 => ( sepSubset
% 0.16/0.39 => ( binunionIL
% 0.16/0.39 => ( upairset2IR
% 0.16/0.39 => ( binunionIR
% 0.16/0.39 => ( binunionEcases
% 0.16/0.39 => ( binunionE
% 0.16/0.39 => ( binunionLsub
% 0.16/0.39 => ( binunionRsub
% 0.16/0.39 => ( binintersectI
% 0.16/0.39 => ( binintersectSubset5
% 0.16/0.39 => ( binintersectEL
% 0.16/0.39 => ( binintersectLsub
% 0.16/0.39 => ( binintersectSubset2
% 0.16/0.39 => ( binintersectSubset3
% 0.16/0.39 => ( binintersectER
% 0.16/0.39 => ( disjointsetsI1
% 0.16/0.39 => ( binintersectRsub
% 0.16/0.39 => ( binintersectSubset4
% 0.16/0.39 => ( binintersectSubset1
% 0.16/0.39 => ( bs114d
% 0.16/0.39 => ( setminusI
% 0.16/0.39 => ( setminusEL
% 0.16/0.39 => ( setminusER
% 0.16/0.39 => ( setminusSubset2
% 0.16/0.39 => ( setminusERneg
% 0.16/0.39 => ( setminusELneg
% 0.16/0.39 => ( setminusILneg
% 0.16/0.39 => ( setminusIRneg
% 0.16/0.39 => ( setminusLsub
% 0.16/0.39 => ( setminusSubset1
% 0.16/0.39 => ( symdiffE
% 0.16/0.39 => ( symdiffI1
% 0.16/0.39 => ( symdiffI2
% 0.16/0.39 => ( symdiffIneg1
% 0.16/0.39 => ( symdiffIneg2
% 0.16/0.39 => ( secondinupair
% 0.16/0.39 => ( setukpairIL
% 0.16/0.39 => ( setukpairIR
% 0.16/0.39 => ( kpairiskpair
% 0.16/0.39 => ( kpairp
% 0.16/0.39 => ( singletonsubset
% 0.16/0.39 => ( singletoninpowerset
% 0.16/0.39 => ( singletoninpowunion
% 0.16/0.39 => ( upairset2E
% 0.16/0.39 => ( upairsubunion
% 0.16/0.39 => ( upairinpowunion
% 0.16/0.39 => ( ubforcartprodlem1
% 0.16/0.39 => ( ubforcartprodlem2
% 0.16/0.39 => ( ubforcartprodlem3
% 0.16/0.39 => ( cartprodpairin
% 0.16/0.39 => ( cartprodmempair1
% 0.16/0.39 => ( cartprodmempair
% 0.16/0.39 => ( setunionE2
% 0.16/0.39 => ( setunionsingleton1
% 0.16/0.39 => ( setunionsingleton2
% 0.16/0.39 => ( setunionsingleton
% 0.16/0.39 => ( singletonprop
% 0.16/0.39 => ( ex1E1
% 0.16/0.39 => ( ex1I
% 0.16/0.39 => ( ex1I2
% 0.16/0.39 => ( singletonsuniq
% 0.16/0.39 => ( setukpairinjL1
% 0.16/0.39 => ( kfstsingleton
% 0.16/0.39 => ( theprop
% 0.16/0.39 => ( kfstpairEq
% 0.16/0.39 => ( cartprodfstin
% 0.16/0.39 => ( setukpairinjL2
% 0.16/0.39 => ( setukpairinjL
% 0.16/0.39 => ( setukpairinjR11
% 0.16/0.39 => ( setukpairinjR12
% 0.16/0.39 => ( setukpairinjR1
% 0.16/0.39 => ( upairequniteq
% 0.16/0.39 => ( setukpairinjR2
% 0.16/0.39 => ( setukpairinjR
% 0.16/0.39 => ( ksndsingleton
% 0.16/0.39 => ( ksndpairEq
% 0.16/0.39 => ( kpairsurjEq
% 0.16/0.39 => ( cartprodsndin
% 0.16/0.39 => ( cartprodpairmemEL
% 0.16/0.39 => ( cartprodpairmemER
% 0.16/0.39 => ( cartprodmempaircEq
% 0.16/0.39 => ( cartprodfstpairEq
% 0.16/0.39 => ( cartprodsndpairEq
% 0.16/0.39 => ( cartprodpairsurjEq
% 0.16/0.39 => ( dpsetconstrI
% 0.16/0.39 => ( dpsetconstrSub
% 0.16/0.39 => ( setOfPairsIsBReln
% 0.16/0.39 => ( dpsetconstrERa
% 0.16/0.39 => ( dpsetconstrEL1
% 0.16/0.39 => ( dpsetconstrEL2
% 0.16/0.39 => ( dpsetconstrER
% 0.16/0.39 => ( funcImageSingleton
% 0.16/0.39 => ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in
% 0.16/0.39 @ ( setunion
% 0.16/0.39 @ ( dsetconstr @ B
% 0.16/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.16/0.39 @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 ------------------------
% 2.59/1.59 ---- Embedded in THF ---
% 2.59/1.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.59/1.59 %%% Generated on Tue Feb 24 12:30:18 EST 2026
% 2.59/1.59 %%% using '$$dhol' embedding, version 1.3.0.
% 2.59/1.59 %%% Logic specification used:
% 2.59/1.59 %%% thf(spec, logic, $$dhol).
% 2.59/1.59
% 2.59/1.59 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2DTF19443.p
% See solution above
% 2.59/1.59 ------------------------
% 2.84/1.63 ---- Cleaned THF ---
% 2.84/1.63 thf(in_type,type,
% 2.84/1.63 in: $i > $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(exu_type,type,
% 2.84/1.63 exu: ( $i > $o ) > $o ).
% 2.84/1.63
% 2.84/1.63 thf(exu,definition,
% 2.84/1.63 ( exu
% 2.84/1.63 = ( ^ [Xphi: $i > $o] :
% 2.84/1.63 ? [Xx: $i] :
% 2.84/1.63 ( ( Xphi @ Xx )
% 2.84/1.63 & ! [Xy: $i] :
% 2.84/1.63 ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setextAx_type,type,
% 2.84/1.63 setextAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setextAx,definition,
% 2.84/1.63 ( setextAx
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 <=> ( in @ Xx @ B ) )
% 2.84/1.63 => ( A = B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptyset_type,type,
% 2.84/1.63 emptyset: $i ).
% 2.84/1.63
% 2.84/1.63 thf(emptysetAx_type,type,
% 2.84/1.63 emptysetAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptysetAx,definition,
% 2.84/1.63 ( emptysetAx
% 2.84/1.63 = ( ! [Xx: $i] :
% 2.84/1.63 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoin_type,type,
% 2.84/1.63 setadjoin: $i > $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinAx_type,type,
% 2.84/1.63 setadjoinAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinAx,definition,
% 2.84/1.63 ( setadjoinAx
% 2.84/1.63 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.63 <=> ( ( Xy = Xx )
% 2.84/1.63 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(powerset_type,type,
% 2.84/1.63 powerset: $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(powersetAx_type,type,
% 2.84/1.63 powersetAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(powersetAx,definition,
% 2.84/1.63 ( powersetAx
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.63 <=> ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setunion_type,type,
% 2.84/1.63 setunion: $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(setunionAx_type,type,
% 2.84/1.63 setunionAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setunionAx,definition,
% 2.84/1.63 ( setunionAx
% 2.84/1.63 = ( ! [A: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.63 <=> ? [B: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 & ( in @ B @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(omega_type,type,
% 2.84/1.63 omega: $i ).
% 2.84/1.63
% 2.84/1.63 thf(omega0Ax_type,type,
% 2.84/1.63 omega0Ax: $o ).
% 2.84/1.63
% 2.84/1.63 thf(omega0Ax,definition,
% 2.84/1.63 ( omega0Ax
% 2.84/1.63 = ( in @ emptyset @ omega ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(omegaSAx_type,type,
% 2.84/1.63 omegaSAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(omegaSAx,definition,
% 2.84/1.63 ( omegaSAx
% 2.84/1.63 = ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ omega )
% 2.84/1.63 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(omegaIndAx_type,type,
% 2.84/1.63 omegaIndAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(omegaIndAx,definition,
% 2.84/1.63 ( omegaIndAx
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ( ( in @ emptyset @ A )
% 2.84/1.63 & ! [Xx: $i] :
% 2.84/1.63 ( ( ( in @ Xx @ omega )
% 2.84/1.63 & ( in @ Xx @ A ) )
% 2.84/1.63 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.84/1.63 => ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ omega )
% 2.84/1.63 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(replAx_type,type,
% 2.84/1.63 replAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(replAx,definition,
% 2.84/1.63 ( replAx
% 2.84/1.63 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( exu
% 2.84/1.63 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.84/1.63 => ? [B: $i] :
% 2.84/1.63 ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 <=> ? [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ A )
% 2.84/1.63 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(foundationAx_type,type,
% 2.84/1.63 foundationAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(foundationAx,definition,
% 2.84/1.63 ( foundationAx
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.84/1.63 => ? [B: $i] :
% 2.84/1.63 ( ( in @ B @ A )
% 2.84/1.63 & ~ ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(wellorderingAx_type,type,
% 2.84/1.63 wellorderingAx: $o ).
% 2.84/1.63
% 2.84/1.63 thf(wellorderingAx,definition,
% 2.84/1.63 ( wellorderingAx
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ? [B: $i] :
% 2.84/1.63 ( ! [C: $i] :
% 2.84/1.63 ( ( in @ C @ B )
% 2.84/1.63 => ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ C )
% 2.84/1.63 => ( in @ Xx @ A ) ) )
% 2.84/1.63 & ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( ( in @ Xx @ A )
% 2.84/1.63 & ( in @ Xy @ A ) )
% 2.84/1.63 => ( ! [C: $i] :
% 2.84/1.63 ( ( in @ C @ B )
% 2.84/1.63 => ( ( in @ Xx @ C )
% 2.84/1.63 <=> ( in @ Xy @ C ) ) )
% 2.84/1.63 => ( Xx = Xy ) ) )
% 2.84/1.63 & ! [C: $i,D: $i] :
% 2.84/1.63 ( ( ( in @ C @ B )
% 2.84/1.63 & ( in @ D @ B ) )
% 2.84/1.63 => ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ C )
% 2.84/1.63 => ( in @ Xx @ D ) )
% 2.84/1.63 | ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ D )
% 2.84/1.63 => ( in @ Xx @ C ) ) ) )
% 2.84/1.63 & ! [C: $i] :
% 2.84/1.63 ( ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ C )
% 2.84/1.63 => ( in @ Xx @ A ) )
% 2.84/1.63 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.84/1.63 => ? [D: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ D @ B )
% 2.84/1.63 & ( in @ Xx @ C )
% 2.84/1.63 & ~ ? [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ D )
% 2.84/1.63 & ( in @ Xy @ C ) )
% 2.84/1.63 & ! [E: $i] :
% 2.84/1.63 ( ( in @ E @ B )
% 2.84/1.63 => ( ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ E )
% 2.84/1.63 => ( in @ Xy @ D ) )
% 2.84/1.63 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(descr_type,type,
% 2.84/1.63 descr: ( $i > $o ) > $i ).
% 2.84/1.63
% 2.84/1.63 thf(descrp_type,type,
% 2.84/1.63 descrp: $o ).
% 2.84/1.63
% 2.84/1.63 thf(descrp,definition,
% 2.84/1.63 ( descrp
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 => ( Xphi
% 2.84/1.63 @ ( descr
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstr_type,type,
% 2.84/1.63 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstrI_type,type,
% 2.84/1.63 dsetconstrI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstrI,definition,
% 2.84/1.63 ( dsetconstrI
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ( Xphi @ Xx )
% 2.84/1.63 => ( in @ Xx
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstrEL_type,type,
% 2.84/1.63 dsetconstrEL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstrEL,definition,
% 2.84/1.63 ( dsetconstrEL
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstrER_type,type,
% 2.84/1.63 dsetconstrER: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstrER,definition,
% 2.84/1.63 ( dsetconstrER
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.63 => ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exuE1_type,type,
% 2.84/1.63 exuE1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exuE1,definition,
% 2.84/1.63 ( exuE1
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 => ? [Xx: $i] :
% 2.84/1.63 ( ( Xphi @ Xx )
% 2.84/1.63 & ! [Xy: $i] :
% 2.84/1.63 ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(prop2set_type,type,
% 2.84/1.63 prop2set: $o > $i ).
% 2.84/1.63
% 2.84/1.63 thf(prop2setE_type,type,
% 2.84/1.63 prop2setE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(prop2setE,definition,
% 2.84/1.63 ( prop2setE
% 2.84/1.63 = ( ! [Xphi: $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.84/1.63 => Xphi ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptysetE_type,type,
% 2.84/1.63 emptysetE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptysetE,definition,
% 2.84/1.63 ( emptysetE
% 2.84/1.63 = ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ emptyset )
% 2.84/1.63 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptysetimpfalse_type,type,
% 2.84/1.63 emptysetimpfalse: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptysetimpfalse,definition,
% 2.84/1.63 ( emptysetimpfalse
% 2.84/1.63 = ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ emptyset )
% 2.84/1.63 => $false ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(notinemptyset_type,type,
% 2.84/1.63 notinemptyset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(notinemptyset,definition,
% 2.84/1.63 ( notinemptyset
% 2.84/1.63 = ( ! [Xx: $i] :
% 2.84/1.63 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exuE3e_type,type,
% 2.84/1.63 exuE3e: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exuE3e,definition,
% 2.84/1.63 ( exuE3e
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setext_type,type,
% 2.84/1.63 setext: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setext,definition,
% 2.84/1.63 ( setext
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ Xx @ B ) )
% 2.84/1.63 => ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ A ) )
% 2.84/1.63 => ( A = B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptyI_type,type,
% 2.84/1.63 emptyI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptyI,definition,
% 2.84/1.63 ( emptyI
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ~ ( in @ Xx @ A )
% 2.84/1.63 => ( A = emptyset ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(noeltsimpempty_type,type,
% 2.84/1.63 noeltsimpempty: $o ).
% 2.84/1.63
% 2.84/1.63 thf(noeltsimpempty,definition,
% 2.84/1.63 ( noeltsimpempty
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ~ ( in @ Xx @ A )
% 2.84/1.63 => ( A = emptyset ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setbeta_type,type,
% 2.84/1.63 setbeta: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setbeta,definition,
% 2.84/1.63 ( setbeta
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ( in @ Xx
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.84/1.63 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(nonempty_type,type,
% 2.84/1.63 nonempty: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(nonempty,definition,
% 2.84/1.63 ( nonempty
% 2.84/1.63 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(nonemptyE1_type,type,
% 2.84/1.63 nonemptyE1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(nonemptyE1,definition,
% 2.84/1.63 ( nonemptyE1
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ( nonempty @ A )
% 2.84/1.63 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(nonemptyI_type,type,
% 2.84/1.63 nonemptyI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(nonemptyI,definition,
% 2.84/1.63 ( nonemptyI
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ( Xphi @ Xx )
% 2.84/1.63 => ( nonempty
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(nonemptyI1_type,type,
% 2.84/1.63 nonemptyI1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(nonemptyI1,definition,
% 2.84/1.63 ( nonemptyI1
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.84/1.63 => ( nonempty @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinIL_type,type,
% 2.84/1.63 setadjoinIL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinIL,definition,
% 2.84/1.63 ( setadjoinIL
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptyinunitempty_type,type,
% 2.84/1.63 emptyinunitempty: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptyinunitempty,definition,
% 2.84/1.63 ( emptyinunitempty
% 2.84/1.63 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinIR_type,type,
% 2.84/1.63 setadjoinIR: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinIR,definition,
% 2.84/1.63 ( setadjoinIR
% 2.84/1.63 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ A )
% 2.84/1.63 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinE_type,type,
% 2.84/1.63 setadjoinE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinE,definition,
% 2.84/1.63 ( setadjoinE
% 2.84/1.63 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.63 => ! [Xphi: $o] :
% 2.84/1.63 ( ( ( Xy = Xx )
% 2.84/1.63 => Xphi )
% 2.84/1.63 => ( ( ( in @ Xy @ A )
% 2.84/1.63 => Xphi )
% 2.84/1.63 => Xphi ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinOr_type,type,
% 2.84/1.63 setadjoinOr: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinOr,definition,
% 2.84/1.63 ( setadjoinOr
% 2.84/1.63 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.84/1.63 => ( ( Xy = Xx )
% 2.84/1.63 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setoftrueEq_type,type,
% 2.84/1.63 setoftrueEq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setoftrueEq,definition,
% 2.84/1.63 ( setoftrueEq
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xx: $i] : $true )
% 2.84/1.63 = A ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(powersetI_type,type,
% 2.84/1.63 powersetI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(powersetI,definition,
% 2.84/1.63 ( powersetI
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ A ) )
% 2.84/1.63 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptyinPowerset_type,type,
% 2.84/1.63 emptyinPowerset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptyinPowerset,definition,
% 2.84/1.63 ( emptyinPowerset
% 2.84/1.63 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptyInPowerset_type,type,
% 2.84/1.63 emptyInPowerset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptyInPowerset,definition,
% 2.84/1.63 ( emptyInPowerset
% 2.84/1.63 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(powersetE_type,type,
% 2.84/1.63 powersetE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(powersetE,definition,
% 2.84/1.63 ( powersetE
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.63 => ( ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setunionI_type,type,
% 2.84/1.63 setunionI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setunionI,definition,
% 2.84/1.63 ( setunionI
% 2.84/1.63 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 => ( ( in @ B @ A )
% 2.84/1.63 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setunionE_type,type,
% 2.84/1.63 setunionE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setunionE,definition,
% 2.84/1.63 ( setunionE
% 2.84/1.63 = ( ! [A: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.63 => ! [Xphi: $o] :
% 2.84/1.63 ( ! [B: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 => ( ( in @ B @ A )
% 2.84/1.63 => Xphi ) )
% 2.84/1.63 => Xphi ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subPowSU_type,type,
% 2.84/1.63 subPowSU: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subPowSU,definition,
% 2.84/1.63 ( subPowSU
% 2.84/1.63 = ( ! [A: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exuE2_type,type,
% 2.84/1.63 exuE2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exuE2,definition,
% 2.84/1.63 ( exuE2
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 => ? [Xx: $i] :
% 2.84/1.63 ! [Xy: $i] :
% 2.84/1.63 ( ( Xphi @ Xy )
% 2.84/1.63 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(nonemptyImpWitness_type,type,
% 2.84/1.63 nonemptyImpWitness: $o ).
% 2.84/1.63
% 2.84/1.63 thf(nonemptyImpWitness,definition,
% 2.84/1.63 ( nonemptyImpWitness
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ( nonempty @ A )
% 2.84/1.63 => ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & $true ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(uniqinunit_type,type,
% 2.84/1.63 uniqinunit: $o ).
% 2.84/1.63
% 2.84/1.63 thf(uniqinunit,definition,
% 2.84/1.63 ( uniqinunit
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(notinsingleton_type,type,
% 2.84/1.63 notinsingleton: $o ).
% 2.84/1.63
% 2.84/1.63 thf(notinsingleton,definition,
% 2.84/1.63 ( notinsingleton
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx != Xy )
% 2.84/1.63 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(eqinunit_type,type,
% 2.84/1.63 eqinunit: $o ).
% 2.84/1.63
% 2.84/1.63 thf(eqinunit,definition,
% 2.84/1.63 ( eqinunit
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx = Xy )
% 2.84/1.63 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(singletonsswitch_type,type,
% 2.84/1.63 singletonsswitch: $o ).
% 2.84/1.63
% 2.84/1.63 thf(singletonsswitch,definition,
% 2.84/1.63 ( singletonsswitch
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.63 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(upairsetE_type,type,
% 2.84/1.63 upairsetE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairsetE,definition,
% 2.84/1.63 ( upairsetE
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.63 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.84/1.63 => ( ( Xz = Xx )
% 2.84/1.63 | ( Xz = Xy ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(upairsetIL_type,type,
% 2.84/1.63 upairsetIL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairsetIL,definition,
% 2.84/1.63 ( upairsetIL
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(upairsetIR_type,type,
% 2.84/1.63 upairsetIR: $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairsetIR,definition,
% 2.84/1.63 ( upairsetIR
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptyE1_type,type,
% 2.84/1.63 emptyE1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptyE1,definition,
% 2.84/1.63 ( emptyE1
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( Xphi @ Xx ) )
% 2.84/1.63 => ( ( ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 = emptyset )
% 2.84/1.63 => $false ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(vacuousDall_type,type,
% 2.84/1.63 vacuousDall: $o ).
% 2.84/1.63
% 2.84/1.63 thf(vacuousDall,definition,
% 2.84/1.63 ( vacuousDall
% 2.84/1.63 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ emptyset )
% 2.84/1.63 => ( Xphi @ Xx ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(quantDeMorgan1_type,type,
% 2.84/1.63 quantDeMorgan1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(quantDeMorgan1,definition,
% 2.84/1.63 ( quantDeMorgan1
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ~ ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( Xphi @ Xx ) )
% 2.84/1.63 => ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(quantDeMorgan2_type,type,
% 2.84/1.63 quantDeMorgan2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(quantDeMorgan2,definition,
% 2.84/1.63 ( quantDeMorgan2
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ~ ( Xphi @ Xx ) )
% 2.84/1.63 => ~ ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(quantDeMorgan3_type,type,
% 2.84/1.63 quantDeMorgan3: $o ).
% 2.84/1.63
% 2.84/1.63 thf(quantDeMorgan3,definition,
% 2.84/1.63 ( quantDeMorgan3
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ~ ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( Xphi @ Xx ) )
% 2.84/1.63 => ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(quantDeMorgan4_type,type,
% 2.84/1.63 quantDeMorgan4: $o ).
% 2.84/1.63
% 2.84/1.63 thf(quantDeMorgan4,definition,
% 2.84/1.63 ( quantDeMorgan4
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ~ ( Xphi @ Xx ) )
% 2.84/1.63 => ~ ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(prop2setI_type,type,
% 2.84/1.63 prop2setI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(prop2setI,definition,
% 2.84/1.63 ( prop2setI
% 2.84/1.63 = ( ! [Xphi: $o] :
% 2.84/1.63 ( Xphi
% 2.84/1.63 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(set2prop_type,type,
% 2.84/1.63 set2prop: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(prop2set2propI_type,type,
% 2.84/1.63 prop2set2propI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(prop2set2propI,definition,
% 2.84/1.63 ( prop2set2propI
% 2.84/1.63 = ( ! [Xphi: $o] :
% 2.84/1.63 ( Xphi
% 2.84/1.63 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(notdexE_type,type,
% 2.84/1.63 notdexE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(notdexE,definition,
% 2.84/1.63 ( notdexE
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ~ ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( Xphi @ Xx ) )
% 2.84/1.63 => ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(notdallE_type,type,
% 2.84/1.63 notdallE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(notdallE,definition,
% 2.84/1.63 ( notdallE
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ~ ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( Xphi @ Xx ) )
% 2.84/1.63 => ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exuI1_type,type,
% 2.84/1.63 exuI1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exuI1,definition,
% 2.84/1.63 ( exuI1
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ? [Xx: $i] :
% 2.84/1.63 ( ( Xphi @ Xx )
% 2.84/1.63 & ! [Xy: $i] :
% 2.84/1.63 ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xx = Xy ) ) )
% 2.84/1.63 => ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exuI3_type,type,
% 2.84/1.63 exuI3: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exuI3,definition,
% 2.84/1.63 ( exuI3
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.84/1.63 => ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xphi @ Xx )
% 2.84/1.63 => ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xx = Xy ) ) )
% 2.84/1.63 => ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exuI2_type,type,
% 2.84/1.63 exuI2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exuI2,definition,
% 2.84/1.63 ( exuI2
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ? [Xx: $i] :
% 2.84/1.63 ! [Xy: $i] :
% 2.84/1.63 ( ( Xphi @ Xy )
% 2.84/1.63 <=> ( Xy = Xx ) )
% 2.84/1.63 => ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(inCongP_type,type,
% 2.84/1.63 inCongP: $o ).
% 2.84/1.63
% 2.84/1.63 thf(inCongP,definition,
% 2.84/1.63 ( inCongP
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( A = B )
% 2.84/1.63 => ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx = Xy )
% 2.84/1.63 => ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(in__Cong_type,type,
% 2.84/1.63 in__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(in__Cong,definition,
% 2.84/1.63 ( in__Cong
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( A = B )
% 2.84/1.63 => ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx = Xy )
% 2.84/1.63 => ( ( in @ Xx @ A )
% 2.84/1.63 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exuE3u_type,type,
% 2.84/1.63 exuE3u: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exuE3u,definition,
% 2.84/1.63 ( exuE3u
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 => ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xphi @ Xx )
% 2.84/1.63 => ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exu__Cong_type,type,
% 2.84/1.63 exu__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exu__Cong,definition,
% 2.84/1.63 ( exu__Cong
% 2.84/1.63 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.63 ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx = Xy )
% 2.84/1.63 => ( ( Xphi @ Xx )
% 2.84/1.63 <=> ( Xpsi @ Xy ) ) )
% 2.84/1.63 => ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 <=> ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptyset__Cong_type,type,
% 2.84/1.63 emptyset__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptyset__Cong,definition,
% 2.84/1.63 ( emptyset__Cong
% 2.84/1.63 = ( emptyset = emptyset ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoin__Cong_type,type,
% 2.84/1.63 setadjoin__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoin__Cong,definition,
% 2.84/1.63 ( setadjoin__Cong
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx = Xy )
% 2.84/1.63 => ! [Xz: $i,Xu: $i] :
% 2.84/1.63 ( ( Xz = Xu )
% 2.84/1.63 => ( ( setadjoin @ Xx @ Xz )
% 2.84/1.63 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(powerset__Cong_type,type,
% 2.84/1.63 powerset__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(powerset__Cong,definition,
% 2.84/1.63 ( powerset__Cong
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( A = B )
% 2.84/1.63 => ( ( powerset @ A )
% 2.84/1.63 = ( powerset @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setunion__Cong_type,type,
% 2.84/1.63 setunion__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setunion__Cong,definition,
% 2.84/1.63 ( setunion__Cong
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( A = B )
% 2.84/1.63 => ( ( setunion @ A )
% 2.84/1.63 = ( setunion @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(omega__Cong_type,type,
% 2.84/1.63 omega__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(omega__Cong,definition,
% 2.84/1.63 ( omega__Cong
% 2.84/1.63 = ( omega = omega ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(exuEu_type,type,
% 2.84/1.63 exuEu: $o ).
% 2.84/1.63
% 2.84/1.63 thf(exuEu,definition,
% 2.84/1.63 ( exuEu
% 2.84/1.63 = ( ! [Xphi: $i > $o] :
% 2.84/1.63 ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 => ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xphi @ Xx )
% 2.84/1.63 => ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(descr__Cong_type,type,
% 2.84/1.63 descr__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(descr__Cong,definition,
% 2.84/1.63 ( descr__Cong
% 2.84/1.63 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.63 ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx = Xy )
% 2.84/1.63 => ( ( Xphi @ Xx )
% 2.84/1.63 <=> ( Xpsi @ Xy ) ) )
% 2.84/1.63 => ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 => ( ( exu
% 2.84/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.84/1.63 => ( ( descr
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 = ( descr
% 2.84/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstr__Cong_type,type,
% 2.84/1.63 dsetconstr__Cong: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dsetconstr__Cong,definition,
% 2.84/1.63 ( dsetconstr__Cong
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( A = B )
% 2.84/1.63 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( ( Xx = Xy )
% 2.84/1.63 => ( ( Xphi @ Xx )
% 2.84/1.63 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.84/1.63 => ( ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 = ( dsetconstr @ B
% 2.84/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subset_type,type,
% 2.84/1.63 subset: $i > $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(disjoint_type,type,
% 2.84/1.63 disjoint: $i > $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(setsmeet_type,type,
% 2.84/1.63 setsmeet: $i > $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(subsetI1_type,type,
% 2.84/1.63 subsetI1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subsetI1,definition,
% 2.84/1.63 ( subsetI1
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ Xx @ B ) )
% 2.84/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(eqimpsubset2_type,type,
% 2.84/1.63 eqimpsubset2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(eqimpsubset2,definition,
% 2.84/1.63 ( eqimpsubset2
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( A = B )
% 2.84/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(eqimpsubset1_type,type,
% 2.84/1.63 eqimpsubset1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(eqimpsubset1,definition,
% 2.84/1.63 ( eqimpsubset1
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( A = B )
% 2.84/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subsetI2_type,type,
% 2.84/1.63 subsetI2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subsetI2,definition,
% 2.84/1.63 ( subsetI2
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ Xx @ B ) )
% 2.84/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(emptysetsubset_type,type,
% 2.84/1.63 emptysetsubset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(emptysetsubset,definition,
% 2.84/1.63 ( emptysetsubset
% 2.84/1.63 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subsetE_type,type,
% 2.84/1.63 subsetE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subsetE,definition,
% 2.84/1.63 ( subsetE
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subsetE2_type,type,
% 2.84/1.63 subsetE2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subsetE2,definition,
% 2.84/1.63 ( subsetE2
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( ~ ( in @ Xx @ B )
% 2.84/1.63 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(notsubsetI_type,type,
% 2.84/1.63 notsubsetI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(notsubsetI,definition,
% 2.84/1.63 ( notsubsetI
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ~ ( in @ Xx @ B )
% 2.84/1.63 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(notequalI1_type,type,
% 2.84/1.63 notequalI1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(notequalI1,definition,
% 2.84/1.63 ( notequalI1
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ~ ( subset @ A @ B )
% 2.84/1.63 => ( A != B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(notequalI2_type,type,
% 2.84/1.63 notequalI2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(notequalI2,definition,
% 2.84/1.63 ( notequalI2
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ~ ( in @ Xx @ B )
% 2.84/1.63 => ( A != B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subsetRefl_type,type,
% 2.84/1.63 subsetRefl: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subsetRefl,definition,
% 2.84/1.63 ( subsetRefl
% 2.84/1.63 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subsetTrans_type,type,
% 2.84/1.63 subsetTrans: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subsetTrans,definition,
% 2.84/1.63 ( subsetTrans
% 2.84/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( ( subset @ B @ C )
% 2.84/1.63 => ( subset @ A @ C ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinSub_type,type,
% 2.84/1.63 setadjoinSub: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinSub,definition,
% 2.84/1.63 ( setadjoinSub
% 2.84/1.63 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinSub2_type,type,
% 2.84/1.63 setadjoinSub2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setadjoinSub2,definition,
% 2.84/1.63 ( setadjoinSub2
% 2.84/1.63 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subset2powerset_type,type,
% 2.84/1.63 subset2powerset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subset2powerset,definition,
% 2.84/1.63 ( subset2powerset
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setextsub_type,type,
% 2.84/1.63 setextsub: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setextsub,definition,
% 2.84/1.63 ( setextsub
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( ( subset @ B @ A )
% 2.84/1.63 => ( A = B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(subsetemptysetimpeq_type,type,
% 2.84/1.63 subsetemptysetimpeq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(subsetemptysetimpeq,definition,
% 2.84/1.63 ( subsetemptysetimpeq
% 2.84/1.63 = ( ! [A: $i] :
% 2.84/1.63 ( ( subset @ A @ emptyset )
% 2.84/1.63 => ( A = emptyset ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(powersetI1_type,type,
% 2.84/1.63 powersetI1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(powersetI1,definition,
% 2.84/1.63 ( powersetI1
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( subset @ B @ A )
% 2.84/1.63 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(powersetE1_type,type,
% 2.84/1.63 powersetE1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(powersetE1,definition,
% 2.84/1.63 ( powersetE1
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( in @ B @ ( powerset @ A ) )
% 2.84/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(inPowerset_type,type,
% 2.84/1.63 inPowerset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(inPowerset,definition,
% 2.84/1.63 ( inPowerset
% 2.84/1.63 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(powersetsubset_type,type,
% 2.84/1.63 powersetsubset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(powersetsubset,definition,
% 2.84/1.63 ( powersetsubset
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(sepInPowerset_type,type,
% 2.84/1.63 sepInPowerset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(sepInPowerset,definition,
% 2.84/1.63 ( sepInPowerset
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( in
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 @ ( powerset @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(sepSubset_type,type,
% 2.84/1.63 sepSubset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(sepSubset,definition,
% 2.84/1.63 ( sepSubset
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( subset
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 @ A ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binunion_type,type,
% 2.84/1.63 binunion: $i > $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(binunionIL_type,type,
% 2.84/1.63 binunionIL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binunionIL,definition,
% 2.84/1.63 ( binunionIL
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(upairset2IR_type,type,
% 2.84/1.63 upairset2IR: $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairset2IR,definition,
% 2.84/1.63 ( upairset2IR
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binunionIR_type,type,
% 2.84/1.63 binunionIR: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binunionIR,definition,
% 2.84/1.63 ( binunionIR
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binunionEcases_type,type,
% 2.84/1.63 binunionEcases: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binunionEcases,definition,
% 2.84/1.63 ( binunionEcases
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.84/1.63 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.84/1.63 => ( ( ( in @ Xx @ A )
% 2.84/1.63 => Xphi )
% 2.84/1.63 => ( ( ( in @ Xx @ B )
% 2.84/1.63 => Xphi )
% 2.84/1.63 => Xphi ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binunionE_type,type,
% 2.84/1.63 binunionE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binunionE,definition,
% 2.84/1.63 ( binunionE
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.84/1.63 => ( ( in @ Xx @ A )
% 2.84/1.63 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binunionLsub_type,type,
% 2.84/1.63 binunionLsub: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binunionLsub,definition,
% 2.84/1.63 ( binunionLsub
% 2.84/1.63 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binunionRsub_type,type,
% 2.84/1.63 binunionRsub: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binunionRsub,definition,
% 2.84/1.63 ( binunionRsub
% 2.84/1.63 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersect_type,type,
% 2.84/1.63 binintersect: $i > $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectI_type,type,
% 2.84/1.63 binintersectI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectI,definition,
% 2.84/1.63 ( binintersectI
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset5_type,type,
% 2.84/1.63 binintersectSubset5: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset5,definition,
% 2.84/1.63 ( binintersectSubset5
% 2.84/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.63 ( ( subset @ C @ A )
% 2.84/1.63 => ( ( subset @ C @ B )
% 2.84/1.63 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectEL_type,type,
% 2.84/1.63 binintersectEL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectEL,definition,
% 2.84/1.63 ( binintersectEL
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.84/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectLsub_type,type,
% 2.84/1.63 binintersectLsub: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectLsub,definition,
% 2.84/1.63 ( binintersectLsub
% 2.84/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset2_type,type,
% 2.84/1.63 binintersectSubset2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset2,definition,
% 2.84/1.63 ( binintersectSubset2
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( ( binintersect @ A @ B )
% 2.84/1.63 = A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset3_type,type,
% 2.84/1.63 binintersectSubset3: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset3,definition,
% 2.84/1.63 ( binintersectSubset3
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( ( binintersect @ A @ B )
% 2.84/1.63 = B )
% 2.84/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectER_type,type,
% 2.84/1.63 binintersectER: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectER,definition,
% 2.84/1.63 ( binintersectER
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.84/1.63 => ( in @ Xx @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(disjointsetsI1_type,type,
% 2.84/1.63 disjointsetsI1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(disjointsetsI1,definition,
% 2.84/1.63 ( disjointsetsI1
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ~ ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( in @ Xx @ B ) )
% 2.84/1.63 => ( ( binintersect @ A @ B )
% 2.84/1.63 = emptyset ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectRsub_type,type,
% 2.84/1.63 binintersectRsub: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectRsub,definition,
% 2.84/1.63 ( binintersectRsub
% 2.84/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset4_type,type,
% 2.84/1.63 binintersectSubset4: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset4,definition,
% 2.84/1.63 ( binintersectSubset4
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( subset @ B @ A )
% 2.84/1.63 => ( ( binintersect @ A @ B )
% 2.84/1.63 = B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset1_type,type,
% 2.84/1.63 binintersectSubset1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(binintersectSubset1,definition,
% 2.84/1.63 ( binintersectSubset1
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( ( binintersect @ A @ B )
% 2.84/1.63 = A )
% 2.84/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(bs114d_type,type,
% 2.84/1.63 bs114d: $o ).
% 2.84/1.63
% 2.84/1.63 thf(bs114d,definition,
% 2.84/1.63 ( bs114d
% 2.84/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.84/1.63 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.84/1.63 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(regular_type,type,
% 2.84/1.63 regular: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminus_type,type,
% 2.84/1.63 setminus: $i > $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(setminusI_type,type,
% 2.84/1.63 setminusI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusI,definition,
% 2.84/1.63 ( setminusI
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ~ ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusEL_type,type,
% 2.84/1.63 setminusEL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusEL,definition,
% 2.84/1.63 ( setminusEL
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusER_type,type,
% 2.84/1.63 setminusER: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusER,definition,
% 2.84/1.63 ( setminusER
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.63 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusSubset2_type,type,
% 2.84/1.63 setminusSubset2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusSubset2,definition,
% 2.84/1.63 ( setminusSubset2
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( subset @ A @ B )
% 2.84/1.63 => ( ( setminus @ A @ B )
% 2.84/1.63 = emptyset ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusERneg_type,type,
% 2.84/1.63 setminusERneg: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusERneg,definition,
% 2.84/1.63 ( setminusERneg
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.63 => ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusELneg_type,type,
% 2.84/1.63 setminusELneg: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusELneg,definition,
% 2.84/1.63 ( setminusELneg
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.84/1.63 => ( ~ ( in @ Xx @ B )
% 2.84/1.63 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusILneg_type,type,
% 2.84/1.63 setminusILneg: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusILneg,definition,
% 2.84/1.63 ( setminusILneg
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ~ ( in @ Xx @ A )
% 2.84/1.63 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusIRneg_type,type,
% 2.84/1.63 setminusIRneg: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusIRneg,definition,
% 2.84/1.63 ( setminusIRneg
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ B )
% 2.84/1.63 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusLsub_type,type,
% 2.84/1.63 setminusLsub: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusLsub,definition,
% 2.84/1.63 ( setminusLsub
% 2.84/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setminusSubset1_type,type,
% 2.84/1.63 setminusSubset1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setminusSubset1,definition,
% 2.84/1.63 ( setminusSubset1
% 2.84/1.63 = ( ! [A: $i,B: $i] :
% 2.84/1.63 ( ( ( setminus @ A @ B )
% 2.84/1.63 = emptyset )
% 2.84/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(symdiff_type,type,
% 2.84/1.63 symdiff: $i > $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffE_type,type,
% 2.84/1.63 symdiffE: $o ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffE,definition,
% 2.84/1.63 ( symdiffE
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.84/1.63 => ! [Xphi: $o] :
% 2.84/1.63 ( ( ( in @ Xx @ A )
% 2.84/1.63 => ( ~ ( in @ Xx @ B )
% 2.84/1.63 => Xphi ) )
% 2.84/1.63 => ( ( ~ ( in @ Xx @ A )
% 2.84/1.63 => ( ( in @ Xx @ B )
% 2.84/1.63 => Xphi ) )
% 2.84/1.63 => Xphi ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffI1_type,type,
% 2.84/1.63 symdiffI1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffI1,definition,
% 2.84/1.63 ( symdiffI1
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ~ ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffI2_type,type,
% 2.84/1.63 symdiffI2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffI2,definition,
% 2.84/1.63 ( symdiffI2
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ~ ( in @ Xx @ A )
% 2.84/1.63 => ( ( in @ Xx @ B )
% 2.84/1.63 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffIneg1_type,type,
% 2.84/1.63 symdiffIneg1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffIneg1,definition,
% 2.84/1.63 ( symdiffIneg1
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ( in @ Xx @ B )
% 2.84/1.63 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffIneg2_type,type,
% 2.84/1.63 symdiffIneg2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(symdiffIneg2,definition,
% 2.84/1.63 ( symdiffIneg2
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ~ ( in @ Xx @ A )
% 2.84/1.63 => ( ~ ( in @ Xx @ B )
% 2.84/1.63 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(iskpair_type,type,
% 2.84/1.63 iskpair: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(secondinupair_type,type,
% 2.84/1.63 secondinupair: $o ).
% 2.84/1.63
% 2.84/1.63 thf(secondinupair,definition,
% 2.84/1.63 ( secondinupair
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairIL_type,type,
% 2.84/1.63 setukpairIL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairIL,definition,
% 2.84/1.63 ( setukpairIL
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairIR_type,type,
% 2.84/1.63 setukpairIR: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairIR,definition,
% 2.84/1.63 ( setukpairIR
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(kpairiskpair_type,type,
% 2.84/1.63 kpairiskpair: $o ).
% 2.84/1.63
% 2.84/1.63 thf(kpairiskpair,definition,
% 2.84/1.63 ( kpairiskpair
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(kpair_type,type,
% 2.84/1.63 kpair: $i > $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(kpairp_type,type,
% 2.84/1.63 kpairp: $o ).
% 2.84/1.63
% 2.84/1.63 thf(kpairp,definition,
% 2.84/1.63 ( kpairp
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprod_type,type,
% 2.84/1.63 cartprod: $i > $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(singletonsubset_type,type,
% 2.84/1.63 singletonsubset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(singletonsubset,definition,
% 2.84/1.63 ( singletonsubset
% 2.84/1.63 = ( ! [A: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(singletoninpowerset_type,type,
% 2.84/1.63 singletoninpowerset: $o ).
% 2.84/1.63
% 2.84/1.63 thf(singletoninpowerset,definition,
% 2.84/1.63 ( singletoninpowerset
% 2.84/1.63 = ( ! [A: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(singletoninpowunion_type,type,
% 2.84/1.63 singletoninpowunion: $o ).
% 2.84/1.63
% 2.84/1.63 thf(singletoninpowunion,definition,
% 2.84/1.63 ( singletoninpowunion
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(upairset2E_type,type,
% 2.84/1.63 upairset2E: $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairset2E,definition,
% 2.84/1.63 ( upairset2E
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.63 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.84/1.63 => ( ( Xz = Xx )
% 2.84/1.63 | ( Xz = Xy ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(upairsubunion_type,type,
% 2.84/1.63 upairsubunion: $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairsubunion,definition,
% 2.84/1.63 ( upairsubunion
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(upairinpowunion_type,type,
% 2.84/1.63 upairinpowunion: $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairinpowunion,definition,
% 2.84/1.63 ( upairinpowunion
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ubforcartprodlem1_type,type,
% 2.84/1.63 ubforcartprodlem1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(ubforcartprodlem1,definition,
% 2.84/1.63 ( ubforcartprodlem1
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ubforcartprodlem2_type,type,
% 2.84/1.63 ubforcartprodlem2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(ubforcartprodlem2,definition,
% 2.84/1.63 ( ubforcartprodlem2
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ubforcartprodlem3_type,type,
% 2.84/1.63 ubforcartprodlem3: $o ).
% 2.84/1.63
% 2.84/1.63 thf(ubforcartprodlem3,definition,
% 2.84/1.63 ( ubforcartprodlem3
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodpairin_type,type,
% 2.84/1.63 cartprodpairin: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodpairin,definition,
% 2.84/1.63 ( cartprodpairin
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodmempair1_type,type,
% 2.84/1.63 cartprodmempair1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodmempair1,definition,
% 2.84/1.63 ( cartprodmempair1
% 2.84/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.63 => ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ? [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 & ( Xu
% 2.84/1.63 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodmempair_type,type,
% 2.84/1.63 cartprodmempair: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodmempair,definition,
% 2.84/1.63 ( cartprodmempair
% 2.84/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.63 => ( iskpair @ Xu ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setunionE2_type,type,
% 2.84/1.63 setunionE2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setunionE2,definition,
% 2.84/1.63 ( setunionE2
% 2.84/1.63 = ( ! [A: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ ( setunion @ A ) )
% 2.84/1.63 => ? [X: $i] :
% 2.84/1.63 ( ( in @ X @ A )
% 2.84/1.63 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setunionsingleton1_type,type,
% 2.84/1.63 setunionsingleton1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setunionsingleton1,definition,
% 2.84/1.63 ( setunionsingleton1
% 2.84/1.63 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setunionsingleton2_type,type,
% 2.84/1.63 setunionsingleton2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setunionsingleton2,definition,
% 2.84/1.63 ( setunionsingleton2
% 2.84/1.63 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setunionsingleton_type,type,
% 2.84/1.63 setunionsingleton: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setunionsingleton,definition,
% 2.84/1.63 ( setunionsingleton
% 2.84/1.63 = ( ! [Xx: $i] :
% 2.84/1.63 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.84/1.63 = Xx ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(singleton_type,type,
% 2.84/1.63 singleton: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(singleton,definition,
% 2.84/1.63 ( singleton
% 2.84/1.63 = ( ^ [A: $i] :
% 2.84/1.63 ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( A
% 2.84/1.63 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(singletonprop_type,type,
% 2.84/1.63 singletonprop: $o ).
% 2.84/1.63
% 2.84/1.63 thf(singletonprop,definition,
% 2.84/1.63 ( singletonprop
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ A )
% 2.84/1.63 => ( ( Xphi @ Xx )
% 2.84/1.63 => ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) )
% 2.84/1.63 => ( ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( Xphi @ Xx ) )
% 2.84/1.63 => ( singleton
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ex1_type,type,
% 2.84/1.63 ex1: $i > ( $i > $o ) > $o ).
% 2.84/1.63
% 2.84/1.63 thf(ex1,definition,
% 2.84/1.63 ( ex1
% 2.84/1.63 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( singleton
% 2.84/1.63 @ ( dsetconstr @ A
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ex1E1_type,type,
% 2.84/1.63 ex1E1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(ex1E1,definition,
% 2.84/1.63 ( ex1E1
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ( ex1 @ A
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.84/1.63 => ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ex1I_type,type,
% 2.84/1.63 ex1I: $o ).
% 2.84/1.63
% 2.84/1.63 thf(ex1I,definition,
% 2.84/1.63 ( ex1I
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ( Xphi @ Xx )
% 2.84/1.63 => ( ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ A )
% 2.84/1.63 => ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xy = Xx ) ) )
% 2.84/1.63 => ( ex1 @ A
% 2.84/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ex1I2_type,type,
% 2.84/1.63 ex1I2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(ex1I2,definition,
% 2.84/1.63 ( ex1I2
% 2.84/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.84/1.63 ( ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ A )
% 2.84/1.63 => ( ( Xphi @ Xx )
% 2.84/1.63 => ( ( Xphi @ Xy )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) )
% 2.84/1.63 => ( ? [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 & ( Xphi @ Xx ) )
% 2.84/1.63 => ( ex1 @ A
% 2.84/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(singletonsuniq_type,type,
% 2.84/1.63 singletonsuniq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(singletonsuniq,definition,
% 2.84/1.63 ( singletonsuniq
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( ( setadjoin @ Xx @ emptyset )
% 2.84/1.63 = ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(atmost1p_type,type,
% 2.84/1.63 atmost1p: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(atleast2p_type,type,
% 2.84/1.63 atleast2p: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(atmost2p_type,type,
% 2.84/1.63 atmost2p: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairsetp_type,type,
% 2.84/1.63 upairsetp: $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjL1_type,type,
% 2.84/1.63 setukpairinjL1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjL1,definition,
% 2.84/1.63 ( setukpairinjL1
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.63 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(kfstsingleton_type,type,
% 2.84/1.63 kfstsingleton: $o ).
% 2.84/1.63
% 2.84/1.63 thf(kfstsingleton,definition,
% 2.84/1.63 ( kfstsingleton
% 2.84/1.63 = ( ! [Xu: $i] :
% 2.84/1.63 ( ( iskpair @ Xu )
% 2.84/1.63 => ( singleton
% 2.84/1.63 @ ( dsetconstr @ ( setunion @ Xu )
% 2.84/1.63 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(theprop_type,type,
% 2.84/1.63 theprop: $o ).
% 2.84/1.63
% 2.84/1.63 thf(theprop,definition,
% 2.84/1.63 ( theprop
% 2.84/1.63 = ( ! [X: $i] :
% 2.84/1.63 ( ( singleton @ X )
% 2.84/1.63 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(kfst_type,type,
% 2.84/1.63 kfst: $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(kfstpairEq_type,type,
% 2.84/1.63 kfstpairEq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(kfstpairEq,definition,
% 2.84/1.63 ( kfstpairEq
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.84/1.63 = Xx ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodfstin_type,type,
% 2.84/1.63 cartprodfstin: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodfstin,definition,
% 2.84/1.63 ( cartprodfstin
% 2.84/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.63 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjL2_type,type,
% 2.84/1.63 setukpairinjL2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjL2,definition,
% 2.84/1.63 ( setukpairinjL2
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjL_type,type,
% 2.84/1.63 setukpairinjL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjL,definition,
% 2.84/1.63 ( setukpairinjL
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.63 ( ( ( kpair @ Xx @ Xy )
% 2.84/1.63 = ( kpair @ Xz @ Xu ) )
% 2.84/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR11_type,type,
% 2.84/1.63 setukpairinjR11: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR11,definition,
% 2.84/1.63 ( setukpairinjR11
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx = Xy )
% 2.84/1.63 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.63 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR12_type,type,
% 2.84/1.63 setukpairinjR12: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR12,definition,
% 2.84/1.63 ( setukpairinjR12
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( Xx = Xy )
% 2.84/1.63 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.63 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR1_type,type,
% 2.84/1.63 setukpairinjR1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR1,definition,
% 2.84/1.63 ( setukpairinjR1
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.63 => ( ( Xz = Xu )
% 2.84/1.63 => ( Xy = Xu ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(upairequniteq_type,type,
% 2.84/1.63 upairequniteq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(upairequniteq,definition,
% 2.84/1.63 ( upairequniteq
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.84/1.63 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.84/1.63 = ( setadjoin @ Xz @ emptyset ) )
% 2.84/1.63 => ( Xx = Xy ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR2_type,type,
% 2.84/1.63 setukpairinjR2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR2,definition,
% 2.84/1.63 ( setukpairinjR2
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.84/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.84/1.63 => ( Xy = Xu ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR_type,type,
% 2.84/1.63 setukpairinjR: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setukpairinjR,definition,
% 2.84/1.63 ( setukpairinjR
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.84/1.63 ( ( ( kpair @ Xx @ Xy )
% 2.84/1.63 = ( kpair @ Xz @ Xu ) )
% 2.84/1.63 => ( Xy = Xu ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ksndsingleton_type,type,
% 2.84/1.63 ksndsingleton: $o ).
% 2.84/1.63
% 2.84/1.63 thf(ksndsingleton,definition,
% 2.84/1.63 ( ksndsingleton
% 2.84/1.63 = ( ! [Xu: $i] :
% 2.84/1.63 ( ( iskpair @ Xu )
% 2.84/1.63 => ( singleton
% 2.84/1.63 @ ( dsetconstr @ ( setunion @ Xu )
% 2.84/1.63 @ ^ [Xx: $i] :
% 2.84/1.63 ( Xu
% 2.84/1.63 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(ksnd_type,type,
% 2.84/1.63 ksnd: $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(ksndpairEq_type,type,
% 2.84/1.63 ksndpairEq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(ksndpairEq,definition,
% 2.84/1.63 ( ksndpairEq
% 2.84/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.84/1.63 = Xy ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(kpairsurjEq_type,type,
% 2.84/1.63 kpairsurjEq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(kpairsurjEq,definition,
% 2.84/1.63 ( kpairsurjEq
% 2.84/1.63 = ( ! [Xu: $i] :
% 2.84/1.63 ( ( iskpair @ Xu )
% 2.84/1.63 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.84/1.63 = Xu ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodsndin_type,type,
% 2.84/1.63 cartprodsndin: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodsndin,definition,
% 2.84/1.63 ( cartprodsndin
% 2.84/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.63 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodpairmemEL_type,type,
% 2.84/1.63 cartprodpairmemEL: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodpairmemEL,definition,
% 2.84/1.63 ( cartprodpairmemEL
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.84/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodpairmemER_type,type,
% 2.84/1.63 cartprodpairmemER: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodpairmemER,definition,
% 2.84/1.63 ( cartprodpairmemER
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.84/1.63 => ( in @ Xy @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodmempaircEq_type,type,
% 2.84/1.63 cartprodmempaircEq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodmempaircEq,definition,
% 2.84/1.63 ( cartprodmempaircEq
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( ( kpair @ Xx @ Xy )
% 2.84/1.63 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodfstpairEq_type,type,
% 2.84/1.63 cartprodfstpairEq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodfstpairEq,definition,
% 2.84/1.63 ( cartprodfstpairEq
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.84/1.63 = Xx ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodsndpairEq_type,type,
% 2.84/1.63 cartprodsndpairEq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodsndpairEq,definition,
% 2.84/1.63 ( cartprodsndpairEq
% 2.84/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.84/1.63 = Xy ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodpairsurjEq_type,type,
% 2.84/1.63 cartprodpairsurjEq: $o ).
% 2.84/1.63
% 2.84/1.63 thf(cartprodpairsurjEq,definition,
% 2.84/1.63 ( cartprodpairsurjEq
% 2.84/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.84/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.84/1.63 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.84/1.63 = Xu ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(breln_type,type,
% 2.84/1.63 breln: $i > $i > $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(breln,definition,
% 2.84/1.63 ( breln
% 2.84/1.63 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstr_type,type,
% 2.84/1.63 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrI_type,type,
% 2.84/1.63 dpsetconstrI: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrI,definition,
% 2.84/1.63 ( dpsetconstrI
% 2.84/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( ( Xphi @ Xx @ Xy )
% 2.84/1.63 => ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.63 @ ( dpsetconstr @ A @ B
% 2.84/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrSub_type,type,
% 2.84/1.63 dpsetconstrSub: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrSub,definition,
% 2.84/1.63 ( dpsetconstrSub
% 2.84/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.84/1.63 ( subset
% 2.84/1.63 @ ( dpsetconstr @ A @ B
% 2.84/1.63 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.84/1.63 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(setOfPairsIsBReln_type,type,
% 2.84/1.63 setOfPairsIsBReln: $o ).
% 2.84/1.63
% 2.84/1.63 thf(setOfPairsIsBReln,definition,
% 2.84/1.63 ( setOfPairsIsBReln
% 2.84/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.84/1.63 ( breln @ A @ B
% 2.84/1.63 @ ( dpsetconstr @ A @ B
% 2.84/1.63 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrERa_type,type,
% 2.84/1.63 dpsetconstrERa: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrERa,definition,
% 2.84/1.63 ( dpsetconstrERa
% 2.84/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ! [Xy: $i] :
% 2.84/1.63 ( ( in @ Xy @ B )
% 2.84/1.63 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.63 @ ( dpsetconstr @ A @ B
% 2.84/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.63 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrEL1_type,type,
% 2.84/1.63 dpsetconstrEL1: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrEL1,definition,
% 2.84/1.63 ( dpsetconstrEL1
% 2.84/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.63 @ ( dpsetconstr @ A @ B
% 2.84/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrEL2_type,type,
% 2.84/1.63 dpsetconstrEL2: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrEL2,definition,
% 2.84/1.63 ( dpsetconstrEL2
% 2.84/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.63 @ ( dpsetconstr @ A @ B
% 2.84/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.63 => ( in @ Xy @ B ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrER_type,type,
% 2.84/1.63 dpsetconstrER: $o ).
% 2.84/1.63
% 2.84/1.63 thf(dpsetconstrER,definition,
% 2.84/1.63 ( dpsetconstrER
% 2.84/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.84/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.84/1.63 @ ( dpsetconstr @ A @ B
% 2.84/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.84/1.63 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(func_type,type,
% 2.84/1.63 func: $i > $i > $i > $o ).
% 2.84/1.63
% 2.84/1.63 thf(func,definition,
% 2.84/1.63 ( func
% 2.84/1.63 = ( ^ [A: $i,B: $i,R: $i] :
% 2.84/1.63 ( ( breln @ A @ B @ R )
% 2.84/1.63 & ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( ex1 @ B
% 2.84/1.63 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(funcSet_type,type,
% 2.84/1.63 funcSet: $i > $i > $i ).
% 2.84/1.63
% 2.84/1.63 thf(funcSet,definition,
% 2.84/1.63 ( funcSet
% 2.84/1.63 = ( ^ [A: $i,B: $i] :
% 2.84/1.63 ( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
% 2.84/1.63 @ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(funcImageSingleton_type,type,
% 2.84/1.63 funcImageSingleton: $o ).
% 2.84/1.63
% 2.84/1.63 thf(funcImageSingleton,definition,
% 2.84/1.63 ( funcImageSingleton
% 2.84/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.63 ( ( func @ A @ B @ Xf )
% 2.84/1.63 => ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( singleton
% 2.84/1.63 @ ( dsetconstr @ B
% 2.84/1.63 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.84/1.63
% 2.84/1.63 thf(apProp,conjecture,
% 2.84/1.63 ( setextAx
% 2.84/1.63 => ( emptysetAx
% 2.84/1.63 => ( setadjoinAx
% 2.84/1.63 => ( powersetAx
% 2.84/1.63 => ( setunionAx
% 2.84/1.63 => ( omega0Ax
% 2.84/1.63 => ( omegaSAx
% 2.84/1.63 => ( omegaIndAx
% 2.84/1.63 => ( replAx
% 2.84/1.63 => ( foundationAx
% 2.84/1.63 => ( wellorderingAx
% 2.84/1.63 => ( descrp
% 2.84/1.63 => ( dsetconstrI
% 2.84/1.63 => ( dsetconstrEL
% 2.84/1.63 => ( dsetconstrER
% 2.84/1.63 => ( exuE1
% 2.84/1.63 => ( prop2setE
% 2.84/1.63 => ( emptysetE
% 2.84/1.63 => ( emptysetimpfalse
% 2.84/1.63 => ( notinemptyset
% 2.84/1.63 => ( exuE3e
% 2.84/1.63 => ( setext
% 2.84/1.63 => ( emptyI
% 2.84/1.63 => ( noeltsimpempty
% 2.84/1.63 => ( setbeta
% 2.84/1.63 => ( nonemptyE1
% 2.84/1.63 => ( nonemptyI
% 2.84/1.63 => ( nonemptyI1
% 2.84/1.63 => ( setadjoinIL
% 2.84/1.63 => ( emptyinunitempty
% 2.84/1.63 => ( setadjoinIR
% 2.84/1.63 => ( setadjoinE
% 2.84/1.63 => ( setadjoinOr
% 2.84/1.63 => ( setoftrueEq
% 2.84/1.63 => ( powersetI
% 2.84/1.63 => ( emptyinPowerset
% 2.84/1.63 => ( emptyInPowerset
% 2.84/1.63 => ( powersetE
% 2.84/1.63 => ( setunionI
% 2.84/1.63 => ( setunionE
% 2.84/1.63 => ( subPowSU
% 2.84/1.63 => ( exuE2
% 2.84/1.63 => ( nonemptyImpWitness
% 2.84/1.63 => ( uniqinunit
% 2.84/1.63 => ( notinsingleton
% 2.84/1.63 => ( eqinunit
% 2.84/1.63 => ( singletonsswitch
% 2.84/1.63 => ( upairsetE
% 2.84/1.63 => ( upairsetIL
% 2.84/1.63 => ( upairsetIR
% 2.84/1.63 => ( emptyE1
% 2.84/1.63 => ( vacuousDall
% 2.84/1.63 => ( quantDeMorgan1
% 2.84/1.63 => ( quantDeMorgan2
% 2.84/1.63 => ( quantDeMorgan3
% 2.84/1.63 => ( quantDeMorgan4
% 2.84/1.63 => ( prop2setI
% 2.84/1.63 => ( prop2set2propI
% 2.84/1.63 => ( notdexE
% 2.84/1.63 => ( notdallE
% 2.84/1.63 => ( exuI1
% 2.84/1.63 => ( exuI3
% 2.84/1.63 => ( exuI2
% 2.84/1.63 => ( inCongP
% 2.84/1.63 => ( in__Cong
% 2.84/1.63 => ( exuE3u
% 2.84/1.63 => ( exu__Cong
% 2.84/1.63 => ( emptyset__Cong
% 2.84/1.63 => ( setadjoin__Cong
% 2.84/1.63 => ( powerset__Cong
% 2.84/1.63 => ( setunion__Cong
% 2.84/1.63 => ( omega__Cong
% 2.84/1.63 => ( exuEu
% 2.84/1.63 => ( descr__Cong
% 2.84/1.63 => ( dsetconstr__Cong
% 2.84/1.63 => ( subsetI1
% 2.84/1.63 => ( eqimpsubset2
% 2.84/1.63 => ( eqimpsubset1
% 2.84/1.63 => ( subsetI2
% 2.84/1.63 => ( emptysetsubset
% 2.84/1.63 => ( subsetE
% 2.84/1.63 => ( subsetE2
% 2.84/1.63 => ( notsubsetI
% 2.84/1.63 => ( notequalI1
% 2.84/1.63 => ( notequalI2
% 2.84/1.63 => ( subsetRefl
% 2.84/1.63 => ( subsetTrans
% 2.84/1.63 => ( setadjoinSub
% 2.84/1.63 => ( setadjoinSub2
% 2.84/1.63 => ( subset2powerset
% 2.84/1.63 => ( setextsub
% 2.84/1.63 => ( subsetemptysetimpeq
% 2.84/1.63 => ( powersetI1
% 2.84/1.63 => ( powersetE1
% 2.84/1.63 => ( inPowerset
% 2.84/1.63 => ( powersetsubset
% 2.84/1.63 => ( sepInPowerset
% 2.84/1.63 => ( sepSubset
% 2.84/1.63 => ( binunionIL
% 2.84/1.63 => ( upairset2IR
% 2.84/1.63 => ( binunionIR
% 2.84/1.63 => ( binunionEcases
% 2.84/1.63 => ( binunionE
% 2.84/1.63 => ( binunionLsub
% 2.84/1.63 => ( binunionRsub
% 2.84/1.63 => ( binintersectI
% 2.84/1.63 => ( binintersectSubset5
% 2.84/1.63 => ( binintersectEL
% 2.84/1.63 => ( binintersectLsub
% 2.84/1.63 => ( binintersectSubset2
% 2.84/1.63 => ( binintersectSubset3
% 2.84/1.63 => ( binintersectER
% 2.84/1.63 => ( disjointsetsI1
% 2.84/1.63 => ( binintersectRsub
% 2.84/1.63 => ( binintersectSubset4
% 2.84/1.63 => ( binintersectSubset1
% 2.84/1.63 => ( bs114d
% 2.84/1.63 => ( setminusI
% 2.84/1.63 => ( setminusEL
% 2.84/1.63 => ( setminusER
% 2.84/1.63 => ( setminusSubset2
% 2.84/1.63 => ( setminusERneg
% 2.84/1.63 => ( setminusELneg
% 2.84/1.63 => ( setminusILneg
% 2.84/1.63 => ( setminusIRneg
% 2.84/1.63 => ( setminusLsub
% 2.84/1.63 => ( setminusSubset1
% 2.84/1.63 => ( symdiffE
% 2.84/1.63 => ( symdiffI1
% 2.84/1.63 => ( symdiffI2
% 2.84/1.63 => ( symdiffIneg1
% 2.84/1.63 => ( symdiffIneg2
% 2.84/1.63 => ( secondinupair
% 2.84/1.63 => ( setukpairIL
% 2.84/1.63 => ( setukpairIR
% 2.84/1.63 => ( kpairiskpair
% 2.84/1.63 => ( kpairp
% 2.84/1.63 => ( singletonsubset
% 2.84/1.63 => ( singletoninpowerset
% 2.84/1.63 => ( singletoninpowunion
% 2.84/1.63 => ( upairset2E
% 2.84/1.63 => ( upairsubunion
% 2.84/1.63 => ( upairinpowunion
% 2.84/1.63 => ( ubforcartprodlem1
% 2.84/1.63 => ( ubforcartprodlem2
% 2.84/1.63 => ( ubforcartprodlem3
% 2.84/1.63 => ( cartprodpairin
% 2.84/1.63 => ( cartprodmempair1
% 2.84/1.63 => ( cartprodmempair
% 2.84/1.63 => ( setunionE2
% 2.84/1.63 => ( setunionsingleton1
% 2.84/1.63 => ( setunionsingleton2
% 2.84/1.63 => ( setunionsingleton
% 2.84/1.63 => ( singletonprop
% 2.84/1.63 => ( ex1E1
% 2.84/1.63 => ( ex1I
% 2.84/1.63 => ( ex1I2
% 2.84/1.63 => ( singletonsuniq
% 2.84/1.63 => ( setukpairinjL1
% 2.84/1.63 => ( kfstsingleton
% 2.84/1.63 => ( theprop
% 2.84/1.63 => ( kfstpairEq
% 2.84/1.63 => ( cartprodfstin
% 2.84/1.63 => ( setukpairinjL2
% 2.84/1.63 => ( setukpairinjL
% 2.84/1.63 => ( setukpairinjR11
% 2.84/1.63 => ( setukpairinjR12
% 2.84/1.63 => ( setukpairinjR1
% 2.84/1.63 => ( upairequniteq
% 2.84/1.63 => ( setukpairinjR2
% 2.84/1.63 => ( setukpairinjR
% 2.84/1.63 => ( ksndsingleton
% 2.84/1.63 => ( ksndpairEq
% 2.84/1.63 => ( kpairsurjEq
% 2.84/1.63 => ( cartprodsndin
% 2.84/1.63 => ( cartprodpairmemEL
% 2.84/1.63 => ( cartprodpairmemER
% 2.84/1.63 => ( cartprodmempaircEq
% 2.84/1.63 => ( cartprodfstpairEq
% 2.84/1.63 => ( cartprodsndpairEq
% 2.84/1.63 => ( cartprodpairsurjEq
% 2.84/1.63 => ( dpsetconstrI
% 2.84/1.63 => ( dpsetconstrSub
% 2.84/1.63 => ( setOfPairsIsBReln
% 2.84/1.63 => ( dpsetconstrERa
% 2.84/1.63 => ( dpsetconstrEL1
% 2.84/1.63 => ( dpsetconstrEL2
% 2.84/1.63 => ( dpsetconstrER
% 2.84/1.63 => ( funcImageSingleton
% 2.84/1.63 => ! [A: $i,B: $i,Xf: $i] :
% 2.84/1.63 ( ( func @ A @ B @ Xf )
% 2.84/1.63 => ! [Xx: $i] :
% 2.84/1.63 ( ( in @ Xx @ A )
% 2.84/1.63 => ( in
% 2.84/1.63 @ ( setunion
% 2.84/1.63 @ ( dsetconstr @ B
% 2.84/1.63 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.84/1.63 @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.84/1.63 ------------------------
% 2.84/1.66 % (19548)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_19443 for (2999ds/275Mi)
% 2.84/1.67 % (19543)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/183Mi)
% 2.84/1.67 % (19544)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_19443 for (2999ds/4Mi)
% 2.84/1.67 % (19549)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/18Mi)
% 2.84/1.67 % (19547)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_19443 for (2999ds/2Mi)
% 2.84/1.67 % (19546)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/2Mi)
% 2.84/1.67 % (19545)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_19443 for (2999ds/27Mi)
% 2.84/1.67 % (19546)Instruction limit reached!
% 2.84/1.67 % (19546)------------------------------
% 2.84/1.67 % (19546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.67 % (19546)Termination reason: Unknown
% 2.84/1.67 % (19546)Termination phase: shuffling
% 2.84/1.67 % (19547)Instruction limit reached!
% 2.84/1.67 % (19547)------------------------------
% 2.84/1.67 % (19547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.67 % (19547)Termination reason: Unknown
% 2.84/1.67 % (19547)Termination phase: shuffling
% 2.84/1.67
% 2.84/1.67 % (19547)Memory used [KB]: 1407
% 2.84/1.67 % (19547)Time elapsed: 0.003 s
% 2.84/1.67 % (19547)Instructions burned: 2 (million)
% 2.84/1.67 % (19547)------------------------------
% 2.84/1.67 % (19547)------------------------------
% 2.84/1.67
% 2.84/1.67 % (19546)Memory used [KB]: 1407
% 2.84/1.67 % (19546)Time elapsed: 0.003 s
% 2.84/1.67 % (19546)Instructions burned: 2 (million)
% 2.84/1.67 % (19546)------------------------------
% 2.84/1.67 % (19546)------------------------------
% 2.84/1.67 % (19544)Instruction limit reached!
% 2.84/1.67 % (19544)------------------------------
% 2.84/1.67 % (19544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.67 % (19544)Termination reason: Unknown
% 2.84/1.67 % (19544)Termination phase: shuffling
% 2.84/1.67
% 2.84/1.67 % (19544)Memory used [KB]: 1407
% 2.84/1.67 % (19544)Time elapsed: 0.004 s
% 2.84/1.67 % (19544)Instructions burned: 4 (million)
% 2.84/1.67 % (19544)------------------------------
% 2.84/1.67 % (19544)------------------------------
% 2.84/1.67 % (19549)Instruction limit reached!
% 2.84/1.67 % (19549)------------------------------
% 2.84/1.67 % (19549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.67 % (19549)Termination reason: Unknown
% 2.84/1.67 % (19549)Termination phase: Property scanning
% 2.84/1.67
% 2.84/1.67 % (19549)Memory used [KB]: 1663
% 2.84/1.67 % (19549)Time elapsed: 0.011 s
% 2.84/1.67 % (19549)Instructions burned: 18 (million)
% 2.84/1.67 % (19549)------------------------------
% 2.84/1.67 % (19549)------------------------------
% 2.84/1.68 % (19545)Instruction limit reached!
% 2.84/1.68 % (19545)------------------------------
% 2.84/1.68 % (19545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.68 % (19545)Termination reason: Unknown
% 2.84/1.68 % (19545)Termination phase: shuffling
% 2.84/1.68
% 2.84/1.68 % (19545)Memory used [KB]: 1918
% 2.84/1.68 % (19545)Time elapsed: 0.015 s
% 2.84/1.68 % (19545)Instructions burned: 28 (million)
% 2.84/1.68 % (19545)------------------------------
% 2.84/1.68 % (19545)------------------------------
% 2.84/1.68 % (19551)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_19443 for (2999ds/37Mi)
% 2.84/1.68 % (19550)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/3Mi)
% 2.84/1.68 % (19552)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_19443 for (2999ds/15Mi)
% 2.84/1.68 % (19550)Instruction limit reached!
% 2.84/1.68 % (19550)------------------------------
% 2.84/1.68 % (19550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.68 % (19550)Termination reason: Unknown
% 2.84/1.68 % (19550)Termination phase: shuffling
% 2.84/1.68
% 2.84/1.68 % (19550)Memory used [KB]: 1407
% 2.84/1.68 % (19550)Time elapsed: 0.003 s
% 2.84/1.68 % (19550)Instructions burned: 3 (million)
% 2.84/1.68 % (19550)------------------------------
% 2.84/1.68 % (19550)------------------------------
% 2.84/1.69 % (19552)Instruction limit reached!
% 2.84/1.69 % (19552)------------------------------
% 2.84/1.69 % (19552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.69 % (19552)Termination reason: Unknown
% 2.84/1.69 % (19552)Termination phase: shuffling
% 2.84/1.69
% 2.84/1.69 % (19552)Memory used [KB]: 1663
% 2.84/1.69 % (19553)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_19443 for (2999ds/3Mi)
% 2.84/1.69 % (19552)Time elapsed: 0.009 s
% 2.84/1.69 % (19552)Instructions burned: 16 (million)
% 2.84/1.69 % (19552)------------------------------
% 2.84/1.69 % (19552)------------------------------
% 2.84/1.69 % (19553)Instruction limit reached!
% 2.84/1.69 % (19553)------------------------------
% 2.84/1.69 % (19553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.69 % (19553)Termination reason: Unknown
% 2.84/1.69 % (19553)Termination phase: shuffling
% 2.84/1.69
% 2.84/1.69 % (19553)Memory used [KB]: 1407
% 2.84/1.69 % (19553)Time elapsed: 0.003 s
% 2.84/1.69 % (19553)Instructions burned: 3 (million)
% 2.84/1.69 % (19553)------------------------------
% 2.84/1.69 % (19553)------------------------------
% 2.84/1.69 % (19554)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_19443 for (2999ds/1041Mi)
% 2.88/1.70 % (19551)Instruction limit reached!
% 2.88/1.70 % (19551)------------------------------
% 2.88/1.70 % (19551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.70 % (19551)Termination reason: Unknown
% 2.88/1.70 % (19551)Termination phase: Property scanning
% 2.88/1.70
% 2.88/1.70 % (19551)Memory used [KB]: 1918
% 2.88/1.70 % (19551)Time elapsed: 0.018 s
% 2.88/1.70 % (19551)Instructions burned: 38 (million)
% 2.88/1.70 % (19551)------------------------------
% 2.88/1.70 % (19551)------------------------------
% 2.88/1.70 % (19555)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/7Mi)
% 2.88/1.70 % (19555)Instruction limit reached!
% 2.88/1.70 % (19555)------------------------------
% 2.88/1.70 % (19555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.70 % (19555)Termination reason: Unknown
% 2.88/1.70 % (19555)Termination phase: shuffling
% 2.88/1.70
% 2.88/1.70 % (19555)Memory used [KB]: 1535
% 2.88/1.70 % (19555)Time elapsed: 0.005 s
% 2.88/1.70 % (19555)Instructions burned: 8 (million)
% 2.88/1.70 % (19555)------------------------------
% 2.88/1.70 % (19555)------------------------------
% 2.88/1.71 % (19556)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/16Mi)
% 2.88/1.71 % (19557)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_19443 for (2999ds/3Mi)
% 2.88/1.71 % (19557)Instruction limit reached!
% 2.88/1.71 % (19557)------------------------------
% 2.88/1.71 % (19557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.71 % (19557)Termination reason: Unknown
% 2.88/1.71 % (19557)Termination phase: shuffling
% 2.88/1.71
% 2.88/1.71 % (19557)Memory used [KB]: 1407
% 2.88/1.71 % (19557)Time elapsed: 0.003 s
% 2.88/1.71 % (19557)Instructions burned: 4 (million)
% 2.88/1.71 % (19557)------------------------------
% 2.88/1.71 % (19557)------------------------------
% 2.88/1.71 % (19556)Instruction limit reached!
% 2.88/1.71 % (19556)------------------------------
% 2.88/1.71 % (19556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.71 % (19556)Termination reason: Unknown
% 2.88/1.71 % (19556)Termination phase: shuffling
% 2.88/1.71
% 2.88/1.71 % (19556)Memory used [KB]: 1663
% 2.88/1.71 % (19556)Time elapsed: 0.009 s
% 2.88/1.71 % (19556)Instructions burned: 17 (million)
% 2.88/1.71 % (19556)------------------------------
% 2.88/1.71 % (19556)------------------------------
% 2.88/1.72 % (19558)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_19443 for (2999ds/3Mi)
% 2.88/1.72 % (19558)Instruction limit reached!
% 2.88/1.72 % (19558)------------------------------
% 2.88/1.72 % (19558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.72 % (19558)Termination reason: Unknown
% 2.88/1.72 % (19558)Termination phase: shuffling
% 2.88/1.72
% 2.88/1.72 % (19558)Memory used [KB]: 1407
% 2.88/1.72 % (19558)Time elapsed: 0.003 s
% 2.88/1.72 % (19558)Instructions burned: 3 (million)
% 2.88/1.72 % (19558)------------------------------
% 2.88/1.72 % (19558)------------------------------
% 2.88/1.72 % (19559)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/7Mi)
% 2.88/1.72 % (19559)Instruction limit reached!
% 2.88/1.72 % (19559)------------------------------
% 2.88/1.72 % (19559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.72 % (19559)Termination reason: Unknown
% 2.88/1.72 % (19559)Termination phase: shuffling
% 2.88/1.72
% 2.88/1.72 % (19559)Memory used [KB]: 1535
% 2.88/1.72 % (19559)Time elapsed: 0.005 s
% 2.88/1.72 % (19559)Instructions burned: 8 (million)
% 2.88/1.72 % (19559)------------------------------
% 2.88/1.72 % (19559)------------------------------
% 2.88/1.73 % (19560)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/3Mi)
% 2.88/1.73 % (19560)Instruction limit reached!
% 2.88/1.73 % (19560)------------------------------
% 2.88/1.73 % (19560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.73 % (19560)Termination reason: Unknown
% 2.88/1.73 % (19560)Termination phase: shuffling
% 2.88/1.73
% 2.88/1.73 % (19560)Memory used [KB]: 1407
% 2.88/1.73 % (19560)Time elapsed: 0.003 s
% 2.88/1.73 % (19560)Instructions burned: 4 (million)
% 2.88/1.73 % (19560)------------------------------
% 2.88/1.73 % (19560)------------------------------
% 2.88/1.73 % (19561)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/4Mi)
% 2.88/1.73 % (19561)Instruction limit reached!
% 2.88/1.73 % (19561)------------------------------
% 2.88/1.73 % (19561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.73 % (19561)Termination reason: Unknown
% 2.88/1.73 % (19561)Termination phase: shuffling
% 2.88/1.73
% 2.88/1.73 % (19561)Memory used [KB]: 1407
% 2.88/1.73 % (19561)Time elapsed: 0.003 s
% 2.88/1.73 % (19561)Instructions burned: 4 (million)
% 2.88/1.73 % (19561)------------------------------
% 2.88/1.73 % (19561)------------------------------
% 2.88/1.73 % (19562)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/18Mi)
% 2.88/1.74 % (19563)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/710Mi)
% 2.88/1.74 % (19562)Instruction limit reached!
% 2.88/1.74 % (19562)------------------------------
% 2.88/1.74 % (19562)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.74 % (19562)Termination reason: Unknown
% 2.88/1.74 % (19562)Termination phase: shuffling
% 2.88/1.74
% 2.88/1.74 % (19562)Memory used [KB]: 1791
% 2.88/1.74 % (19562)Time elapsed: 0.010 s
% 2.88/1.74 % (19562)Instructions burned: 18 (million)
% 2.88/1.74 % (19562)------------------------------
% 2.88/1.74 % (19562)------------------------------
% 2.88/1.74 % (19564)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/6Mi)
% 2.88/1.75 % (19564)Instruction limit reached!
% 2.88/1.75 % (19564)------------------------------
% 2.88/1.75 % (19564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.75 % (19564)Termination reason: Unknown
% 2.88/1.75 % (19564)Termination phase: shuffling
% 2.88/1.75
% 2.88/1.75 % (19564)Memory used [KB]: 1407
% 2.88/1.75 % (19564)Time elapsed: 0.004 s
% 2.88/1.75 % (19564)Instructions burned: 6 (million)
% 2.88/1.75 % (19564)------------------------------
% 2.88/1.75 % (19564)------------------------------
% 2.88/1.75 % (19565)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_19443 for (2999ds/902Mi)
% 2.88/1.75 % (19543)Instruction limit reached!
% 2.88/1.75 % (19543)------------------------------
% 2.88/1.75 % (19543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.75 % (19543)Termination reason: Unknown
% 2.88/1.75 % (19543)Termination phase: Saturation
% 2.88/1.75
% 2.88/1.75 % (19543)Memory used [KB]: 7547
% 2.88/1.75 % (19543)Time elapsed: 0.085 s
% 2.88/1.75 % (19543)Instructions burned: 184 (million)
% 2.88/1.75 % (19543)------------------------------
% 2.88/1.75 % (19543)------------------------------
% 2.88/1.76 % (19566)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_19443 for (2999ds/21Mi)
% 2.88/1.76 % (19567)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_19443 for (2999ds/5Mi)
% 2.88/1.76 % (19567)Instruction limit reached!
% 2.88/1.76 % (19567)------------------------------
% 2.88/1.76 % (19567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.76 % (19567)Termination reason: Unknown
% 2.88/1.76 % (19567)Termination phase: shuffling
% 2.88/1.76
% 2.88/1.76 % (19567)Memory used [KB]: 1407
% 2.88/1.76 % (19567)Time elapsed: 0.004 s
% 2.88/1.76 % (19567)Instructions burned: 5 (million)
% 2.88/1.76 % (19567)------------------------------
% 2.88/1.76 % (19567)------------------------------
% 2.88/1.76 % (19568)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_19443 for (2998ds/6Mi)
% 2.88/1.77 % (19566)Instruction limit reached!
% 2.88/1.77 % (19566)------------------------------
% 2.88/1.77 % (19566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.77 % (19566)Termination reason: Unknown
% 2.88/1.77 % (19566)Termination phase: shuffling
% 2.88/1.77
% 2.88/1.77 % (19566)Memory used [KB]: 1791
% 2.88/1.77 % (19566)Time elapsed: 0.012 s
% 2.88/1.77 % (19566)Instructions burned: 22 (million)
% 2.88/1.77 % (19566)------------------------------
% 2.88/1.77 % (19566)------------------------------
% 2.88/1.77 % (19568)Instruction limit reached!
% 2.88/1.77 % (19568)------------------------------
% 2.88/1.77 % (19568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.77 % (19568)Termination reason: Unknown
% 2.88/1.77 % (19568)Termination phase: shuffling
% 2.88/1.77
% 2.88/1.77 % (19568)Memory used [KB]: 1535
% 2.88/1.77 % (19568)Time elapsed: 0.005 s
% 2.88/1.77 % (19568)Instructions burned: 7 (million)
% 2.88/1.77 % (19568)------------------------------
% 2.88/1.77 % (19568)------------------------------
% 2.88/1.78 % (19569)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_19443 for (2998ds/377Mi)
% 2.88/1.78 % (19571)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_19443 for (2998ds/19Mi)
% 2.88/1.78 % (19570)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_19443 for (2998ds/779Mi)
% 2.88/1.79 % (19571)Instruction limit reached!
% 2.88/1.79 % (19571)------------------------------
% 2.88/1.79 % (19571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.79 % (19571)Termination reason: Unknown
% 2.88/1.79 % (19571)Termination phase: shuffling
% 2.88/1.79
% 2.88/1.79 % (19571)Memory used [KB]: 1791
% 2.88/1.79 % (19571)Time elapsed: 0.011 s
% 2.88/1.79 % (19571)Instructions burned: 19 (million)
% 2.88/1.79 % (19571)------------------------------
% 2.88/1.79 % (19571)------------------------------
% 2.88/1.80 % (19548)Instruction limit reached!
% 2.88/1.80 % (19548)------------------------------
% 2.88/1.80 % (19548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.80 % (19548)Termination reason: Unknown
% 2.88/1.80 % (19548)Termination phase: Saturation
% 2.88/1.80
% 2.88/1.80 % (19548)Memory used [KB]: 8443
% 2.88/1.80 % (19548)Time elapsed: 0.137 s
% 2.88/1.80 % (19548)Instructions burned: 275 (million)
% 2.88/1.80 % (19548)------------------------------
% 2.88/1.80 % (19548)------------------------------
% 2.88/1.81 % (19572)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_19443 for (2998ds/879Mi)
% 2.88/1.82 % (19573)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_19443 for (2998ds/17Mi)
% 2.88/1.82 % (19573)Instruction limit reached!
% 2.88/1.82 % (19573)------------------------------
% 2.88/1.82 % (19573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.82 % (19573)Termination reason: Unknown
% 2.88/1.82 % (19573)Termination phase: shuffling
% 2.88/1.82
% 2.88/1.82 % (19573)Memory used [KB]: 1663
% 2.88/1.82 % (19573)Time elapsed: 0.010 s
% 2.88/1.82 % (19573)Instructions burned: 17 (million)
% 2.88/1.82 % (19573)------------------------------
% 2.88/1.82 % (19573)------------------------------
% 2.88/1.84 % (19574)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_19443 for (2998ds/3Mi)
% 2.88/1.84 % (19574)Instruction limit reached!
% 2.88/1.84 % (19574)------------------------------
% 2.88/1.84 % (19574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.84 % (19574)Termination reason: Unknown
% 2.88/1.84 % (19574)Termination phase: shuffling
% 2.88/1.84
% 2.88/1.84 % (19574)Memory used [KB]: 1407
% 2.88/1.84 % (19574)Time elapsed: 0.004 s
% 2.88/1.84 % (19574)Instructions burned: 5 (million)
% 2.88/1.84 % (19574)------------------------------
% 2.88/1.84 % (19574)------------------------------
% 2.88/1.86 % (19575)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_19443 for (2998ds/30Mi)
% 2.88/1.87 % (19575)Instruction limit reached!
% 2.88/1.87 % (19575)------------------------------
% 2.88/1.87 % (19575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.87 % (19575)Termination reason: Unknown
% 2.88/1.87 % (19575)Termination phase: Property scanning
% 2.88/1.87
% 2.88/1.87 % (19575)Memory used [KB]: 1918
% 2.88/1.87 % (19575)Time elapsed: 0.016 s
% 2.88/1.87 % (19575)Instructions burned: 31 (million)
% 2.88/1.87 % (19575)------------------------------
% 2.88/1.87 % (19575)------------------------------
% 2.88/1.89 % (19576)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_19443 for (2997ds/127Mi)
% 2.88/1.94 % (19576)Instruction limit reached!
% 2.88/1.94 % (19576)------------------------------
% 2.88/1.94 % (19576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.94 % (19576)Termination reason: Unknown
% 2.88/1.94 % (19576)Termination phase: Saturation
% 2.88/1.94
% 2.88/1.94 % (19576)Memory used [KB]: 6908
% 2.88/1.94 % (19576)Time elapsed: 0.057 s
% 2.88/1.94 % (19576)Instructions burned: 127 (million)
% 2.88/1.94 % (19576)------------------------------
% 2.88/1.94 % (19576)------------------------------
% 2.88/1.95 % (19569)Instruction limit reached!
% 2.88/1.95 % (19569)------------------------------
% 2.88/1.95 % (19569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.95 % (19569)Termination reason: Unknown
% 2.88/1.95 % (19569)Termination phase: Saturation
% 2.88/1.95
% 2.88/1.95 % (19569)Memory used [KB]: 8443
% 2.88/1.95 % (19569)Time elapsed: 0.197 s
% 2.88/1.95 % (19569)Instructions burned: 377 (million)
% 2.88/1.95 % (19569)------------------------------
% 2.88/1.95 % (19569)------------------------------
% 2.88/1.96 % (19577)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_19443 for (2997ds/100Mi)
% 2.88/1.97 % (19578)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_19443 for (2996ds/3Mi)
% 2.88/1.97 % (19578)Instruction limit reached!
% 2.88/1.97 % (19578)------------------------------
% 2.88/1.97 % (19578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.97 % (19578)Termination reason: Unknown
% 2.88/1.97 % (19578)Termination phase: shuffling
% 2.88/1.97
% 2.88/1.97 % (19578)Memory used [KB]: 1407
% 2.88/1.97 % (19578)Time elapsed: 0.004 s
% 2.88/1.97 % (19578)Instructions burned: 4 (million)
% 2.88/1.97 % (19578)------------------------------
% 2.88/1.97 % (19578)------------------------------
% 2.88/1.98 % (19579)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_19443 for (2996ds/20Mi)
% 2.88/1.99 % (19579)Instruction limit reached!
% 2.88/1.99 % (19579)------------------------------
% 2.88/1.99 % (19579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.99 % (19579)Termination reason: Unknown
% 2.88/1.99 % (19579)Termination phase: Property scanning
% 2.88/1.99
% 2.88/1.99 % (19579)Memory used [KB]: 1663
% 2.88/1.99 % (19579)Time elapsed: 0.012 s
% 2.88/1.99 % (19579)Instructions burned: 21 (million)
% 2.88/1.99 % (19579)------------------------------
% 2.88/1.99 % (19579)------------------------------
% 2.88/2.00 % (19577)Instruction limit reached!
% 2.88/2.00 % (19577)------------------------------
% 2.88/2.00 % (19577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/2.00 % (19577)Termination reason: Unknown
% 2.88/2.00 % (19577)Termination phase: Saturation
% 2.88/2.00
% 2.88/2.00 % (19577)Memory used [KB]: 6780
% 2.88/2.00 % (19577)Time elapsed: 0.045 s
% 2.88/2.00 % (19577)Instructions burned: 100 (million)
% 2.88/2.00 % (19577)------------------------------
% 2.88/2.00 % (19577)------------------------------
% 2.88/2.01 % (19580)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_19443 for (2996ds/86Mi)
% 2.88/2.02 % (19581)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_19443 for (2996ds/107Mi)
% 2.88/2.02 % (19565)First to succeed.
% 2.88/2.03 % (19565)Refutation found. Thanks to Tanya!
% 2.88/2.03 % SZS status Theorem for DTF2THF_19443
% 2.88/2.03 % SZS output start Proof for DTF2THF_19443
% 2.88/2.03 thf(func_def_0, type, in: $i > $i > $o).
% 2.88/2.03 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.88/2.03 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.88/2.03 thf(func_def_8, type, powerset: $i > $i).
% 2.88/2.03 thf(func_def_10, type, setunion: $i > $i).
% 2.88/2.03 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.88/2.03 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.88/2.03 thf(func_def_26, type, prop2set: $o > $i).
% 2.88/2.03 thf(func_def_36, type, nonempty: $i > $o).
% 2.88/2.03 thf(func_def_69, type, set2prop: $i > $o).
% 2.88/2.03 thf(func_def_88, type, subset: $i > $i > $o).
% 2.88/2.03 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.88/2.03 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.88/2.03 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.88/2.03 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.88/2.03 thf(func_def_135, type, regular: $i > $o).
% 2.88/2.03 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.88/2.03 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.88/2.03 thf(func_def_153, type, iskpair: $i > $o).
% 2.88/2.03 thf(func_def_158, type, kpair: $i > $i > $i).
% 5.42/2.03 thf(func_def_160, type, cartprod: $i > $i > $i).
% 5.42/2.03 thf(func_def_177, type, singleton: $i > $o).
% 5.42/2.03 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 5.42/2.03 thf(func_def_184, type, atmost1p: $i > $o).
% 5.42/2.03 thf(func_def_185, type, atleast2p: $i > $o).
% 5.42/2.03 thf(func_def_186, type, atmost2p: $i > $o).
% 5.42/2.03 thf(func_def_187, type, upairsetp: $i > $o).
% 5.42/2.03 thf(func_def_191, type, kfst: $i > $i).
% 5.42/2.03 thf(func_def_203, type, ksnd: $i > $i).
% 5.42/2.03 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 5.42/2.03 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 5.42/2.03 thf(func_def_222, type, func: $i > $i > $i > $o).
% 5.42/2.03 thf(func_def_223, type, funcSet: $i > $i > $i).
% 5.42/2.03 thf(func_def_239, type, sP1: $i > $i > $o).
% 5.42/2.03 thf(func_def_240, type, sP2: $i > $o).
% 5.42/2.03 thf(func_def_241, type, sP3: $i > $i > $o).
% 5.42/2.03 thf(func_def_242, type, sP4: $i > $i > $o).
% 5.42/2.03 thf(func_def_244, type, sP6: $i > $o > $i > $i > $o).
% 5.42/2.03 thf(func_def_246, type, sK8: $i > $o).
% 5.42/2.03 thf(func_def_260, type, sK22: $i > $o).
% 5.42/2.03 thf(func_def_261, type, sK23: ($i > $o) > $i).
% 5.42/2.03 thf(func_def_272, type, sK34: $i > $i).
% 5.42/2.03 thf(func_def_276, type, sK38: $i > $i > $o).
% 5.42/2.03 thf(func_def_281, type, sK43: $i > $i > $o).
% 5.42/2.03 thf(func_def_283, type, sK45: $i > $i).
% 5.42/2.03 thf(func_def_284, type, sK46: $i > $i).
% 5.42/2.03 thf(func_def_285, type, sK47: $i > ($i > $i > $o) > $i).
% 5.42/2.03 thf(func_def_286, type, sK48: $i > $i > ($i > $i > $o) > $i).
% 5.42/2.03 thf(func_def_287, type, sK49: $i > ($i > $i > $o) > $i).
% 5.42/2.03 thf(func_def_292, type, sK54: $i > $i).
% 5.42/2.03 thf(func_def_300, type, sK62: $i > $o).
% 5.42/2.03 thf(func_def_305, type, sK67: $i > $i > $i > $i).
% 5.42/2.03 thf(func_def_306, type, sK68: $i > $i > $i > $i).
% 5.42/2.03 thf(func_def_309, type, sK71: $i > $i > $o).
% 5.42/2.03 thf(func_def_329, type, sK91: $i > $i > $i).
% 5.42/2.03 thf(func_def_330, type, sK92: $i > $i > $i).
% 5.42/2.03 thf(func_def_336, type, sK98: $i > $i).
% 5.42/2.03 thf(func_def_337, type, sK99: ($i > $o) > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_338, type, sK100: ($i > $o) > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_339, type, sK101: $i > $o).
% 5.42/2.03 thf(func_def_340, type, sK102: $i > $o).
% 5.42/2.03 thf(func_def_343, type, sK105: $i > $o).
% 5.42/2.03 thf(func_def_344, type, sK106: $i > $i).
% 5.42/2.03 thf(func_def_345, type, sK107: ($i > $o) > $i).
% 5.42/2.03 thf(func_def_357, type, sK119: $i > $i).
% 5.42/2.03 thf(func_def_358, type, sK120: $i > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_359, type, sK121: $i > $o).
% 5.42/2.03 thf(func_def_386, type, sK148: $i > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_387, type, sK149: $i > $o).
% 5.42/2.03 thf(func_def_392, type, sK154: $i > $i > $i).
% 5.42/2.03 thf(func_def_393, type, sK155: $i > $i > $i).
% 5.42/2.03 thf(func_def_394, type, sK156: $i > $i > $i).
% 5.42/2.03 thf(func_def_395, type, sK157: $i > $i > $i).
% 5.42/2.03 thf(func_def_396, type, sK158: $i > $i > $i > $i).
% 5.42/2.03 thf(func_def_397, type, sK159: $i > $i).
% 5.42/2.03 thf(func_def_398, type, sK160: $i > $i).
% 5.42/2.03 thf(func_def_399, type, sK161: $i > $i).
% 5.42/2.03 thf(func_def_400, type, sK162: $i > $i).
% 5.42/2.03 thf(func_def_401, type, sK163: $i > $i > $i).
% 5.42/2.03 thf(func_def_402, type, sK164: $i > $i > $i > $i).
% 5.42/2.03 thf(func_def_403, type, sK165: $i > $i > $i > $i).
% 5.42/2.03 thf(func_def_404, type, sK166: $i > $i > $i).
% 5.42/2.03 thf(func_def_405, type, sK167: $i > $i > $i).
% 5.42/2.03 thf(func_def_407, type, sK169: $i > $i).
% 5.42/2.03 thf(func_def_408, type, sK170: $i > $i).
% 5.42/2.03 thf(func_def_409, type, sK171: $i > $i).
% 5.42/2.03 thf(func_def_410, type, sK172: $i > $i > $i).
% 5.42/2.03 thf(func_def_412, type, sK174: $i > $o).
% 5.42/2.03 thf(func_def_426, type, sK188: $i > $i > $i).
% 5.42/2.03 thf(func_def_437, type, sK199: $i > $o).
% 5.42/2.03 thf(func_def_439, type, sK201: $i > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_440, type, sK202: $i > $i > $i).
% 5.42/2.03 thf(func_def_456, type, sK218: ($i > $o) > ($i > $o) > $i > $i > $i).
% 5.42/2.03 thf(func_def_457, type, sK219: ($i > $o) > ($i > $o) > $i > $i > $i).
% 5.42/2.03 thf(func_def_460, type, sK222: $i > $o).
% 5.42/2.03 thf(func_def_461, type, sK223: $i > $o).
% 5.42/2.03 thf(func_def_469, type, sK231: $i > $o).
% 5.42/2.03 thf(func_def_472, type, sK234: $i > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_473, type, sK235: $i > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_480, type, sK242: $i > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_481, type, sK243: $i > $o).
% 5.42/2.03 thf(func_def_483, type, sK245: $i > $o).
% 5.42/2.03 thf(func_def_487, type, sK249: $i > $o).
% 5.42/2.03 thf(func_def_510, type, sK272: $i > $o).
% 5.42/2.03 thf(func_def_514, type, sK276: $i > $o).
% 5.42/2.03 thf(func_def_517, type, sK279: $i > $o).
% 5.42/2.03 thf(func_def_532, type, sK294: $i > $i > $o).
% 5.42/2.03 thf(func_def_536, type, sK298: ($i > $o) > $i).
% 5.42/2.03 thf(func_def_537, type, sK299: ($i > $o) > $i).
% 5.42/2.03 thf(func_def_538, type, sK300: $i > $o).
% 5.42/2.03 thf(func_def_547, type, sK309: $i > $o).
% 5.42/2.03 thf(func_def_548, type, sK310: $i > $i).
% 5.42/2.03 thf(func_def_549, type, sK311: ($i > $o) > $i).
% 5.42/2.03 thf(func_def_561, type, sK323: $o > $i > $i > $i).
% 5.42/2.03 thf(func_def_568, type, sK330: $i > $i > $o).
% 5.42/2.03 thf(func_def_593, type, sK355: $i > $o).
% 5.42/2.03 thf(func_def_595, type, sK357: $i > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_596, type, sK358: $i > $i).
% 5.42/2.03 thf(func_def_605, type, sK367: $i > $i > $i).
% 5.42/2.03 thf(func_def_609, type, sK371: $i > $i > $i).
% 5.42/2.03 thf(func_def_617, type, sK379: $i > $o).
% 5.42/2.03 thf(func_def_622, type, sK384: $i > $i > $i).
% 5.42/2.03 thf(func_def_646, type, sK408: $i > $o).
% 5.42/2.03 thf(func_def_648, type, sK410: ($i > $o) > $i > $i).
% 5.42/2.03 thf(func_def_657, type, sK419: $i > $i > $o).
% 5.42/2.03 thf(func_def_662, type, sK424: $i > $o).
% 5.42/2.03 thf(func_def_664, type, sK426: $i > $o).
% 5.42/2.03 thf(func_def_668, type, sK430: ($i > $o) > $i > $i).
% 5.42/2.03 thf(func_def_670, type, sK432: $i > $o).
% 5.42/2.03 thf(func_def_681, type, sK443: ($i > $o) > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_682, type, sK444: ($i > $o) > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_683, type, sK445: $i > $o).
% 5.42/2.03 thf(func_def_684, type, sK446: $i > $o).
% 5.42/2.03 thf(func_def_686, type, sK448: $i > $o).
% 5.42/2.03 thf(func_def_688, type, sK450: ($i > $o) > $i > $i).
% 5.42/2.03 thf(func_def_689, type, sK451: ($i > $o) > $i > $i).
% 5.42/2.03 thf(func_def_702, type, sK464: $i > $i > $i).
% 5.42/2.03 thf(func_def_709, type, sK471: $i > $i > $o).
% 5.42/2.03 thf(func_def_710, type, sK472: $i > $o).
% 5.42/2.03 thf(func_def_723, type, sK485: $i > $i > $i).
% 5.42/2.03 thf(func_def_724, type, sK486: $i > $i > $i).
% 5.42/2.03 thf(func_def_725, type, sK487: ($i > $o) > $i > $i > $i).
% 5.42/2.03 thf(func_def_728, type, sK490: $i > $o).
% 5.42/2.03 thf(func_def_750, type, sK512: $i > $i > $o).
% 5.42/2.03 thf(func_def_771, type, sK533: $i > $o).
% 5.42/2.03 thf(func_def_773, type, sK535: ($i > $o) > $i > $i).
% 5.42/2.03 thf(func_def_776, type, sK538: $i > $i).
% 5.42/2.03 thf(func_def_777, type, sK539: $i > $i).
% 5.42/2.03 thf(func_def_779, type, sK541: $i > $o).
% 5.42/2.03 thf(func_def_782, type, sK544: $i > ($i > $o) > $i).
% 5.42/2.03 thf(func_def_787, type, ph549: !>[X0: $tType]:(X0)).
% 5.42/2.03 thf(f5580,plain,(
% 5.42/2.03 $false),
% 5.42/2.03 inference(avatar_sat_refutation,[status(thm)],[f3500,f5169,f5573])).
% 5.42/2.03 thf(f5573,plain,(
% 5.42/2.03 spl548_10),
% 5.42/2.03 inference(avatar_contradiction_clause,[status(thm)],[f5572])).
% 5.42/2.03 thf(f5572,plain,(
% 5.42/2.03 $false | spl548_10),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f5561])).
% 5.42/2.03 thf(f5561,plain,(
% 5.42/2.03 ($true != $true) | spl548_10),
% 5.42/2.03 inference(superposition,[status(thm)],[f5357,f2347])).
% 5.42/2.03 thf(f2347,plain,(
% 5.42/2.03 ((in @ sK399 @ sK397) = $true)),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1457])).
% 5.42/2.03 thf(f1457,plain,(
% 5.42/2.03 (omegaIndAx = $true) & (omega__Cong = $true) & (emptyset__Cong = $true) & (ubforcartprodlem1 = $true) & (binunionIL = $true) & (binintersectSubset1 = $true) & (binunionE = $true) & (kfstsingleton = $true) & (kpairsurjEq = $true) & (setoftrueEq = $true) & (binintersectSubset5 = $true) & (binintersectER = $true) & (binintersectRsub = $true) & (in__Cong = $true) & (upairset2IR = $true) & (upairsubunion = $true) & (cartprodmempair1 = $true) & (ksndpairEq = $true) & (setukpairinjR1 = $true) & (powersetsubset = $true) & (singletonsuniq = $true) & (setadjoinIL = $true) & (eqimpsubset2 = $true) & (setunionI = $true) & (setminusERneg = $true) & (nonemptyI1 = $true) & (setminusSubset1 = $true) & (setunionsingleton2 = $true) & (upairset2E = $true) & (inCongP = $true) & (notequalI1 = $true) & (setadjoinAx = $true) & (kpairp = $true) & (symdiffI1 = $true) & (setukpairIR = $true) & (exuE2 = $true) & (exu__Cong = $true) & (upairsetIR = $true) & (setminusIRneg = $true) & (dpsetconstrER = $true) & (notinsingleton = $true) & (replAx = $true) & (ubforcartprodlem2 = $true) & (bs114d = $true) & (theprop = $true) & (dpsetconstrI = $true) & (quantDeMorgan3 = $true) & (quantDeMorgan2 = $true) & (emptysetsubset = $true) & (emptyinPowerset = $true) & (ex1I2 = $true) & (ksndsingleton = $true) & (dpsetconstrEL1 = $true) & (setminusLsub = $true) & (upairequniteq = $true) & (exuI2 = $true) & (subsetemptysetimpeq = $true) & (setminusSubset2 = $true) & (((func @ sK397 @ sK396 @ sK398) = $true) & (((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) @ sK396) != $true) & ((in @ sK399 @ sK397) = $true))) & (sepInPowerset = $true) & (dpsetconstrSub = $true) & (notdexE = $true) & (emptysetE = $true) & (emptyE1 = $true) & (prop2set2propI = $true) & (subPowSU = $true) & (setunionAx = $true) & (powersetE1 = $true) & (setextAx = $true) & (uniqinunit = $true) & (powersetE = $true) & (setextsub = $true) & (setukpairinjL2 = $true) & (exuI3 = $true) & (powersetAx = $true) & (singletonsswitch = $true) & (notsubsetI = $true) & (vacuousDall = $true) & (setukpairinjL = $true) & (subsetE2 = $true) & (ubforcartprodlem3 = $true) & (symdiffE = $true) & (emptyinunitempty = $true) & (powersetI1 = $true) & (binintersectI = $true) & (powersetI = $true) & (foundationAx = $true) & (setunion__Cong = $true) & (setbeta = $true) & (setunionE = $true) & (cartprodpairsurjEq = $true) & (symdiffI2 = $true) & (setminusER = $true) & (eqinunit = $true) & (setunionE2 = $true) & (cartprodpairmemEL = $true) & (upairsetIL = $true) & (cartprodfstin = $true) & (subsetI2 = $true) & (singletonsubset = $true) & (setminusILneg = $true) & (eqimpsubset1 = $true) & (setukpairinjR2 = $true) & (setukpairIL = $true) & (quantDeMorgan4 = $true) & (sepSubset = $true) & (dsetconstrEL = $true) & (singletoninpowerset = $true) & (binintersectSubset4 = $true) & (exuE1 = $true) & (setukpairinjR12 = $true) & (dsetconstrI = $true) & (ex1E1 = $true) & (exuI1 = $true) & (subsetE = $true) & (singletonprop = $true) & (setadjoinSub = $true) & (binunionRsub = $true) & (setukpairinjR11 = $true) & (dpsetconstrEL2 = $true) & (cartprodmempaircEq = $true) & (setadjoinSub2 = $true) & (ex1I = $true) & (descr__Cong = $true) & (binunionEcases = $true) & (nonemptyImpWitness = $true) & (cartprodsndin = $true) & (prop2setE = $true) & (setadjoinOr = $true) & (binintersectLsub = $true) & (setunionsingleton = $true) & (kpairiskpair = $true) & (setadjoin__Cong = $true) & (setminusI = $true) & (cartprodpairin = $true) & (setext = $true) & (binunionLsub = $true) & (cartprodpairmemER = $true) & (disjointsetsI1 = $true) & (quantDeMorgan1 = $true) & (cartprodsndpairEq = $true) & (singletoninpowunion = $true) & (binintersectSubset2 = $true) & (cartprodmempair = $true) & (dpsetconstrERa = $true) & (setOfPairsIsBReln = $true) & (nonemptyI = $true) & (setunionsingleton1 = $true) & (emptysetimpfalse = $true) & (subsetRefl = $true) & (exuE3u = $true) & (cartprodfstpairEq = $true) & (omegaSAx = $true) & (setukpairinjR = $true) & (notinemptyset = $true) & (kfstpairEq = $true) & (dsetconstrER = $true) & (emptysetAx = $true) & (subsetI1 = $true) & (setadjoinIR = $true) & (notequalI2 = $true) & (wellorderingAx = $true) & (prop2setI = $true) & (subset2powerset = $true) & (binintersectSubset3 = $true) & (exuE3e = $true) & (emptyI = $true) & (symdiffIneg2 = $true) & (setminusEL = $true) & (emptyInPowerset = $true) & (setadjoinE = $true) & (notdallE = $true) & (exuEu = $true) & (setukpairinjL1 = $true) & (setminusELneg = $true) & (descrp = $true) & (funcImageSingleton = $true) & (binunionIR = $true) & (subsetTrans = $true) & (dsetconstr__Cong = $true) & (binintersectEL = $true) & (noeltsimpempty = $true) & (secondinupair = $true) & (symdiffIneg1 = $true) & (upairsetE = $true) & (omega0Ax = $true) & (upairinpowunion = $true) & (inPowerset = $true) & (nonemptyE1 = $true) & (powerset__Cong = $true)),
% 5.42/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK396,sK397,sK398,sK399])],[f785,f1456,f1455])).
% 5.42/2.03 thf(f1455,plain,(
% 5.42/2.03 ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) = $true) & ? [X3 : $i] : (((in @ (setunion @ (dsetconstr @ X0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X2)))) @ X0) != $true) & ((in @ X3 @ X1) = $true))) => (((func @ sK397 @ sK396 @ sK398) = $true) & ? [X3 : $i] : (((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ sK398)))) @ sK396) != $true) & ($true = (in @ X3 @ sK397))))),
% 5.42/2.03 introduced(definition,[],[choice_axiom])).
% 5.42/2.03 thf(f1456,plain,(
% 5.42/2.03 ? [X3 : $i] : (((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ sK398)))) @ sK396) != $true) & ($true = (in @ X3 @ sK397))) => (((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) @ sK396) != $true) & ((in @ sK399 @ sK397) = $true))),
% 5.42/2.03 introduced(definition,[],[choice_axiom])).
% 5.42/2.03 thf(f785,plain,(
% 5.42/2.03 (omegaIndAx = $true) & (omega__Cong = $true) & (emptyset__Cong = $true) & (ubforcartprodlem1 = $true) & (binunionIL = $true) & (binintersectSubset1 = $true) & (binunionE = $true) & (kfstsingleton = $true) & (kpairsurjEq = $true) & (setoftrueEq = $true) & (binintersectSubset5 = $true) & (binintersectER = $true) & (binintersectRsub = $true) & (in__Cong = $true) & (upairset2IR = $true) & (upairsubunion = $true) & (cartprodmempair1 = $true) & (ksndpairEq = $true) & (setukpairinjR1 = $true) & (powersetsubset = $true) & (singletonsuniq = $true) & (setadjoinIL = $true) & (eqimpsubset2 = $true) & (setunionI = $true) & (setminusERneg = $true) & (nonemptyI1 = $true) & (setminusSubset1 = $true) & (setunionsingleton2 = $true) & (upairset2E = $true) & (inCongP = $true) & (notequalI1 = $true) & (setadjoinAx = $true) & (kpairp = $true) & (symdiffI1 = $true) & (setukpairIR = $true) & (exuE2 = $true) & (exu__Cong = $true) & (upairsetIR = $true) & (setminusIRneg = $true) & (dpsetconstrER = $true) & (notinsingleton = $true) & (replAx = $true) & (ubforcartprodlem2 = $true) & (bs114d = $true) & (theprop = $true) & (dpsetconstrI = $true) & (quantDeMorgan3 = $true) & (quantDeMorgan2 = $true) & (emptysetsubset = $true) & (emptyinPowerset = $true) & (ex1I2 = $true) & (ksndsingleton = $true) & (dpsetconstrEL1 = $true) & (setminusLsub = $true) & (upairequniteq = $true) & (exuI2 = $true) & (subsetemptysetimpeq = $true) & (setminusSubset2 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) = $true) & ? [X3 : $i] : (((in @ (setunion @ (dsetconstr @ X0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X2)))) @ X0) != $true) & ((in @ X3 @ X1) = $true))) & (sepInPowerset = $true) & (dpsetconstrSub = $true) & (notdexE = $true) & (emptysetE = $true) & (emptyE1 = $true) & (prop2set2propI = $true) & (subPowSU = $true) & (setunionAx = $true) & (powersetE1 = $true) & (setextAx = $true) & (uniqinunit = $true) & (powersetE = $true) & (setextsub = $true) & (setukpairinjL2 = $true) & (exuI3 = $true) & (powersetAx = $true) & (singletonsswitch = $true) & (notsubsetI = $true) & (vacuousDall = $true) & (setukpairinjL = $true) & (subsetE2 = $true) & (ubforcartprodlem3 = $true) & (symdiffE = $true) & (emptyinunitempty = $true) & (powersetI1 = $true) & (binintersectI = $true) & (powersetI = $true) & (foundationAx = $true) & (setunion__Cong = $true) & (setbeta = $true) & (setunionE = $true) & (cartprodpairsurjEq = $true) & (symdiffI2 = $true) & (setminusER = $true) & (eqinunit = $true) & (setunionE2 = $true) & (cartprodpairmemEL = $true) & (upairsetIL = $true) & (cartprodfstin = $true) & (subsetI2 = $true) & (singletonsubset = $true) & (setminusILneg = $true) & (eqimpsubset1 = $true) & (setukpairinjR2 = $true) & (setukpairIL = $true) & (quantDeMorgan4 = $true) & (sepSubset = $true) & (dsetconstrEL = $true) & (singletoninpowerset = $true) & (binintersectSubset4 = $true) & (exuE1 = $true) & (setukpairinjR12 = $true) & (dsetconstrI = $true) & (ex1E1 = $true) & (exuI1 = $true) & (subsetE = $true) & (singletonprop = $true) & (setadjoinSub = $true) & (binunionRsub = $true) & (setukpairinjR11 = $true) & (dpsetconstrEL2 = $true) & (cartprodmempaircEq = $true) & (setadjoinSub2 = $true) & (ex1I = $true) & (descr__Cong = $true) & (binunionEcases = $true) & (nonemptyImpWitness = $true) & (cartprodsndin = $true) & (prop2setE = $true) & (setadjoinOr = $true) & (binintersectLsub = $true) & (setunionsingleton = $true) & (kpairiskpair = $true) & (setadjoin__Cong = $true) & (setminusI = $true) & (cartprodpairin = $true) & (setext = $true) & (binunionLsub = $true) & (cartprodpairmemER = $true) & (disjointsetsI1 = $true) & (quantDeMorgan1 = $true) & (cartprodsndpairEq = $true) & (singletoninpowunion = $true) & (binintersectSubset2 = $true) & (cartprodmempair = $true) & (dpsetconstrERa = $true) & (setOfPairsIsBReln = $true) & (nonemptyI = $true) & (setunionsingleton1 = $true) & (emptysetimpfalse = $true) & (subsetRefl = $true) & (exuE3u = $true) & (cartprodfstpairEq = $true) & (omegaSAx = $true) & (setukpairinjR = $true) & (notinemptyset = $true) & (kfstpairEq = $true) & (dsetconstrER = $true) & (emptysetAx = $true) & (subsetI1 = $true) & (setadjoinIR = $true) & (notequalI2 = $true) & (wellorderingAx = $true) & (prop2setI = $true) & (subset2powerset = $true) & (binintersectSubset3 = $true) & (exuE3e = $true) & (emptyI = $true) & (symdiffIneg2 = $true) & (setminusEL = $true) & (emptyInPowerset = $true) & (setadjoinE = $true) & (notdallE = $true) & (exuEu = $true) & (setukpairinjL1 = $true) & (setminusELneg = $true) & (descrp = $true) & (funcImageSingleton = $true) & (binunionIR = $true) & (subsetTrans = $true) & (dsetconstr__Cong = $true) & (binintersectEL = $true) & (noeltsimpempty = $true) & (secondinupair = $true) & (symdiffIneg1 = $true) & (upairsetE = $true) & (omega0Ax = $true) & (upairinpowunion = $true) & (inPowerset = $true) & (nonemptyE1 = $true) & (powerset__Cong = $true)),
% 5.42/2.03 inference(flattening,[status(thm)],[f784])).
% 5.42/2.03 thf(f784,plain,(
% 5.42/2.03 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) = $true) & ? [X3 : $i] : (((in @ (setunion @ (dsetconstr @ X0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X2)))) @ X0) != $true) & ((in @ X3 @ X1) = $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)),
% 5.42/2.03 inference(ennf_transformation,[status(thm)],[f581])).
% 5.42/2.03 thf(f581,plain,(
% 5.42/2.03 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ((in @ (setunion @ (dsetconstr @ X0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X2)))) @ X0) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.42/2.03 inference(fool_elimination,[status(thm)],[f580])).
% 5.42/2.03 thf(f580,plain,(
% 5.42/2.03 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X1 @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (setunion @ (dsetconstr @ X0 @ (^[X4 : $i] : (in @ (kpair @ X3 @ X4) @ X2)))) @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.42/2.03 inference(rectify,[status(thm)],[f198])).
% 5.42/2.03 thf(f198,negated_conjecture,(
% 5.42/2.03 ~(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 => ! [X4 : $i,X3 : $i,X13 : $i] : ((func @ X3 @ X4 @ X13) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (setunion @ (dsetconstr @ X4 @ (^[X2 : $i] : (in @ (kpair @ X1 @ X2) @ X13)))) @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.42/2.03 inference(negated_conjecture,[status(cth)],[f197])).
% 5.42/2.03 thf(f197,conjecture,(
% 5.42/2.03 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 => ! [X4 : $i,X3 : $i,X13 : $i] : ((func @ X3 @ X4 @ X13) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (setunion @ (dsetconstr @ X4 @ (^[X2 : $i] : (in @ (kpair @ X1 @ X2) @ X13)))) @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.42/2.03 file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',apProp)).
% 5.42/2.03 thf(f5357,plain,(
% 5.42/2.03 ((in @ sK399 @ sK397) != $true) | spl548_10),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f5356])).
% 5.42/2.03 thf(f5356,plain,(
% 5.42/2.03 ((in @ sK399 @ sK397) != $true) | ($true != $true) | spl548_10),
% 5.42/2.03 inference(superposition,[status(thm)],[f4946,f2349])).
% 5.42/2.03 thf(f2349,plain,(
% 5.42/2.03 ((func @ sK397 @ sK396 @ sK398) = $true)),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1457])).
% 5.42/2.03 thf(f4946,plain,(
% 5.42/2.03 ( ! [X0 : $i] : (((func @ X0 @ sK396 @ sK398) != $true) | ((in @ sK399 @ X0) != $true)) ) | spl548_10),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f4945])).
% 5.42/2.03 thf(f4945,plain,(
% 5.42/2.03 ( ! [X0 : $i] : (((in @ sK399 @ X0) != $true) | ((func @ X0 @ sK396 @ sK398) != $true) | ($true != $true)) ) | spl548_10),
% 5.42/2.03 inference(forward_demodulation,[status(thm)],[f4939,f2229])).
% 5.42/2.03 thf(f2229,plain,(
% 5.42/2.03 (funcImageSingleton = $true)),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1457])).
% 5.42/2.03 thf(f4939,plain,(
% 5.42/2.03 ( ! [X0 : $i] : (((in @ sK399 @ X0) != $true) | (funcImageSingleton != $true) | ((func @ X0 @ sK396 @ sK398) != $true)) ) | spl548_10),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f4932])).
% 5.42/2.03 thf(f4932,plain,(
% 5.42/2.03 ( ! [X0 : $i] : ((funcImageSingleton != $true) | ($true != $true) | ((in @ sK399 @ X0) != $true) | ((func @ X0 @ sK396 @ sK398) != $true)) ) | spl548_10),
% 5.42/2.03 inference(superposition,[status(thm)],[f2998,f1839])).
% 5.42/2.03 thf(f1839,plain,(
% 5.42/2.03 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((singleton @ (dsetconstr @ X6 @ (^[Y0 : $i]: (in @ (kpair @ X7 @ Y0) @ X5)))) = $true) | ((in @ X7 @ X4) != $true) | ($true != (func @ X4 @ X6 @ X5)) | (funcImageSingleton != $true)) )),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1014])).
% 5.42/2.03 thf(f1014,plain,(
% 5.42/2.03 ((funcImageSingleton = $true) | (((func @ sK111 @ sK113 @ sK112) = $true) & (($true = (in @ sK114 @ sK111)) & ($true != (singleton @ (dsetconstr @ sK113 @ (^[Y0 : $i]: (in @ (kpair @ sK114 @ Y0) @ sK112)))))))) & (! [X4 : $i,X5 : $i,X6 : $i] : (($true != (func @ X4 @ X6 @ X5)) | ! [X7 : $i] : (((in @ X7 @ X4) != $true) | ((singleton @ (dsetconstr @ X6 @ (^[Y0 : $i]: (in @ (kpair @ X7 @ Y0) @ X5)))) = $true))) | (funcImageSingleton != $true))),
% 5.42/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK111,sK112,sK113,sK114])],[f1011,f1013,f1012])).
% 5.42/2.03 thf(f1012,plain,(
% 5.42/2.03 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X0 @ X2 @ X1)) & ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ((singleton @ (dsetconstr @ X2 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1)))) != $true))) => (((func @ sK111 @ sK113 @ sK112) = $true) & ? [X3 : $i] : (((in @ X3 @ sK111) = $true) & ((singleton @ (dsetconstr @ sK113 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ sK112)))) != $true)))),
% 5.42/2.03 introduced(definition,[],[choice_axiom])).
% 5.42/2.03 thf(f1013,plain,(
% 5.42/2.03 ? [X3 : $i] : (((in @ X3 @ sK111) = $true) & ((singleton @ (dsetconstr @ sK113 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ sK112)))) != $true)) => (($true = (in @ sK114 @ sK111)) & ($true != (singleton @ (dsetconstr @ sK113 @ (^[Y0 : $i]: (in @ (kpair @ sK114 @ Y0) @ sK112))))))),
% 5.42/2.03 introduced(definition,[],[choice_axiom])).
% 5.42/2.03 thf(f1011,plain,(
% 5.42/2.03 ((funcImageSingleton = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X0 @ X2 @ X1)) & ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ((singleton @ (dsetconstr @ X2 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1)))) != $true)))) & (! [X4 : $i,X5 : $i,X6 : $i] : (($true != (func @ X4 @ X6 @ X5)) | ! [X7 : $i] : (((in @ X7 @ X4) != $true) | ((singleton @ (dsetconstr @ X6 @ (^[Y0 : $i]: (in @ (kpair @ X7 @ Y0) @ X5)))) = $true))) | (funcImageSingleton != $true))),
% 5.42/2.03 inference(rectify,[status(thm)],[f1010])).
% 5.42/2.03 thf(f1010,plain,(
% 5.42/2.03 ((funcImageSingleton = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X0 @ X2 @ X1)) & ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ((singleton @ (dsetconstr @ X2 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1)))) != $true)))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (func @ X0 @ X2 @ X1)) | ! [X3 : $i] : (($true != (in @ X3 @ X0)) | ((singleton @ (dsetconstr @ X2 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1)))) = $true))) | (funcImageSingleton != $true))),
% 5.42/2.03 inference(nnf_transformation,[status(thm)],[f679])).
% 5.42/2.03 thf(f679,plain,(
% 5.42/2.03 (funcImageSingleton = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (func @ X0 @ X2 @ X1)) | ! [X3 : $i] : (($true != (in @ X3 @ X0)) | ((singleton @ (dsetconstr @ X2 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1)))) = $true)))),
% 5.42/2.03 inference(ennf_transformation,[status(thm)],[f374])).
% 5.42/2.03 thf(f374,plain,(
% 5.42/2.03 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X0 @ X2 @ X1)) => ! [X3 : $i] : (($true = (in @ X3 @ X0)) => ((singleton @ (dsetconstr @ X2 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1)))) = $true))) <=> (funcImageSingleton = $true)),
% 5.42/2.03 inference(fool_elimination,[status(thm)],[f373])).
% 5.42/2.03 thf(f373,plain,(
% 5.42/2.03 (! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X0 @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X0) => (singleton @ (dsetconstr @ X2 @ (^[X4 : $i] : (in @ (kpair @ X3 @ X4) @ X1)))))) = funcImageSingleton)),
% 5.42/2.03 inference(rectify,[status(thm)],[f196])).
% 5.42/2.03 thf(f196,axiom,(
% 5.42/2.03 (! [X3 : $i,X13 : $i,X4 : $i] : ((func @ X3 @ X4 @ X13) => ! [X1 : $i] : ((in @ X1 @ X3) => (singleton @ (dsetconstr @ X4 @ (^[X2 : $i] : (in @ (kpair @ X1 @ X2) @ X13)))))) = funcImageSingleton)),
% 5.42/2.03 file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',funcImageSingleton)).
% 5.42/2.03 thf(f2998,plain,(
% 5.42/2.03 ((singleton @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) != $true) | spl548_10),
% 5.42/2.03 inference(avatar_component_clause,[status(thm)],[f2996])).
% 5.42/2.03 thf(f2996,definition,(
% 5.42/2.03 spl548_10 <=> ((singleton @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) = $true)),
% 5.42/2.03 introduced(definition,[new_symbols(naming,[spl548_10])],[avatar_definition])).
% 5.42/2.03 thf(f5169,plain,(
% 5.42/2.03 spl548_9),
% 5.42/2.03 inference(avatar_contradiction_clause,[status(thm)],[f5168])).
% 5.42/2.03 thf(f5168,plain,(
% 5.42/2.03 $false | spl548_9),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f5167])).
% 5.42/2.03 thf(f5167,plain,(
% 5.42/2.03 ($true != $true) | spl548_9),
% 5.42/2.03 inference(superposition,[status(thm)],[f4585,f2300])).
% 5.42/2.03 thf(f2300,plain,(
% 5.42/2.03 (sepSubset = $true)),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1457])).
% 5.42/2.03 thf(f4585,plain,(
% 5.42/2.03 (sepSubset != $true) | spl548_9),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f4577])).
% 5.42/2.03 thf(f4577,plain,(
% 5.42/2.03 ($true != $true) | (sepSubset != $true) | spl548_9),
% 5.42/2.03 inference(superposition,[status(thm)],[f2994,f2659])).
% 5.42/2.03 thf(f2659,plain,(
% 5.42/2.03 ( ! [X0 : $i > $o,X1 : $i] : (((subset @ (dsetconstr @ X1 @ X0) @ X1) = $true) | (sepSubset != $true)) )),
% 5.42/2.03 inference(beta_eta_normalization,[status(thm)],[f1700])).
% 5.42/2.03 thf(f1700,plain,(
% 5.42/2.03 ( ! [X0 : $i > $o,X1 : $i] : (((subset @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0))) @ X1) = $true) | (sepSubset != $true)) )),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f835])).
% 5.42/2.03 thf(f835,plain,(
% 5.42/2.03 (! [X0 : $i > $o,X1 : $i] : ((subset @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0))) @ X1) = $true) | (sepSubset != $true)) & ((sepSubset = $true) | ((subset @ (dsetconstr @ sK9 @ (^[Y0 : $i]: (sK8 @ Y0))) @ sK9) != $true))),
% 5.42/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9])],[f833,f834])).
% 5.42/2.03 thf(f834,plain,(
% 5.42/2.03 ? [X2 : $i > $o,X3 : $i] : ((subset @ (dsetconstr @ X3 @ (^[Y0 : $i]: (X2 @ Y0))) @ X3) != $true) => ((subset @ (dsetconstr @ sK9 @ (^[Y0 : $i]: (sK8 @ Y0))) @ sK9) != $true)),
% 5.42/2.03 introduced(definition,[],[choice_axiom])).
% 5.42/2.03 thf(f833,plain,(
% 5.42/2.03 (! [X0 : $i > $o,X1 : $i] : ((subset @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0))) @ X1) = $true) | (sepSubset != $true)) & ((sepSubset = $true) | ? [X2 : $i > $o,X3 : $i] : ((subset @ (dsetconstr @ X3 @ (^[Y0 : $i]: (X2 @ Y0))) @ X3) != $true))),
% 5.42/2.03 inference(rectify,[status(thm)],[f832])).
% 5.42/2.03 thf(f832,plain,(
% 5.42/2.03 (! [X0 : $i > $o,X1 : $i] : ((subset @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0))) @ X1) = $true) | (sepSubset != $true)) & ((sepSubset = $true) | ? [X0 : $i > $o,X1 : $i] : ((subset @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0))) @ X1) != $true))),
% 5.42/2.03 inference(nnf_transformation,[status(thm)],[f441])).
% 5.42/2.03 thf(f441,plain,(
% 5.42/2.03 ! [X0 : $i > $o,X1 : $i] : ((subset @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0))) @ X1) = $true) <=> (sepSubset = $true)),
% 5.42/2.03 inference(fool_elimination,[status(thm)],[f440])).
% 5.42/2.03 thf(f440,plain,(
% 5.42/2.03 (sepSubset = ! [X0 : $i > $o,X1 : $i] : (subset @ (dsetconstr @ X1 @ (^[X2 : $i] : (X0 @ X2))) @ X1))),
% 5.42/2.03 inference(rectify,[status(thm)],[f100])).
% 5.42/2.03 thf(f100,axiom,(
% 5.42/2.03 (sepSubset = ! [X0 : $i > $o,X3 : $i] : (subset @ (dsetconstr @ X3 @ (^[X1 : $i] : (X0 @ X1))) @ X3))),
% 5.42/2.03 file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',sepSubset)).
% 5.42/2.03 thf(f2994,plain,(
% 5.42/2.03 ((subset @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398))) @ sK396) != $true) | spl548_9),
% 5.42/2.03 inference(avatar_component_clause,[status(thm)],[f2992])).
% 5.42/2.03 thf(f2992,definition,(
% 5.42/2.03 spl548_9 <=> ((subset @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398))) @ sK396) = $true)),
% 5.42/2.03 introduced(definition,[new_symbols(naming,[spl548_9])],[avatar_definition])).
% 5.42/2.03 thf(f3500,plain,(
% 5.42/2.03 ~spl548_10 | ~spl548_9),
% 5.42/2.03 inference(avatar_split_clause,[status(thm)],[f3499,f2992,f2996])).
% 5.42/2.03 thf(f3499,plain,(
% 5.42/2.03 ((subset @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398))) @ sK396) != $true) | ((singleton @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) != $true)),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f3498])).
% 5.42/2.03 thf(f3498,plain,(
% 5.42/2.03 ((singleton @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) != $true) | ((subset @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398))) @ sK396) != $true) | ($true != $true)),
% 5.42/2.03 inference(forward_demodulation,[status(thm)],[f3422,f2363])).
% 5.42/2.03 thf(f2363,plain,(
% 5.42/2.03 (theprop = $true)),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1457])).
% 5.42/2.03 thf(f3422,plain,(
% 5.42/2.03 ((singleton @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) != $true) | (theprop != $true) | ((subset @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398))) @ sK396) != $true)),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f3327])).
% 5.42/2.03 thf(f3327,plain,(
% 5.42/2.03 ((singleton @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) != $true) | ((subset @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398))) @ sK396) != $true) | ($true != $true) | (theprop != $true)),
% 5.42/2.03 inference(superposition,[status(thm)],[f2782,f2541])).
% 5.42/2.03 thf(f2541,plain,(
% 5.42/2.03 ( ! [X1 : $i] : (((in @ (setunion @ X1) @ X1) = $true) | (theprop != $true) | ($true != (singleton @ X1))) )),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1615])).
% 5.42/2.03 thf(f1615,plain,(
% 5.42/2.03 ((theprop = $true) | (((singleton @ sK495) = $true) & ($true != (in @ (setunion @ sK495) @ sK495)))) & (! [X1 : $i] : (($true != (singleton @ X1)) | ((in @ (setunion @ X1) @ X1) = $true)) | (theprop != $true))),
% 5.42/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK495])],[f1613,f1614])).
% 5.42/2.03 thf(f1614,plain,(
% 5.42/2.03 ? [X0 : $i] : (((singleton @ X0) = $true) & ((in @ (setunion @ X0) @ X0) != $true)) => (((singleton @ sK495) = $true) & ($true != (in @ (setunion @ sK495) @ sK495)))),
% 5.42/2.03 introduced(definition,[],[choice_axiom])).
% 5.42/2.03 thf(f1613,plain,(
% 5.42/2.03 ((theprop = $true) | ? [X0 : $i] : (((singleton @ X0) = $true) & ((in @ (setunion @ X0) @ X0) != $true))) & (! [X1 : $i] : (($true != (singleton @ X1)) | ((in @ (setunion @ X1) @ X1) = $true)) | (theprop != $true))),
% 5.42/2.03 inference(rectify,[status(thm)],[f1612])).
% 5.42/2.03 thf(f1612,plain,(
% 5.42/2.03 ((theprop = $true) | ? [X0 : $i] : (((singleton @ X0) = $true) & ((in @ (setunion @ X0) @ X0) != $true))) & (! [X0 : $i] : (((singleton @ X0) != $true) | ((in @ (setunion @ X0) @ X0) = $true)) | (theprop != $true))),
% 5.42/2.03 inference(nnf_transformation,[status(thm)],[f805])).
% 5.42/2.03 thf(f805,plain,(
% 5.42/2.03 (theprop = $true) <=> ! [X0 : $i] : (((singleton @ X0) != $true) | ((in @ (setunion @ X0) @ X0) = $true))),
% 5.42/2.03 inference(ennf_transformation,[status(thm)],[f262])).
% 5.42/2.03 thf(f262,plain,(
% 5.42/2.03 ! [X0 : $i] : (((singleton @ X0) = $true) => ((in @ (setunion @ X0) @ X0) = $true)) <=> (theprop = $true)),
% 5.42/2.03 inference(fool_elimination,[status(thm)],[f261])).
% 5.42/2.03 thf(f261,plain,(
% 5.42/2.03 (theprop = ! [X0 : $i] : ((singleton @ X0) => (in @ (setunion @ X0) @ X0)))),
% 5.42/2.03 inference(rectify,[status(thm)],[f165])).
% 5.42/2.03 thf(f165,axiom,(
% 5.42/2.03 (theprop = ! [X11 : $i] : ((singleton @ X11) => (in @ (setunion @ X11) @ X11)))),
% 5.42/2.03 file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',theprop)).
% 5.42/2.03 thf(f2782,plain,(
% 5.42/2.03 ( ! [X0 : $i] : (((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) @ X0) != $true) | ($true != (subset @ X0 @ sK396))) )),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f2781])).
% 5.42/2.03 thf(f2781,plain,(
% 5.42/2.03 ( ! [X0 : $i] : (((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) @ X0) != $true) | ($true != (subset @ X0 @ sK396)) | ($true != $true)) )),
% 5.42/2.03 inference(forward_demodulation,[status(thm)],[f2759,f2326])).
% 5.42/2.03 thf(f2326,plain,(
% 5.42/2.03 (subsetE2 = $true)),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1457])).
% 5.42/2.03 thf(f2759,plain,(
% 5.42/2.03 ( ! [X0 : $i] : (((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) @ X0) != $true) | ($true != (subset @ X0 @ sK396)) | (subsetE2 != $true)) )),
% 5.42/2.03 inference(trivial_inequality_removal,[status(thm)],[f2757])).
% 5.42/2.03 thf(f2757,plain,(
% 5.42/2.03 ( ! [X0 : $i] : (($true != $true) | (subsetE2 != $true) | ($true != (subset @ X0 @ sK396)) | ((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) @ X0) != $true)) )),
% 5.42/2.03 inference(superposition,[status(thm)],[f2348,f2011])).
% 5.42/2.03 thf(f2011,plain,(
% 5.42/2.03 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X5)) | (subsetE2 != $true) | ((in @ X3 @ X4) != $true) | ((subset @ X4 @ X5) != $true)) )),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1211])).
% 5.42/2.03 thf(f1211,plain,(
% 5.42/2.03 ((subsetE2 = $true) | (((in @ sK236 @ sK237) = $true) & ((in @ sK236 @ sK238) != $true) & ($true = (subset @ sK237 @ sK238)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X4) != $true) | ($true = (in @ X3 @ X5)) | ((subset @ X4 @ X5) != $true)) | (subsetE2 != $true))),
% 5.42/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK236,sK237,sK238])],[f1209,f1210])).
% 5.42/2.03 thf(f1210,plain,(
% 5.42/2.03 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ X0 @ X2)) & ((subset @ X1 @ X2) = $true)) => (((in @ sK236 @ sK237) = $true) & ((in @ sK236 @ sK238) != $true) & ($true = (subset @ sK237 @ sK238)))),
% 5.42/2.03 introduced(definition,[],[choice_axiom])).
% 5.42/2.03 thf(f1209,plain,(
% 5.42/2.03 ((subsetE2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ X0 @ X2)) & ((subset @ X1 @ X2) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X4) != $true) | ($true = (in @ X3 @ X5)) | ((subset @ X4 @ X5) != $true)) | (subsetE2 != $true))),
% 5.42/2.03 inference(rectify,[status(thm)],[f1208])).
% 5.42/2.03 thf(f1208,plain,(
% 5.42/2.03 ((subsetE2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ X0 @ X2)) & ((subset @ X1 @ X2) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ X2)) | ((subset @ X1 @ X2) != $true)) | (subsetE2 != $true))),
% 5.42/2.03 inference(nnf_transformation,[status(thm)],[f634])).
% 5.42/2.03 thf(f634,plain,(
% 5.42/2.03 (subsetE2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ X2)) | ((subset @ X1 @ X2) != $true))),
% 5.42/2.03 inference(flattening,[status(thm)],[f633])).
% 5.42/2.03 thf(f633,plain,(
% 5.42/2.03 ! [X0 : $i,X1 : $i,X2 : $i] : ((($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ X2))) | ((subset @ X1 @ X2) != $true)) <=> (subsetE2 = $true)),
% 5.42/2.03 inference(ennf_transformation,[status(thm)],[f615])).
% 5.42/2.03 thf(f615,plain,(
% 5.42/2.03 ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ X2) = $true) => (($true != (in @ X0 @ X2)) => ($true != (in @ X0 @ X1)))) <=> (subsetE2 = $true)),
% 5.42/2.03 inference(flattening,[status(thm)],[f426])).
% 5.42/2.03 thf(f426,plain,(
% 5.42/2.03 (subsetE2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ X2) = $true) => (~($true = (in @ X0 @ X2)) => ~($true = (in @ X0 @ X1))))),
% 5.42/2.03 inference(fool_elimination,[status(thm)],[f425])).
% 5.42/2.03 thf(f425,plain,(
% 5.42/2.03 (subsetE2 = ! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X1 @ X2) => (~(in @ X0 @ X2) => ~(in @ X0 @ X1))))),
% 5.42/2.03 inference(rectify,[status(thm)],[f84])).
% 5.42/2.03 thf(f84,axiom,(
% 5.42/2.03 (subsetE2 = ! [X1 : $i,X3 : $i,X4 : $i] : ((subset @ X3 @ X4) => (~(in @ X1 @ X4) => ~(in @ X1 @ X3))))),
% 5.42/2.03 file('/export/starexec/sandbox/tmp/tmp.2yNPKVECum/DTF2THF_19443.p',subsetE2)).
% 5.42/2.03 thf(f2348,plain,(
% 5.42/2.03 ((in @ (setunion @ (dsetconstr @ sK396 @ (^[Y0 : $i]: (in @ (kpair @ sK399 @ Y0) @ sK398)))) @ sK396) != $true)),
% 5.42/2.03 inference(cnf_transformation,[status(thm)],[f1457])).
% 5.42/2.03 % SZS output end Proof for DTF2THF_19443
% 5.42/2.03 % (19565)------------------------------
% 5.42/2.03 % (19565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.42/2.03 % (19565)Termination reason: Refutation
% 5.42/2.03
% 5.42/2.03 % (19565)Memory used [KB]: 9466
% 5.42/2.03 % (19565)Time elapsed: 0.300 s
% 5.42/2.03 % (19565)Instructions burned: 565 (million)
% 5.42/2.03 % (19565)------------------------------
% 5.42/2.03 % (19565)------------------------------
% 5.42/2.03 % (19542)Success in time 0.384 s
%------------------------------------------------------------------------------